ON GEOMETRIC FORMULATION OF DYNAMICAL SYSTEMS WITH HIDDEN CONSTRAINTS. PART I – BASIC THEORY

Andrzej Szatkowski*

Constrained dynamics is a cornerstone of theoretical physics. Simulation of multi-body systems of mass points is one of the fields of interest. Constrained dynamics is also one of these basic mathematical tools which are widely explored in theoretical considerations and simulations in various areas of engineering. The formulation of inconsistent higher-order differential dynamical systems is developed. A differential dynamical system is essentially inconsistent, if one can deduce the existence of hidden constraints by analysing the description of the constitutive space of the system. Hidden constraints arise as additional algebraic and lower-order differential relations which must be satisfied by the dynamic variables along the trajectories of the system. They are observed in the case when the configuration space is too large in order to describe the real degree of freedom of a system, as well as in over-determined systems, and in singular systems. The sub-bundles of the tangent bundles formalism is used in the considerations. A completely intrinsic geometric solution to the problem of reduction of a general differential dynamical system to an equivalent consistent system is given. The projection-prolongation and the prolongation algorithms for reduction of a differential dynamical system to an equivalent consistent system are presented. The relations between these algorithms are studied. Also the examples of a mass point system and a control system are given. In the part two, the geometric theory of electrical networks is discussed. Electrical networks are considered as the interesting examples of general dynamical systems which are possibly essentially inconsistent.

Keywords: differential dynamical systems; constrained dynamics; hidden constraints; inconsistent systems; reduction algorithms; differential algebraic equations; electrical networks; mass point and control systems

*Andrzej Szatkowski, Department of Mechatronics, High School of Computer Science and Management in Olsztyn, Artyleryjska Str. 3f, 10-165 Olsztyn, Poland. Email: [email protected]

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1. INTRODUCTION We develop geometric frameworks for the treatment of differential dynamical systems whose dynamics is given by the solutions of ordinary differential inclusions. Both, first-order and higher-order differential dynamical systems are analysed. Implicit ordinary differential equations and differential algebraic equations are considered as the examples. In geometric approach to dynamical systems, the dynamics of a system is considered as motion occurring in the solution space embedded as a subset in the ambient space, which is the space where the entire system is defined [Beitrag… 1990], [Differential-algebraic… 2006], [Differential-algebraic… 2008], [Gracia, Pons 1992], [Involution… 2010], [Kunkel, Mehrmann 2004], [Mendella et al. 1995], [Rabier, Rheinboldt 1991, 1994], [Reich 1990, 1991], [Reissig 1996], [Seiler 1998], [Szatkowski 1989, 1990a, 1990b, 1992, 2001, 2002a, 2003]. The ambient space of a dynamical system has, by assumption, the structure of a differentiable manifold. The constitutive space of a differential dynamical system , abbreviated d.d.s. , is the space of possible positions, generalized velocities, accelerations etc. It is given by a set of differential algebraic constraints and defines constitution of the system. The constitutive space contains implicitly the information which is necessary in order to extract the solution space and the infinitesimal generator for the solutions of the system. In geometric formulation of the differential dynamical systems, the constitutive space of a d.d.s. has the structure of a sub-bundle of the tangent bundle, or of the higher-order tangent bundle to the ambient space of the system. The infinitesimal generator for the solutions of a d.d.s. is a tangent sub-bundle of the constitutive bundle of the d.d.s. whose trajectories define the evolution of the system in time. The solutions of a d.d.s. are the solutions of the differential inclusion which corresponds to the tangent sub-bundle being the infinitesimal generator of the system. The configuration space of a d.d.s. is the base space of the constitutive bundle of the system. The base space of the tangent part to its base space of the constitutive sub-bundle of a d.d.s. is called the proper configuration space of the system. The solution space of a d.d.s. is a subset of its proper configuration space which is filled by the solution curves of the system. A d.d.s. is consistent iff its constitutive sub-bundle is a tangent sub-bundle. In the other case a d.d.s. is called an inconsistent system. As an example, consider the following differential algebraic equation

dx1 dx 2 dx = 0, = x1, 3 = x 2 , x3 = 0 dt dt dt

(1)

which is defined in 3 . The independent variable t is interpreted as ‘time’, and the dependent variables x1, x 2 , and x3 are the state variables. In the following, for a differential algebraic equation there is used the abbreviation d.a.e. It is easily observed that there are no solutions to the d.a.e. (1) which would be passing through the points in 3 \ {0}. A function x = x(t ), where x = ( x1, x 2 , x3 ), is the solution of (1) iff x(t ) = 0 for all t ∈ Dom x(⋅) . The solution space of the d.a.e. (1) is the point x = 0 which is the equilibrium point of (1). Thus, the integrable part of (1) is the following d.a.e.

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dx = 0 , x = 0 . dt

(2)

In this example, one observes reduction of the given d.a.e. (1) to the equivalent d.a.e. (2). The configuration space of the reduced d.a.e. (2), which is defined by the constraint x = 0 , is the proper subset of the configuration space of the d.a.e. (1), which is given by the constraint x3 = 0 , but the sets of the solutions of the d.a. equations (1) and (2) are the same sets of functions. The problem of reduction of a given d.d.s. to its equivalent proper sub-system, which is as simple as possible, is the subject of the considerations in this article. In the approach which is developed, the reduced d.d.s. is called the consistent part of the given system. For most d.d. systems being the subject of interest in the applications, the consistent part and the integrable part of a d.d.s. are equal. Let us note here that recently there have been also considered inconsistent discrete-time systems [Rieger, Schlacher 2011]. However, the name ‘hidden restrictions’ has been used rather improperly in the paper cited. In fact, the authors consider the discrete-time dynamical systems whose configuration spaces are given by the infinite systems of equations defined in the infinite dimensional spaces. The restrictions are known, but not all of them are taken into considerations at the begin, when the entire constitution of a discrete-time system is being defined. Denote by = ( 1, 2, 3 ) a tangent vector to the space 3 at a point x. The vector is the velocity vector at x of a point moving along a differentiable trajectory in

3

which is passing through the point x. The space of all pairs ( x, ), where 3

3

a tangent vector to at the point x ∈ , defines the total space T tangent bundle to 3 . The constraints given by the d.a.e. (1) define the following subset

= {( x1, x 2, x3, 1,

2, 3) ∈ T

3≅

6:

x3 = 0,

1=

0,

2

= x1, and

3

of the

3

= x2 }

is

(3)

of the state-velocity space T 3 . Here, the symbol has been used to denote the d.a.e. (1). The subset defines the total space of the constitutive bundle of the d.d.s. which corresponds to the d.a.e. (1). The bundle which corresponds to the set of all states and velocities which have been constrained by the d.a.e (1) is denoted by . The fibre of above the point ( x1, x 2 , x3 ) is the vector ( 1, Thus, the constitutive bundle

2, 3 )

= ( 0, x 2 , x3 ) .

(4)

is the vector field. It is shown in Figure 1.a.

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Fig. 1. (a) The constitutive bundle . (b) The tangent part of part to its base space of . The base space of the constitutive bundle

, denoted by B (

3

. (c) The tangent

) ,. is the

projection of on the space of points x. The base space of the bundle the configuration space of the d.d.s. ,

= {( x1, x 2 , x3 ) ∈

3:

x3 = 0 } .

is

(5)

Let us observe that not all vectors of the constitutive bundle are the tangent vectors to the configuration space of the d.d.s. . Thus, is not a tangent vector field on . In general, the constitutive bundle of the integrable part of a d.d.s. is necessarily a tangent sub-bundle. For most systems being the subject of interest, the constitutive bundle of the integrable part of a d.d.s. equals the tangent part to its base space of the constitutive bundle of the system. That is, the integrable part of a d.d.s. equals its consistent part. Thus, the algorithms for extraction of the tangent part to its base space of a sub-bundle are of the basic importance when the problem of extraction of the integrable part of a d.d.s. is under considerations. They are the subject of this article. The integrable part of the d.a.e. (1) is given in (2). The constitutive space of the d.d.s. which corresponds to the d.a.e. (2) is the point ( x, ) = (0, 0) in T 3 . Let us observe that the sub-bundle which is the constitutive bundle of the d.d.s. being the integrable part of the system is the largest tangent sub-bundle of . It equals the tangent part to its base space of shown in Figure 1.c. The following iterations have been performed in order to extract the tangent part to its base space of the bundle of vectors given in Figure 1.a. At the step number one, there has been extracted the tangent part of the bundle which is the vector field shown in Figure 1.b. However, the vector field in Figure 1.b. is not a

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tangent vector field. Hence, the iterations were continued at the step number two. The total space of the tangent part of the bundle of vectors which has been obtained at step number one of the iterations is the pair ( x, ) = (0, 0) . This is exactly the desired result. The extraction process has been terminated at the step number two. In result of the iterations shown in Figure 1, there has been obtained the constitutive bundle of the integrable part of the d.d.s. . An extension of this example which concerns an electrical network will be presented in Section 4. The main subject of the considerations in this article are the algorithms for extraction of the consistent part of an inconsistent d.d.s. A d.d.s. is essentially inconsistent, if the proper configuration space of the system is a proper subset of its configuration space. One can deduce the existence of hidden constraints in an essentially inconsistent d.d.s. by analysing description of the constitutive space of the system. Hidden constraints, which imply reduction of the configuration space of the system to its proper subspace, arise as additional algebraic and lower-order differential relations which must be satisfied by the dynamic variables along the trajectories of the system. Hidden constraints are observed in the case when the configuration space is too large in order to describe the real degree of freedom of a system, as well as in over-determined systems, and in singular systems. The procedure of completing the constraints which define the constitutive sub-bundle of a d.d.s. by the extracted hidden constraints is called the reduction of the d.d.s. to its consistent form. The geometric index of a given d.d.s. has been defined as equal to the number of iterations of the projection-prolongation reduction algorithm which are necessary in order to extract the consistent part of the system [Beitrag… 1990], [Involution… 2010], [Rabier, Rheinboldt 1994], [Reich 1990, 1991], [Seiler 1998], [Szatkowski 1989, 1990b, 1992]. Here prolongation means an extension by the differentiation. Next, there has been proposed the definition of the structural index of a sub-bundle, [Szatkowski 2001, 2002a, 2002b, 2003]. It has been shown that the geometric index of a d.d.s. and the structural index of the constitutive bundle of the system are the same numbers. First general solutions to the problem of completion of the differential systems to the equivalent consistent systems were presented for partial differential equations in the Janet-Riquier theory [Janet 1920] and the Cartan-Kähler theory [Les Systémes… 1945]. The development of these theories is contained in [Involution… 2010] and [Systems of partial… 1978]. The Cartan-Kuranishi process was the first general algorithm for completion of an implicit partial differential equation to an equivalent involutive one which is necessarily consistent. It stems from the Cartan-Kuranishi theorem given e.g. in [Kuranishi 1957]. The Cartan-Kuranishi theorem is one of the main results of the formal theory of differential equations which is devoted to the problem of construction of a formal power series solution of an implicit partial differential equation. In the articles [Fesser et al. 2002], [Seiler 1998], and in [Involution… 2010], the jet bundles formalism was used in the study of the projection-prolongation algorithm which is the basic process for reduction of a general d.d.s. to an equivalent consistent system. In the cited references, the reduction algorithms are discussed in connection with the problems of the formal theory of differential equations. We formulate the projection-prolongation reduction process within the sub-bundles of the tangent bundles formalism continuing the line of research of [Beitrag… 1990], [Mendella et

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al. 1995], [Rabier, Rheinboldt 1994], [Reich 1990, 1991], [Szatkowski 1989, 1990a, 1990b, 1992, 2001, 2002a, 2002b, 2003]. One can discuss the projection-prolongation algorithm in connection with the Cartan-Kuranishi algorithm, in the case when there is considered the problem of reduction of a d.d.s. to an equivalent consistent system. However, the projectionprolongation algorithm possesses generalization which one can use to partial differential inclusions, for the completion of the constraints by the hidden constraints [Szatkowski 2001]. It is to be noted that the generalization of the projectionprolongation algorithm defines the basic process for construction of all hidden constraints also in a partial differential inclusion, see Supplement. The Dirac algorithm was proposed as the geometrically motivated process for construction of all hidden constraints in the constrained Hamiltonian systems, [Dirac 1950, 1958], [Fesser et al. 2002], [Involution… 2010], [Mendella et al. 1995], [Seiler 1998]. Next, the great advances were made when there have been applied the methods of modern differential geometry for describing dynamics of singular Lagrangian and Hamiltonian systems in which hidden constraints may arise, see [Carinena 1990], [Gotay, Nester 1979, 1980], [Gracia, Pons 1992]. In [Szatkowski 1990a, 2002a], the reduction process was described with use of the projectionprolongation algorithm, for the equations of motion for a broad class of nonholonomic systems of mass points with the generalized Appell-Tshetajev constraints. In [Mendella et al. 1995], the Dirac algorithm was formulated as a special version of the projection-prolongation process. Nevertheless, the Lagrangian and Hamiltonian equations are particular classes of differential equations. A completely intrinsic geometric solution to the problem of reduction of a general d.d.s. to an equivalent consistent system was given in [Szatkowski 1989, 1990b, 1992, 2001, 2002a, 2003]. Also the examples concerning electrical networks and systems of mass points were discussed in these articles. The reduction process of a general inconsistent implicit ordinary differential equation to an equivalent consistent ordinary differential equation was studied comprehensively in [Systems of partial… 1978], [Beitrag… 1990], [Rabier, Rheinboldt 1991, 1994], [Reich 1990, 1991]. In [Gracia, Pons 1992], [Munoz-Lecanda, Roman-Roy 1999] the projection-prolongation algorithm was used in the study of quasilinear differential equations with hidden constraints. In [Munoz-Lecanda, Roman-Roy 1999], the examples were analysed which concerned the sliding control systems. Another solution concerning the problem of extracting the consistent part of a first-order implicit ordinary differential equation was developed in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995]. The reduction algorithm which is described in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995] uses the differentiation procedure. A given implicit differential equation is differentiated until the step at which the corresponding system of derivative array equations uniquely determines the derivative x' of the dependent variable x as a function of x. However, this termination condition of the prolongation algorithm does not assure that the resulting vector field is tangent to its base space. Thus, an additional procedure is necessary in order to verify whether the resulting implicit differential equation is a consistent ordinary differential equation. We shall prove that if the map which defines constraints for the constitutive subbundle of an implicit differential equation is sufficiently submersive, and the additional solvability conditions are satisfied, then one can consider the prolongation algorithm as the calculation procedure which makes it possible to perform the

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successive steps of the projection-prolongation process, under assumption that the properly defined termination condition has been used for the prolongation iterations. In conclusion, although the prolongation algorithm in its formulation developed in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995] is very important in the calculations, the general theory of inconsistent differential dynamical systems can not be based on the prolongation process. The basic procedure for reduction of a general d.d.s. to an equivalent consistent system is given by the projection-prolongation process. The class of d.d. systems, such that the tangent part to its base space of the constitutive bundle of a system defines the constitutive bundle of an integrable dynamical system is considered most often in the modelling of real systems. In that case, the tangent part to its base space of the constitutive bundle of a d.d.s. is equal to the infinitesimal generator for the solutions of , and the proper is equal to the solution space of . We will call configuration space of the d.d.s. this class of d.d. systems the relatively well-posed or, respectively, the well-posed systems. A d.d.s. is well-posed if the infinitesimal generator of the system is equal to the entire constitutive bundle of the system. If is a relatively well-posed d.d.s., including a well-posed system, then both the infinitesimal generator and the solution space of the system are known, if the has been extracted. Then a consistent initialization is possible, consistent part of which is the first step in numerical integration of the differential inclusion whose trajectories define the evolution of the system in time. A d.d.s. is called the regular system, if it is the well-posed or the relatively wellposed system, and the infinitesimal generator for the solutions of the system is a vector field [Beitrag… 1990], [Reich 1990, 1991]. Incomplete and singular d.d. systems are discussed from a geometric point of view. A d.d.s. is an incomplete system if it is not observed by the solutions in its entire configuration space. A dynamical system whose solution space is not a topologically closed subspace of the ambient space of the system is called the singular dynamical system. The points in the topological closure of the configuration space that are not contained in the solution space of the d.d.s. are called the impasse points of the system. The points in the topological closure of the solution space that are not contained in the solution space of the dynamical system are called the singularity points of the system. The systematic classification of the impasse points in dynamical systems which is presented in the next section extends the classification of the impasse points in systems defined by the differential algebraic equations [Reissig 1996]. The classification of d.d. systems is also proposed which includes inconsistent, incomplete, singular, well-posed, and relatively well-posed systems. The subject of the considerations in the part two of the paper is the geometric theory of electrical networks. Electrical networks are discussed as the interesting and important examples of general dynamical systems which are possibly essentially inconsistent.

2. GEOMETRIC MODELLING OF DIFFERENTIAL DYNAMICAL SYSTEMS The dynamics of a d.d.s. is considered as motion occurring in the solution space embedded in the ambient space which is the space where the entire system is defined. The ambient space of a d.d.s. of the order k, k being a positive natural number, has the structure of a C k -manifold. The constitutive space of a d.d.s. of the order k is a -7-

subset of the total space of the tangent bundle of the order k to the ambient manifold. The constitutive space of a d.d.s. is usually given by a set of differential-algebraic relations which define the constitution of the system. On the constitutive space of a d.d.s. there is defined the structure of a sub-bundle which is inherited from the ambient tangent bundle. General Banach manifolds are considered, unless it is explicitly assumed that the manifold is modelled on Euclidean space, [Nonlinear functional… 1988]. Let M be a C k -manifold. The tangent bundle of the order k to M is denoted by T kM. Let T kM denote the total space of the bundle T kM , and ( x, ) a point in T kM , where x ∈ M , and

is a vector in the tangent space TxkM to M at x ∈ M .

There are also used the notations

x

and ( x' , x'' , ..., x ( k ) ) for a vector in TxkM . The

projection map from T kM onto M is denoted by px(⋅) , px (⋅) : T kM ∋ ( x, ) → x ∈ M . A sub-bundle triplet

of the tangent bundle T kM to a differentiable manifold M is a

=(

,

x (⋅), B (

)) ,

(6)

where: is a subset of T kM called the total space of , x (⋅) is the restriction of the projection map px (⋅) to , and B ( ) = x ( ) is the base space of the sub. The empty sub-bundle is denoted by ∅ and is considered as the empty bundle subset of T kM . Let T kM denote the set of all sub-bundles of T kM , where M is a C k -manifold. The empty subset of T kM is included as the empty sub-bundle ∅ . = ( , x (⋅), B ( ) ) and a point x ∈ B ( ) , the subset x For a sub-bundle k of Tx M of all vectors such that ( x, ) ∈ is the fibre of above x. Let x(⋅) : I ∋ → x( ) ∈ M be a C k - function, I being an open and nonempty interval in the space of real numbers. The map

~ D k(⋅) : x(⋅) → ( x , x' , ..., x ( k ) ) (⋅) ,

(7)

~ ~ where Dom D k x(⋅) = I , and D k x( ) = ( x ( ), x' ( ), ..., x ( k ) ( )), defines k th order prolongation of x(⋅) ; x' (⋅) is the derivative of x(⋅) , and x (i) (⋅) is the derivative of ~ the order i of x(⋅) . For ∈ I , p (D k x ( ) ) ∈ Txk( ) M , where p (⋅) : T kM ∋ ( x , ) → ∈Txk M . Definition 1. A differential dynamical system of the order k, denoted by , is a triplet ( M , , ) , where M is a C k - Banach manifold, is a sub-bundle of k the tangent bundle T M to M, and is a set of functions called the solutions to . Each function x(⋅) ∈ is defined on an open and nonempty interval in the space of reals and takes values in M, x(⋅) is k times differentiable, and ~k Im D x(⋅) ⊆ .

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The manifold M is the (ambient) space of the d.d.s. , is the constitutive bundle of the system , is the constitutive space of , and the base space B( ), denoted by , is the configuration space of . We call x the state variable of the system . In the constructive differential-geometric approach to the modelling of real systems, the constitutive space of a dynamical system of the order k is given by a set of the constitutive differential-algebraic relations defined in the tangent bundle T kM to the manifold M being the ambient space of the system. If the relations which define the constitutive space of a d.d.s. are given by a set of constraint equations, then the dynamics of the system is given by the solutions of the corresponding differential algebraic equation. of a d.d.s. contains implicitly all the necessary The constitutive bundle information which is sufficient in order to extract a tangent sub-bundle whose trajectories define the solutions of the system. Generally, the dynamics of a d.d.s. is given by the solutions of a differential inclusion. Remark 1. The following is assumed for a d.d.s. = ( M , , ) which is called a non-autonomous d.d.s., and is explicitly considered as a nonautonomous system: M =

i).

(

t∈

( t , y) ) ,

y ∈Mˆ t

(8)

where t is the variable which denotes ‘time’, t is the time axis, subset of t , and t × Mˆ t is a subset of M, for each t ∈ . ii).

p t(

, t, y) )

=1

at each point x = ( t , y ) in the configuration space

is a nonempty

(9) of the d.d.s.

. The symbol

p t (⋅) in the equation (9) denotes the projection map from the tangent space T( kt , y ) M onto the tangent space Tt

t

to

t

at t.

The variable t, which denotes time in the expressions (8) and (9), is considered as equivalent to the independent variable which has been used in the definition of a solution of a d.d.s. The equivalence of the variables t and means that − t has a constant value along any fixed solution of a d.d.s. Higher-order derivatives of the coordinate t = t ( ) of a solution x = x( ) of a higher-order nonautonomous system, which are equal to zero, are usually not taken into account in the analysis of the dynamic behaviour of the d.d.s. If a d.d.s. is considered explicitly as an autonomous system then we put y for x. In that case, the time t variable is not taken into account as one of the state variables and the equation t' = 1 is being omitted. In general case, the solution curves of a d.d.s. do not fill to the full extent the configuration space , or even the proper configuration space of the system . The solution space of a d.d.s. = ( M , , ) , called also the dynamic space of the system (see Figure 2), is a subset of the configuration space of the system plaited by its solution curves,

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=

x (⋅) ∈

Im x(⋅) .

(10)

The dynamics of a d.d.s. is given by the trajectories of a tangent sub-bundle defined on the solution space of . A point x which is contained in the solution space of a d.d.s. defines the dynamic state x for the system , and is the set of all dynamic states x for . The following definitions and mathematical remarks are used in the considerations. Let M be a C 1 - manifold, A a nonempty subset of M, and x a point in A. A vector ∈Tx M is tangent to A at the point x, if there is a zero or one-dimensional

C 1 - submanifold P of M such that: x∈ P, ∈Tx P ( is tangent to P at the point x∈ P ), and P is contained in A. The tangent space to A at a point x∈ A , denoted by Tx A , is the subspace of all vectors in Tx M which are tangent to A at x. The tangent bundle TA to a nonempty subset A of M is considered as a sub-bundle of TM. The total space TA of the subbundle TA is given by TA =

x ∈A

( x, Tx A ) .

(11)

For A = ∅ , TA is defined as the empty set, and TA is the empty sub-bundle. Let A be a nonempty subset of a C k - manifold M. The prolongation T kA of the order k of the set A is a sub-bundle of T kM which is defined in the following way. A vector ∈ TxkM is contained in the fibre TxkA if there is a zero or one-dimensional

C k - submanifold P of M such that: x∈ P, ∈TxkP , and P is contained in A. The total space of the sub-bundle T kA is denoted by T kA . For A = ∅ , T kA is defined as the empty set, and T kA is the empty sub-bundle. To a sub-bundle ∈ T kM , the map ∂B (⋅) : T kM ⊇ assigns the sub-bundle ∂ B



∩ T kB(

of all these vectors of

)

(12)

which are contained in the

k

prolongation T B( ) . The total space of the sub-bundle ∂ B is denoted by ∂B . A sub-bundle ∈ T kM , such that ∂ B = is called the ∂ B - invariant subbundle. A sub-bundle A sub-bundle



is ∂ B - invariant iff T kM

⊆ T kB(

).

is considered as a tangent sub-bundle of T kM iff

⊆ T kB( ) . In equivalent formulation, a sub-bundle is a tangent sub-bundle iff it is a ∂ B - invariant sub-bundle. The set of all tangent sub-bundles of the tangent bundle T kM is denoted by t T kM . Definition 2. The tangent part to its base space of a sub-bundle ∈ T kM , denoted by t.p.B , is the largest tangent sub-bundle of T kM which is contained as a subset in , the largest in the sense of inclusion relation. there is being used the symbol t.p.B . For the space of the sub-bundle t.p.B

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Definition 3. The infinitesimal generator for the solutions of a d.d.s. = (M , , ) of the order k is a tangent sub-bundle of T kM , such that =

x (⋅) ∈

~ Im D kx (⋅) .

(13)

One obtains that = B ( ). A point ( x, ) which is contained in the total space of the infinitesimal generator of a d.d.s. defines the dynamic state ( x, ) is the set of all dynamic states ( x, ) for . for the system , and Let = ( M , , ) be a d.d.s. and

G (⋅) :

∋x→

x

⊆ Tx

(14)

the fibre valued map, such that

= Graph G (⋅) .

(15)

Note that the set of the solutions of the d.d.s. solutions of the differential inclusion ( x' , ..., x ( k ) ) ( ) ∈ G ( x( )) ,

is equal to the set of the (16)

where the independent variable for the solutions of the inclusion (16) is considered in open intervals in . Definition 4. There is given a d.d.s. = ( M , , ). = (M , , ) is called the integrable part of the system . The d.d.s. It follows from the definitions that

=

.

Well-posed and relatively well-posed d.d. systems are considered most often in the applications. Definition 5. A d.d.s. is relatively well-posed iff = t.p.B and t

∉ T kM . A d.d.s. is well-posed iff = . One obtains that = B ( t.p.B ) for a relatively well-posed d.d.s. , and = , where = B( ) , for a well-posed system. Definition 6. A well-posed or a relatively well-posed d.d.s., such that the infinitesimal generator for the solutions of the system is a vector field is called the regular d.d.s. [Beitrag… 1990], [Reich 1990, 1991]. For a relatively well-posed d.d.s. , including a well-posed system , the problem of extracting the infinitesimal generator for the solutions of the system reduces to the problem of extracting the sub-bundle t.p.B of the constitutive bundle of the system. The basic algorithm which enables extraction of the sub-bundle t.p.B for a given d.d.s. is discussed in Section 3. t

Definition 7. A d.d.s. is a consistent system iff ∈ T kM . In the other case a d.d.s. is called an inconsistent system. Definition 8. A d.d.s. is essentially inconsistent iff the set B ( t.p.B ), denoted by # , is a proper subset of . The set # (see Figure 2) is called the proper configuration space of the system , [Szatkowski 2003].

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Essentially inconsistent first-order d.d. systems are called the algebraic incomplete systems.

Fig. 2. The manifold M is the ambient space of a first-order d.d.s. whose constitutive bundle is defined as a vector field on the configuration space of . # The vector field on \ does not contain a tangent sub-bundle of vectors, the vector field on # \ the vector field on

is a tangent vector field which defines nowhere a flow, and is a tangent vector field which defines a flow at least locally.

Definition 9. The points in the subset \ # of the configuration space of an essentially inconsistent d.d.s. = ( M , , ) are called the impasse points of the first kind of the system (abbreviated 1-i.p.’s), [Reissig 1996], [Szatkowski 2003]. Definition 10. There is given a d.d.s. . The d.d.s. c = ( M , t.p.B , c ) is called the consistent part of the system . It follows from the definitions that = . c Let us note that is a well-posed or a relatively well-posed d.d.s., iff = c. Definition 11. A d.d.s. = ( M , , ) is complete iff = . Definition 12. A d.d.s. = ( M , , ) is incomplete iff the solution space of the system is a proper subset of the configuration space of . Definition 13. A d.d.s. is strongly incomplete iff the solution space of the system is a proper subset of the proper configuration space # of , [Szatkowski 2003]. In the other words, a d.d.s. is incomplete or, respectively, a d.d.s. is strongly incomplete iff the configuration space or, respectively, the proper is not filled to the full extent by the configuration space # of the system solution curves of the system. Thus, one can observe reduction of the configuration - 12 -

space

of an incomplete d.d.s.

. If

is a strongly incomplete d.d.s., then there

is observed reduction of the proper configuration space

#

of the system.

#

Definition 14. A point in the subset \ of the proper configuration space # , ) is called an impasse point of the of the strongly incomplete d.d.s. = ( M , (abbreviated 2-i.p.). second kind of the d.d.s. Definition 15. If the configuration space of a d.d.s. = ( M , , ) is not a closed subset of the ambient space M of the system , then a point in cl \ is called an impasse point of the third kind of the d.d.s. (abbreviated 3-i.p.). Definition 16. A point in the topological closure cl of the configuration space of a d.d.s. = ( M , , ) which is an impasse point of the first kind, or an impasse point of the second kind, or an impasse point of the third kind is called an impasse point of the d.d.s. . By taking into account the definitions of the impasse points, one obtains the following decomposition

cl where: cl

\

= ( cl

\

)∪(

\

#

is the set of all 3–i.p.’s,

the set of all 2–i.p.’s, and

#

)∪( \

#

\

)∪

,

(17)

is the set of all 1–i.p.’s,

is the solution space of a given d.d.s.

Fig. 3. Illustration to the definitions of impasse points. A point x' ∈

#

\

is

(see Figure 3).

\

#

is an

is an impasse point of the second impasse point of the first kind, and x'' ∈ # \ kind of a d.d.s. . The definition of a singular d.d.s., which is used in the considerations, is the following. , ) is singular iff the solution space of Definition 17. A d.d.s. = ( M , the system is a proper subset of the topological closure cl of in the ambient space M of . Definition 18. For a singular d.d.s. , the points in cl \ are called the singularity points of . The inconsistent, the incomplete, and the singular ordinary differential algebraic systems are comprehensively discussed for instance in [Beitrag… 1990], [Campbell 1993], [Campbell, Gear 1995], [Differential-algebraic… 2006], [Differentialalgebraic… 2008], [Kunkel, Mehrmann 2004], [Mendella et al. 1995], [Rabier, Rheinboldt 1991, 1994], [Reich 1990, 1991], [Reissig 1996], [Seiler 1998], [Szatkowski 1989, 1990b, 1992, 2001, 2002a, 2003]. - 13 -

Let us consider the following example. Example 1. Let = ( M , , ) be a d.d.s. whose constitutive sub-bundle ) = . Denote by is a vector field defined in the following way on B ( )x , where x ∈ , is a vector x ∈Tx the set of rational numbers. The fibre ( which is given by the formula

x

=

0 , if x ∈

,

1 , if x ∈

\

.

is consistent. However, it is the strongly incomplete system, because The d.d.s. there are no solutions to the d.d.s. which would be passing through the points in the subset \ of the proper configuration space # = . The solution curves of = of the are the equilibrium points which fill the entire solution space are the impasse points of the second kind of the system system. The points in \ . Note also that \ is the set of the singularity points of . The relations between the classes of consistent, inconsistent, incomplete, and singular d.d. systems are illustrated in the diagram in Figure 4. The well-posed and the relatively well-posed d.d. systems have been also taken into account.

Fig. 4. Classification of the differential dynamical systems. Let M be a C k - manifold. The following definition is used. Definition 19. A sub-bundle of T kM (including the empty sub-bundle) is locally vector continuous (abbreviated l.v.c.), iff for each ( x 0, 0 ) ∈ there exists a sub-bundle of , such that: B( ) is either zero or one-dimensional

C k - submanifold of M, the total space of is the image of the prolongation of the order k of a C k - function x(⋅) : I ∋ → x( ) ∈ B ( ), I being an open and nonempty interval in

, and ( x 0,

0

)∈ .

- 14 -

Denote by

l.v.c. T kM

the set of all locally vector continuous sub-bundles of T kM ,

where M is a C k - manifold. It follows from the definitions that

l.v.c. T kM



t T kM .

Definition 20. The locally vector continuous part l.v.c. of a sub-bundle ∈ T kM is the largest l.v.c. sub-bundle which is contained in . For the total space of the sub-bundle l.v.c. there is used the symbol l.v.c. Let us note that

l.v.c.

= t.p.B

,

.

(18)

for a broad class of d.d. systems which are considered most often in the applications. , ) fulfils the If the constitutive sub-bundle of a d.d.s. = ( M , relation (18), and



t T kM ,

Definition 5 in this section). If

then the system



l.v.c. T kM ,

is relatively well-posed (the

then the system

is well-posed.

For a relatively well-posed d.d.s. = ( M , , ) , the problem of extraction of the infinitesimal generator reduces to the problem of extraction of the tangent of the constitutive bundle of the system. For a part to its base space t.p.B = . well-posed d.d.s. , The problem of extraction of the tangent part to its base space of a sub-bundle was solved in the articles [Szatkowski 1992, 2001, 2003]. The formulation of the projection-prolongation iterations, called also the ∂ B - reduction iterations, for extraction of the sub-bundle t.p.B of a sub-bundle is presented in the following Section 3. In result of the ∂ B - reduction process, it is possible to find the infinitesimal generator and the solution space for a relatively well-posed d.d.s. . The general projection-prolongation iterations are the extension of the ∂ B - reduction process which has been proposed in order to obtain transformation of an inconsistent implicit ordinary differential equation into an equivalent consistent ordinary differential equation [Beitrag… 1990], [Mendella et al. 1995], [Rabier, Rheinboldt 1994], [Reich 1990, 1991].

3. PROJECTION-PROLONGATION PROCESS The tangent part to its base space t.p.B of the constitutive sub-bundle of a d.d.s. defines the constitutive sub-bundle of the consistent part c of the system. Next, because for a broad class of inconsistent d.d. systems, which are the subject of interest in the applications, the consistent part c of a d.d.s. is equal to the integrable part of , then the problem of extraction of the tangent part to its base space of a sub-bundle is of the basic importance when integrability conditions are considered for d.d. systems with hidden constraints. Let us formulate the iterative projection-prolongation reduction process (the ∂ B - reduction process) for extracting the tangent part to its base space of a subbundle. The formulation of the projection-prolongation iterations which is proposed for higher-order sub-bundles in this section is the extension of the iterations defined for the sub-bundles of the first-order tangent bundles in [Szatkowski 1992, 2001,

- 15 -

2002a, 2003]. In general, the reduction process does not stabilize after a finite number of steps which implies that transfinite iterations are to be considered. Let N be the set {0, 1, 2 ,...} of natural numbers, and the ordinal number of the set N with natural ordering relation of numbers in N, [Set theory 1968]. Let ∈ T kM . The successive steps of the ∂ B - reduction process for extracting the tangent part to its base space of a sub-bundle are given by (0)

=

,

(0)

=

∩ T kB(

), (1) ) , (m ) ) ,

= ∂B

(1) (2)

= ∂B

(1)

=

∩ T kB (

( m +1)

= ∂B

( m)

=

∩ T kB(

( )

=

()

∈N

(19)

,

( + 1)

= ∂B

( )

=

∩ T kB(

( )) ,

( + 1)

= ∂B

( )

=

∩ T kB (

( )) ,

( ).

The ∂ B - reduction iterations terminate at the step , where is the smallest . ordinal number such that ( ) becomes a ∂ B - invariant sub-bundle of It has been taken into account in (19) that ( + 1)

=

( )

∩ T kB(

for each ordinal number . Theorem 1. The following equation Proof. Note that the sub-bundle

( ))

( )

( + 1) ,

=

∩ T kB(

= t.p.B

( )) ,

is true.

which one obtains at the successive step

+ 1 of the iterations (19), contains all these vectors of the sub-bundle are tangent to the base space of sub-bundle

( ),

for

( ).

Next, for

t.p.B



( ),

for each ordinal number

tangent sub-bundle of

( ),

which

being a limit ordinal number the

≠ 0 , contains all these vectors in

the base spaces of each of the sub-bundles

( )

for

which are tangent to

< . It follows then that

. By taking into account that

, one finally obtains that

( )

= t.p.B

( )

is a

.

Definition 21. ([Szatkowski 2001, 2002a, 2002b, 2003]). The ordinal number of steps of the ∂ B - reduction process which are necessary in order to extract the ∈ T kM is called the ∂ B - structural tangent part to its base space of a sub-bundle . index of the sub-bundle

- 16 -

The ∂ B - structural index of a sub-bundle ∈ T kM is denoted by st . The ∂ B - reduction iterations (19) have been called here the projectionprolongation iterations, because at each successive step + 1 one first projects on M, next B( ( + 1)

=

( )

( ))

is prolonged to T kB (

∩ T kB (

( )) ,

( )

and one finally defines

( )) .

Analytic conditions have been formulated for sub-bundles of the first-order tangent bundles which ensure that the ∂ B - iterations terminate at the successive step + 1 of the ∂ B - reduction process. In [Szatkowski 1992, 2001], the transversality condition has been proposed which may be used to a broad class of sub-bundles of the first-order tangent bundles. According to the theorems proved in [Szatkowski 1992, 2001], if the locally defined transversality relation is satisfied for the fibres and for the tangent spaces to the base space of the sub-bundle which has been obtained at the step of the iterations, then the ∂ B - reduction process terminates at the successive step + 1 , and the resulting sub-bundle ( + 1 ) = t.p.B is locally vector continuous. If is the constitutive sub-bundle of a d.d.s. , then in that case the sub-bundle t.p.B is the infinitesimal generator for the solutions of the system . The conditions which ensure that a d.d.s. is regular (see the Definition 6 in Section 2) were given in [Beitrag… 1990], [Rabier, Rheinboldt 1994], [Reich 1990, 1991] for a broad class of systems defined by implicit ordinary differential algebraic equations. Generally, the ∂ B - structural index of a sub-bundle may be not a finite number. Hence, transfinite ∂ B - reduction iterations are to be considered. However, we note that the values of the ∂ B - structural index are finite numbers for the constitutive subbundles of the d.d. systems which are the subject of interest in applications. Transfinite ∂ B - iterations may be of interest in algebraic-topological studies of the sub-bundles. , ) be a d.d.s. We assign the sequence Let = ( M , ( )

= ( M,

, ( ),

,( ) ) ,

= 0,1, 2,... st

,

(20)

of the d.d. systems to the system . The constitutive bundle , ( ) of the system of the reduction iterations (19), where one should set ( ) is given at the step for

. Note that

,( )

=

, for each

= 0,1, 2,... st

, and

( st

)

=

c,

= ( M , t.p.B , ) being the consistent part of the system . This process of generating the sequence (0) , (1) , (2) ,... ( st ) is called the reduction of a given c

system to a consistent system, where the reduction process corresponds to the projection-prolongation iterations (19) for extracting the tangent part to its base space of a sub-bundle being the constitutive bundle of a given system. The consistent part is the end element of the sequence (20). c of a d.d.s Let us note that the described projection-prolongation reduction process is always effective. of the constitutive bundle of a d.d.s. defines The structural index st the geometric index of the system [Beitrag… 1990], [Involution… 2010],

- 17 -

[Rabier, Rheinboldt 1994], [Reich 1990, 1991], [Seiler 1998], [Szatkowski 1989, 1990b, 1992, 2001, 2002a, 2003]. Definition 22. The geometric index of a d.d.s. = ( M , , ) is defined as equal to the ∂ B - structural index of the constitutive bundle of the system , and it is denoted by g . The geometric index values were studied for implicit ordinary differential equations in [Beitrag… 1990], [Rabier, Rheinboldt 1994], [Reich 1990, 1991]. For a general d.d.s. , the index g was considered in [Szatkowski 1989, 1990b, 1992, 2001, 2002a, 2003]. Let us also note that sometimes as the constitutive bundle of a d.d.s. being analysed there is being taken into considerations the sub-bundle which one obtains at the step number one of the projection-prolongation algorithm. With use of the geometric index g of a d.d.s, one can characterize consistent, inconsistent, and essentially inconsistent d.d. systems in the following way. , ) is a consistent system iff g = 0 . A d.d.s. is an A d.d.s. = ( M , inconsistent system iff

g

> 0 . If

g

>1, then the d.d.s.

is essentially inconsistent.

The projection-prolongation reduction process is the basic procedure which makes it possible to construct all hidden constraints in a d.d.s. If is a relatively well-posed d.d.s., including a well-posed system, then the projection-prolongation process yields the integrable part of the system which in that case is the end element of the sequence (20). In its special formulations, the projection-prolongation algorithm was first used for extracting the consistent parts of the constrained Hamiltonian and Lagrangian systems (see, for instance, [Dirac 1950, 1958], [Gotay, Nester 1979, 1980], [Gracia, Pons 1992]). In general formulation, it was applied in the study of inconsistent mechanical systems in [Mendella et al. 1995], [Szatkowski 1990a, 1990b, 2002a]. The projection-prolongation procedure was also applied in the theory of electrical networks in order to perform reduction of a set of state variables in a network [Beitrag… 1990], [Reich 1990, 1991], [Szatkowski 1988, 1989, 1990b, 2002a]. In [Munoz-Lecanda, Roman-Roy 1999], the projection-prolongation algorithm was used in the theory of sliding mode control systems with hidden constraints. In its general formulation, the projection-prolongation process was described in [Beitrag… 1990], [Mendella et al. 1995], [Rabier, Rheinboldt 1994], [Reich 1990, 1991], [Szatkowski 1990b, 1992, 2001, 2002a, 2003]. Connection of this reduction process with the Cartan-Kuranishi algorithm formulated for implicit partial differential equations is discussed in [Fesser et al. 2002], [Involution… 2010], [Seiler 1998]. Remark 2. One can generalize the projection-prolongation reduction iterations (19) and the definition of the ∂ B - structural index of a sub-bundle of the tangent bundle to the manifold for a sub-bundle of the jet bundle. Next, one can define the geometric index of a partial differential inclusion as equal to the structural index of the constitutive jet sub-bundle of the inclusion, see Supplement. One can also define the projection-prolongation reduction iterations in the general formulation for the sub-bundles of the co-tangent bundles to the manifolds, see [Szatkowski 2001].

- 18 -

4. EXAMPLES Let us illustrate the theory by the examples. In the following, the independent variable in a d.d.s. is defined to be the time t variable of the system being analysed. Example 2. In Figure 5.a there is given the circuit diagram of an electrical built of n capacitors and n − 1 linear voltage controlled current sources. network The network contains also one singular element called nullator.

Fig. 5. (a) Electrical network . (b) The constitutive relation of the voltage controlled current source contained in the segment j of the network . (c) The constitutive relation of the nullator. The capacitance of the capacitor numbered by j equals C j , C j > 0. The constitutive relation of the capacitor C j is given by the equations:

dqC j dt

= iC j ,

and qC j = C j ⋅ vC j , where: qC j is the charge associated with the element C j , vC j is the voltage across C j , and iC j is the current through C j .

Continuing the description, the constitutive relation of the linear voltage controlled current source numbered by j, see Figure 5.b, is given by the equation i j = g j ⋅ vj ,

where: g j ≠ 0 is the transconductance of the source, vj is the voltage across the capacitor C j -1 , and i j is the current through the element g j , j = 2, 3,..., n . - 19 -

The constitutive relation of the nullator, see Figure 5.c, is given by the constraints: vo = 0, and io = 0 . We use the voltage values vC 1, vC 2 ,..., vC n across the capacitors as the state variables in the description of the behaviour of the network . Put vC = ( vC 1, vC 2 ,..., vC n ) . By taking into account the constitutive relations of the elements, and the Kirchhoff equations, one obtains the following differential algebraic equation dvC n g n dvC 1 dvC 2 g 2 = 0, = ⋅ vC 1 , …, = ⋅v , v =0 , dt dt dt C2 Cn C n - 1 C n whose solutions describe the dynamics of the network

n



2n

: vC n = 0,

The configuration space of the d.d.s.

1=

0,

be the d.d.s.

n

which corresponds to the equation (21). That is, total space of the constitutive sub-bundle = {(vC , ) ∈ T

. Let

=( , , is given by 2

=

g2 ⋅ v ,..., C2 C 1

is the subspace

n

(21)

n

=

), where the gn ⋅v } . (22) Cn C n - 1

= { vC ∈

n

: vC n = 0 }

of the system. of the space Note that there are no solutions to the d.a.e. (21) which would be passing through the points in \{0}. Thus, is an incomplete system. Because the constitutive sub-bundle is the vector field which is not tangent to its base space , then is not a consistent system. By using the projectionprolongation algorithm one obtains the sequence of sub-bundles and constrained subsets which terminates at the step = n − 1. Hence, g = st = n − 1. The total space

, ( n - 1)

of the resulting sub-bundle

(vC , ) = (0, 0) in T

n

. The solution space

, ( n - 1)

= t.p.B

is the point

is the point vC = 0 which is the

equilibrium point of the system . = t.p.B which implies the equation We conclude that , = ,c . Let us now analyse the behaviour of the network . In order to do this, let initially be vC = 0 . Suppose next that there has been given a nonzero value to the voltage vC 1 at a time

o>

0 . The following might occur in the real network at

= o : the nullator became the short circuit branch, or it became the open circuit branch. In the first case, the behaviour of the network was no longer defined by dvC n g n the solutions of the d.a.e. (21). In fact, the equation = ⋅v must have dt Cn C n - 1 dvC n been replaced by the equation = 0. In the other case, if the nullator was dt transformed into the open circuit branch, then the constraint vC n = 0 would not have been taken into account in the description of the constitutive space of the network which was observed for ≥ o. Identification of the dynamic space and the state equation of an electrical network is one of the basic problems which are the subject of considerations in electrical circuits theory [Beitrag… 1990], [Brayton, Moser 1964], [Desoer, Wu - 20 -

1972], [Haggman, Bryant 1984], [Matsumoto et al. 1981], [Szatkowski 1982, 1983, 1988]. Both, the projection-prolongation process [Reich 1990, 1991], [Szatkowski 1988, 1989, 1990b, 2002a], as well as the prolongation algorithm [Differentialalgebraic… 2008], [Kunkel, Mehrmann 2004], [Schwarz, Tischendorf 2000] have been used in order to find a set of algebraic independent state variables for an electrical network. Electrical networks are studied in the part two of the paper. Both the complete networks as well as the networks, such that there is observed reduction of the configuration space of a network to its proper subset being the solution space (the dynamic space) of the network are analysed. Example 3. Consider the following differential algebraic equation dx 2 = − x1 , x 2 = ( x1 - a )3 dt 2

which is defined in T

. The d.a.e. (23) is denoted by

constitutive sub-bundle

2) ∈T

The configuration space , (1)

,

,

. The d.d.s.

) , where the total space

which of the

is given by

= {( x1, x 2, 1,

total space

2

is the triplet (

corresponds to

(23)

2

: x 2 = ( x1 - a )3, and

2

= − x1 } .

(24)

is the curve x 2 = ( x1 - a ) 3 in

of the d.d.s.

of the sub-bundle ∂ B

=

∩ T B(

2

. The

) is given by the

constraint equations x 2 = ( x1 - a ) 3 , 2

in T

2

= − x1 , and

2 = 3 ( x1 - a )

2⋅

1

(25)

. Note that for x1 = a and a ≠ 0 the equations (25) do not have a solution.

Thus, for a ≠ 0

, (1)

is a proper sub-bundle of

. This denotes that in that case

is an inconsistent system. Let a ≠ 0 . Then B ( sub-bundle

, (1)

, (1))

, (1)) .

2

: x2 = ( x1 - a ) 3, and x1 ≠ a } . The

is the tangent vector field given by

− x1 , and 3 ( x1 - a ) 2

1=

on B(

= {( x1, x 2 ) ∈

Because

, (1)

2

= − x1

is the ∂ B - invariant sub-bundle of

, then the

projection-prolongation iterations terminate at the step = 1 . Thus: , st = 1 , and g = 1 . Finally, because , (1) = t.p.B , (1) is the smooth vector field, then

, (1)

=

. Hence,

- 21 -

,

=

,c.

Fig. 6. Trajectory behaviour of the system

.

Let us analyse the behaviour of the solutions of the d.d.s. has been assumed that a > 0 . For any initial state x(0) ∈

(see Figure 6). It

= B(

,

that x1(0) > a the point x(t ) in the trajectory of the system

x = (a, 0) at a finite time

, (1)) ,

such

approaches the point

t . For t ≥ t the behaviour of the system

is not

defined. In the other case, if x1(0) < a , x(t ) approaches the equilibrium point

(0, − a 3 ) as t → ∞ . Example 4. To generalize the Examples 2 and 3, consider an over-determined differential algebraic equation

dy = h(t , y ) , dt

(26)

g (t , y ) = 0 , +1 where: t ∈ , y ∈ n , and the functions h(⋅) : → and g (⋅) : sufficiently differentiable. The d.d.s. corresponding to the d.a.e.

n

n

n +1

j

→ are given in (26)

is usually inconsistent [Beitrag… 1990], [Etchechoury, Muravchik 2003], [Involution… 2010], [Reich 1990, 1991], [Seiler 1998], [Szatkowski 2003]. Define the function h (⋅) : h (t , y ) = (1, h (t , y )). Where i is a positive natural i

number, let (L h g )(⋅) denote the directional derivative of the order i defined i +1

i

recursively by: (L h g )(⋅) = ( Dg ( h ))(⋅) , and (L h g )(⋅) = ( L h (L h g ))(⋅) . Assume that the geometric index By the definition, the index which corresponds to

g

g

of the given d.a.e. (26) is a finite number.

of a d.a.e.

equals the geometric index of the d.d.s.

.

It is easily verified that if g (⋅) is the C g - map, for C

g

-1

- map, for

g

g

≥ 1, h(⋅) is the

≥ 2, and each of the maps

g (⋅), ( g , L h g )(⋅), ( g , L h g , L h2 g )(⋅), ..., ( g , L h g , L h2 g ,..., L h

g

- 22 -

-1

g )(⋅) ,

(27)

for

g n +1

≥ 1, is submersive on its kernel, for the independent (t , y ) considered in

, then:

B(

n +1

, ( m ) ) = { (t , y ) ∈

m

: g (t , y ) = 0, (L h g )(t , y ) = 0,..., and ( L h g )(t , y ) = 0 } ,

and , (m ) =

( t , y, 1, h (t , y )) , (t , y ) ∈ B (

at the step number m, m = 0, 1, 2,..., process.

g,

, ( m ))

of the projection-prolongation reduction

Let us note here that if the C 1- map ( g , L h g , L h2 g ,..., L h

g

-1

g )(⋅) , for

submersion on its kernel, for the independent (t , y ) considered in

g

≥ 1, is the

n +1

, and

-1

ker ( g , L h g , L h2 g ,..., L h g )(⋅) ≠ ∅ , then there is satisfied the inequality g

g⋅ j

≤ n +1.

This example possesses the following generalization to infinite dimensional overdetermined d.a. equations. Assume that y ∈ Y, where Y is a Banach space. Set X = × Y. Let h(⋅) : X → Y, and g (⋅) : X → Z , where Z is a Banach space, and let the geometric index g of the corresponding d.a.e. (26) be a finite number. If g (⋅) is the C g - map, for of the maps in (27), for

g

g

≥ 1, h(⋅) is the C

g

-1

- map, for

g

≥ 2, and each

≥ 1, is submersive on its kernel, for the independent

x = (t , y ) considered in X, then the expressions for

, ( m ) and

B(

, ( m ) ) which

have been given for a finite dimensional d.a.e. (26) remain valid for the d.a.e. (26) defined in a Banach space. Where y∈Y and Im g (⋅) ⊆ Z , Y, Z being the Banach spaces, the C 1- map ( g , L h g , L h2 g ,..., L h

g

-1

g )(⋅) , for

≥ 1, is submersive on its kernel, for the

g

g

independent x considered in X, iff the derivative ( D g , D (L h g ),..., D ( L h

-1

g ))x (⋅) of

-1

this map is surjective for all x ∈ ker ( g , L h g , L h2 g ,..., L h g )(⋅) , and g

g

ker ( D g , D (L h g ),..., D ( L h

-1

g ))x (⋅) splits X [Nonlinear functional… 1988]. The g

splitting property is satisfied automatically for ker ( D g , D (L h g ),..., D ( L h g

dim ker ( D g , D (L h g ),..., D ( L h

g ))x (⋅) < ∞ or

g

-1

g ))x (⋅) < ∞ . The second condition is always

satisfied if Z is finite dimensional. Note that in that case the inequality g

g ))x (⋅) if

-1

codim ker ( D g , D (L h g ),..., D ( L h is always true, for

-1

g⋅ j

≤ dim X

being a finite number, and for X being an infinite dimensional

Banach space. Example 5. Consider the following d.a.e. denoted by

- 23 -

,

d z = h(t , z, a) , dt

(28)

g (t, z, a ) = 0 , where: t ∈ , z ∈

n1

all points x is denoted by of

n +1

n +1

. Set: x = (t , z , a), and n = n1 + n 2 . The space of

, h(⋅) is a map of

n +1

n1

into

, and g (⋅) is a map

l

into . The total space =(

n2

, and a ∈

n +1

,

,

of the constitutive sub-bundle ) which corresponds to the d.a.e.

= {( x , x' ) = (t , z , a, t', z', a' ) ∈ T

n +1

of the d.d.s. is given by

: g ( x) = 0, t' = 1, and z' = h( x)} .

(29)

The following three cases are analysed. 5.1. Let g (⋅) be a C 1- map, ker g (⋅) ≠ ∅ , and l = n 2 . Assume that the Jacobi matrix ga(x) is nonsingular for all x ∈ ker g (⋅). Then the equation g ( x) = 0 defines an (n1 + 1) -dimensional C 1- submanifold P of

n +1

which is locally

C 1- parameterized by the variables t and z. The total space , (1) of the constitutive sub-bundle

, (1)

of the d.d.s.

, (1)

obtained at the step number one of the projection-prolongation reduction process for extracting the consistent part is given by: , c of the d.d.s. , (1)

n +1

= {( x , x' ) = ( t , z , a, t', z', a' ) ∈ T : g ( x) = 0, t' = 1, z' = h( x), T and g t ( x) + gz ( x) ⋅ ( z' ) + ga ( x) ⋅ ( a' )T = 0 }.

(30)

It follows from (30), by taking into account nonsingularity of the Jacobi matrix ga(x), for all x ∈ P, that , (1) is the sub-bundle of vectors which are tangent to P. For a point x ∈ P, the fibre (

, (1) )x

is the vector whose co-ordinates are given by

( t', z', a' )( x ) = (1, h( x ), - ( ga- 1 ⋅ ( gt + gz ⋅ h T ))T( x) ) .

Thus, the geometric index d.d.s.

, (1)

g

of the d.a.e.

is the consistent part of

(of the d.d.s.

(31) ) equals one. The

.

The d.a.e. which has been considered in this example is called the Hessenberg index one d.a.e. 5.2. Let us now generalize the example concerning the Hesseberg index one d.a.e. . Assume that: g (⋅) is a C 1- map, ker g (⋅) ≠ ∅ , l ≤ n 2 , and

rank ga( x) = l ,

(32)

for all x ∈ ker g (⋅). Then the equation g ( x) = 0 defines an (n + 1 − l ) -dimensional

C 1- submanifold P of n + 1 which possesses a C 1- atlas {(U , f -1(⋅))} ∈ , being a set of indices, such that f -1(⋅) = ( p t, z , a ( ) U )(⋅) , where a ( ) = ( ai1( ) , ai 2( ) ,..., ain 2 - l ( ) ), i1( ) < i 2 ( ) < ... < in 2 − l ( ) . For

- 24 -



∈ , let a ( ) be

the complementary subsequence of the co-ordinates of a with respect to a ( ). With no loss of generality, we assume that rank ga ( )( x) = l , for each ∈ and all

x∈U . The total space

, (1)

of the constitutive sub-bundle

is given by the formulae (30). Let { } ∈ be a family of open subsets of each ∈ . One obtains , (1) ∩ T

= {( x , x' ) = (t , z , a, t', z', a' ) ∈ T and g t ( x) + gz

( x) ⋅( z' )T + g

, (1)

of the d.d.s.

n +1

, such that U =

∩ P, for

: g ( x) = 0, t' =1, z' = h( x),

a ( ) ( x )⋅( a' (

))T + ga ( ) ( x) ⋅( a ' ( ))T = 0 } , (33)

for each ∈ . Equivalently, by taking into account that for each matrix ga ( ) ( x) is nonsingular for all x∈U ,



the Jacobi

, (1) ∩ T

= {( x , x' ) = (t , z , a, t', z', a' ) ∈ T : g ( x) = 0, t' = 1, z' = h( x), and a ' ( ) = - ( g a- 1( )( x) ⋅ ( g t ( x) + gz ( x) ⋅ h T ( x) + g a ( ) ( x) ⋅ ( a' ( ))T ))T } .

It follows from (30) and (34), that The dimension of the fibres of the d.a.e. ,c

of

(of the d.d.s.

, (1)

, (1)

, (1)

(34)

is the affine tangent sub-bundle to P.

equals n 2 - l . Thus, the geometric index

) equals one. The d.d.s.

, (1)

g

of

is the consistent part

.

For an open neighbourhood U of P, let V be the projection of U on the space n +1- l

whose co-ordinates are: t , z1,..., zn1, ai1( ) , ai 2( ) ,..., and ain 2 - l ( ) . The

solutions of the d.a.e. differential inclusion

in an open neighbourhood V are the solutions of the dz = h( f (t , z, a ( ) ) , dt da ( ) n -l ∈ 2 . dt

(35)

5.3. In the mathematical modelling of physical systems, there are also considered the d.a. equations which are given by the formulae (28), such that the constraint g (t , z , a) = 0 defines a submanifold of n + 1 which is not locally a graph with respect +1 whose set of the co-ordinates contains all the co-ordinates to a subspace of t , z1,..., and zn1 . The systems of the constitutive differential algebraic equations of n

electrical networks containing loops consisting of capacitors and independent voltage sources and/or cutsets consisting of inductors and independent current sources are the examples of these type of d.a. equations [Desoer, Wu 1972], [Differentialalgebraic… 2008], [Haggman, Bryant 1984], [Szatkowski 1982, 1983, 1988, 1990b]. Thus, let us generalize the example 5.1 in the following way. Let g (⋅) be a

C 1- map, and ker g (⋅) ≠ ∅ . Assume that there exist: a subsequence z = ( zi1, zi 2 ,..., zi r ), i1 < i 2 < ... < ir , of the sequence ( z1, z 2 ,..., zn1 ) of the co-

- 25 -

ordinates of a point z, and a subsequence a = ( ak 1, ak 2 ,..., aku ), k1 < k 2 < ... < k u , of

the sequence (a1, a 2,..., an 2 ) of the co-ordinates of a point a, where r + u = l , such

that det gz , a ( x) ≠ 0

(36)

for all x ∈ ker g (⋅). Let z = ( zj1, zj 2 ,..., zj r ), where j1 < j 2 < ... < j r , be the complementary subsequence of the co-ordinates of z with respect to z , and a = (al1, al 2,..., al u ), where l1 < l 2 < ... < l u , the complementary subsequence of the co-ordinates of a with respect to a . By the assumption (36) being satisfied, the n +1 equation g ( x) = 0 defines an (n + 1 − l ) -dimensional C 1- submanifold P of which is locally C 1- parameterized by the variables t , z , and a . The total space , (1) of the constitutive sub-bundle of the d.d.s.

, (1)

is given

by, , (1)

If

n +1

= {( x , x' ) = (t , z , a, t', z', a' ) ∈ T : g ( x) = 0, t' =1, z' = h( x), T T and g t ( x) + g z ( x) ⋅( z' ) + g z ( x) ⋅( z' ) + g a ( x) ⋅( a' )T + ga ( x)⋅( a' )T = 0 } .

, (1)

(37)

is a tangent sub-bundle, then the projection-prolongation iterations

terminate at the step number one. In the other case, one should continue the extraction process at the step number two. It follows from (37), by the assumption (36) being satisfied, that z' T −1 = − [ gz ga ]I x ⋅ ( g t ( x) + g z ( x) ⋅( z' ) T + g a ( x) ⋅( a' ) T ) at ( x , x' ) ∈ T a'

, (1) .

(38)

Where x ∈ P, we partition the matrix [ gz ga ]I x , denoted by E (x), according −1

to

E ( x) =

E1( x ) E 2( x )

}r }u

.

(39)

Define: h ( x) = (hi1, hi 2,..., hir )T( x) ,

(40)

h ( x) = (hj1, hj 2,..., hjr )T ( x) .

(41)

and

It is now easily observed that the base space of the sub-bundle

, (1)

is given

by, B(

, (1) )

= {x ∈

n +1

: g ( x) = 0, and there exists a' ∈

u

,

such that h ( x) = - E1( x) ⋅ ( g t ( x) + g z ( x) ⋅ h ( x) + g a ( x) ⋅ ( a' )T )}.

- 26 -

(42)

In this example, the total space

, (2)

of the sub-bundle

, (2)

which one

obtains at the step number two of the projection-prolongation algorithm for extraction of the sub-bundle t.p.B is defined by, , (2)

If

, (2)

= { ( x , x' ) ∈ T B(

, (1) )

: t' =1, and z' = h( x)} .

(43)

is a tangent sub-bundle, then the projection-prolongation iterations

terminate at the step number two. In that case, the geometric index

g

of the d.d.s.

equals two. The examples of the index two d.a. equations which are given by the formulae (28) are analysed in the part two of the paper, where the subject of the considerations are the dynamic equations of electrical networks. Example 6. Let us consider a mass point having the inert mass m > 0 and observed in a chosen inertial reference system 3 whose co-ordinates are the time co-ordinate t , and the position co-ordinates x1 and x 2 . The Euclidean space 2 of points x = ( x1, x 2 ) is the position space of the mass point . The total space 4 of the tangent bundle T 2 is the position-velocity space of . The co-ordinates in T 2 are x and , where = ( 1, 2 ) denotes velocity of . Let the space and velocity co-ordinates of the mass point the equation (

2

− a ) 3 − ( x 1 − b) = 0 ,

be constrained by

(44)

where a and b are given numbers. The constraint equation (44) defines a sub-bundle of T 2. The fibre above the point x ∈ B ( ) , where , x of

B(

)=

2

, is the one-dimensional affine subspace of Tx 2

The total space of

= 3 x1 − b + a .

of the sub-bundle 2

given by the equation

(45)

is the constrained position-velocity space

. There is also given a function F (⋅) : T

force acting in T

2

2

∋ ( x, ) →

2

which defines the

. In this example, let F ( x, ) = (− x1 + b, 0) .

The second tangent bundle T 2 2 is necessary in the following, in order to formulate the equation of motion of the mass point . The co-ordinates in T 2 2 are x and , where = ( , ). The co-ordinate , where = ( 1, 2 ), defines the acceleration of . The generalized d’Alembert principle, called also the Appell-Tshetajev principle, is assumed for the reaction force of the perfectly smooth nonholonomic constraints imposed on the space and velocity co-ordinates of . The formulation of this constraints reaction principle is contained in the following Definition 23. Definition 23. Let x(⋅) : t → 2 be a two times differentiable function defined on an open and nonempty interval Dom x(⋅) in . The function x(⋅) defines the ~ trajectory of the mass point , if Im Dx(⋅) ⊆ , and for each t ∈ Dom x(⋅) there exists a reaction force vector R = R( x, ), where x = x(t ) and = x' (t), which is

- 27 -

⊥ contained in the orthogonal complement (T , x ) of the tangent space T , x at ∈ , x , and which satisfies the following equation

m ⋅x'' (t ) = F ( x(t ), ( t)) + R( x(t ), (t )) . The orthogonal complement (T

, x)



,x

(46) 2

is defined in the space

of vectors . 2

, let pII(⋅) be the projection map from the space For a fixed ( x, ) ∈ vectors onto the tangent space T ,x. The total space

=(

2

,

,

of the constitutive sub-bundle

2

: ( x, ) ∈

of

of the d.d.s.

) whose solutions define the trajectories of

= { ( x, , ) ∈ T 2

to

is given by

, and m ⋅ pII( ) = pII( F ( x, ))} .

(47)

By taking into account (44), (45) and (46), and the expression for F ( x, ) , one obtains = { ( x1, x 2 , 1,

2,

1,

2

2 2 ) ∈T

We shall prove that the d.d.s.

:(

2 − a)

3 − (x

1 − b)

= 0 , and

is not consistent. Put

(0)

1=

=

− x1 + b }. m (48) at the begin

of the projection-prolongation process (19). The constraints which define the total space (1) of the sub-bundle (1) = (0) ∩ T 2 2 are given by the equations

(

2 − a)

3 − (x

1 − b)

= 0,

1=

− x1 + b , and 3 ⋅( m

It is easy to see that for 2 = a and not have a solution with respect to

2

− a)2 ⋅

2



1=

0.

(49)

1≠

0 the equation 3⋅ ( 2 − a )2 ⋅ 2 − 1 = 0 does (1) is a proper sub-bundle of (0) . 2 . Thus,

Let us analyse geometric structure of the sub-bundle Figure 7) defined as the projection of

(1)

(1) .

The set

(1)

on the position-velocity space T

(see 2

of

is a proper subset of the constrained position-velocity space of . It is easily verified that (1) is the union of two disjoint subsets ('1) and (''1) , where ('1) is 2

a one-dimensional submanifold of T

x1 = b , and

1= 0,

given by the constraints and

2

=a,

'' is a three-dimensional submanifold of T

(1)

x1 ≠ b , and The base space of the sub-bundle the line x1 = b in

2

2

'

(1)

2

= 3 x1 − b + a .

(50) given by (51)

of the position-velocity sub-bundle

, and B ( '('1) ) is the complement of B(

- 28 -

'

(1) )

in

2

.

is

Fig. 7. Geometric structure of the sub-bundle '

The fibre ''

(1) , x

above the point (b, x 2 ) ∈ B(

is the vector (0, a), and the fibre

(1) )

'(' ) ) is the one-dimensional affine subspace of Tx 1

above the point x ∈ B(

(1) , x

'

which is defined in (45). The fibres of (1) are given by: for a point x ∈ B( (1) , x

= { ( 1,

(1) .

2 2 , 1, 2 ) ∈ Tx

2

:

1 = 0,

2

2

'

(1) ) ,

= a , and

1 = 0},

(52)

and (1) , x

2 2 , 1, 2 ) ∈ Tx

= { ( 1,

2

:

at x ∈ B ( '('1) ) . It is easy to see that

2

= 3 x1 − b + a ,

(1)

=

∩T2

− x1 + b , and m

2

=

3 ⋅(

is not a tangent sub-bundle of T 2

because there does not exist a vector = ( 1, ∈ (1) , x at x ∈ B( ('1) ) , and 2 ≠ 0 . By continuing reduction of the d.d.s.

1=

2 , 1, 2 )

in Tx2

2

1

2 2 − a) (53)

2

,

, such that

to its consistent part one defines

2

at the step number two of the projection-prolongation process. Note that each of the fibres (1) , x , for x ∈ B( ('1) ) , has been reduced to its subset (2)

(2) , x

(1)

= { ( 1,

of the fibres

2 2 , 1, 2 ) ∈ Tx

2

: 1 = 0, 2 = a , 1 = 0, and 2 = 0 } whereas each '' (1) , x , for x ∈ B ( (1) ) , remained the same. Note also that the

on the position-velocity space T 2 equals (1) . The reaction force of the constraints at (b, x , 0, a ) ∈ ('1) is the zero vector, and projection

(2)

of

(2)

2

(0, m

1

( x1, x 2 ,

3 3 ( x1 − b)2 ) is the reaction force vector at the point 1,

3

x1 − b + a ) ∈

''

(1) .

- 29 -

}

It is easy to see that the sub-bundle

(2)

is the tangent sub-bundle of T 2

projection-prolongation iterations terminate at the step = 2. One obtains in conclusion that: (2) = t.p.B = 2 , and , st d.d.s.

, (2)

=(

2

,

) is the consistent part

(2) ,

noting that the sub-bundle

(2)

,c

g

,c

. The

= 2 . The

. Next, by

of the d.d.s.

is l.v.c., one concludes that

2

is the integrable

part . , of Let us now analyse the behaviour of the trajectories of the mass point . The dynamics of is given on ('1) by the solutions of the following system dx1 dx 2 d 1 d = 0, = a, = 0 , and 2 = 0 dt dt dt dt

(54)

of the differential equations. On the other hand, dx1 = dt and

dx 2 = dt

3

d 1 − x1 + b = , m dt

1,

x1 − b + a , and

are the equations of motion of

on

''

(55)

d 2 = dt 3⋅(

2

Let ( x o,

o)

(1) .

1

− a)2

(56)

be a point in

''

(1) ,

being the

at t = 0 . By solving the system (55) of the differential equations, initial state of one obtains that the trajectory of which is passing through the point ( x o, o) at t = 0 approaches the line x1 = b (the set B ( ('1) ) ) at a finite time t given by

t = m ( - arc ctg

m 1o ). x 1o − b

At the time t , when the trajectory approached the line x1 = b , the value of

1

has

been reduced immediately to zero. The value of 2 became constant and equal to a. For t > t , the mass point continues motion along the line x1 = b with constant velocity 2 = a . Let us also note that lim - 2 ( t) = ∞ . t→t

In conclusion, the mass point

, which started in

''

(1) ,

behaved as if it was

approaching the transporting band at a finite time. The transporting band is moving with a constant speed a along the line x1 = b . Let us note here, that the projection-prolongation process for extraction of the consistent part of the dynamic equation of a system of mass points with the generalized Appell-Tshetajev constraints was described in its general formulation in [Szatkowski 1990a, 2002a]. Example 7. This example concerns hidden constraints in sliding mode control systems. The theory of variable structure control systems and their associated sliding regimes are presented e.g. in [Sira-Ramirez 1988], [Slotine, Sastry 1983], [Slotine 1984], [Utkin 1977]. Hidden constraints in quasilinear sliding mode control systems are discussed in [Munoz-Lecanda, Roman-Roy 1999].

- 30 -

Let P be a nonempty subset of a differentiable Banach manifold M, with the topology and the differential structure inherited from M. A point x ∈ M is considered as a controlled point. Let E be a Banach space. A control is assumed to be a map u (⋅) : M ∋ x → u ( x) ∈ Ux , where Ux is a nonempty subset of E. There is also defined a map

f (⋅) :

x ∈M

x × Ux ∋ ( x, u ) → f ( x, u ) ∈ Tx M

(57)

which is given in the description of the control system. Let u be the tangent sub-bundle of TM whose total space u

=

x ∈M

( x, Im f ( x, ⋅) ) .

u

is given by

(58)

There is considered the differential inclusion x' (t ) ∈

and the corresponding d.d.s. constitutive bundle

u

u

= (M ,

of the d.d.s.

(59)

u , x(t) u

u

,

u

) . The total space of the

is defined in (58). For a given control u (⋅) ,

the differential inclusion (59) becomes the differential equation

x' = f ( x, u ( x))

(60)

which is defined on M. Next, there is considered the constrained system where u IP

uIP

= (M ,

uIP ,

u IP

),

uIP

is the sub-bundle u with the restricted base space to P. Note that = { x(⋅) ∈ : Im x(⋅) ⊆ P } . The question is to seek a set s of inputs u (⋅) , u

such that for each u (⋅) ∈

s

the evolution of the constrained control system

x' = f ( x, u ( x)) , x∈ P ,

(61)

takes place in a possibly large subset of P on which the dynamics can be observed. This is the problem which is considered in the sliding control. We note that the maximal set on which the sliding dynamics can be observed is necessarily a subset of B ( t.p.B ( u I P )) . To solve this generalized problem which is considered in the sliding mode control we use the projection-prolongation algorithm (19), with = u I P , in order to extract the sub-bundle t.p.B ( u I P ) . In result, one obtains the consistent part ( u I P )c = ( M , t.p.B ( u I P ) , uIP . u IP ) of the d.d.s. The sequence P(0) , P(1) , P(2) ,... P( st ( u IP )) of the base spaces of the successive sub-bundles (

u I P ) (0)

=(

u IP ) ,

(

u I P ) (1) ,

given explicitly by:

- 31 -

(

u I P ) (2) ,…

(

u I P ) ( st (

u IP ) )

is

P(0) = P , P(1) = { x ∈ P(0) : P( 2) = { x ∈ P(1) : P( m + 1) = { x ∈ P( m) : P( where P( st (

u IP ) )

+ 1)

u , x ∩ Tx P(1 ) ≠ ∅ } ,…,

u , x ∩ Tx P(m ) ≠ ∅ } ,…

= { x ∈ P(

= B ( t.p.B (

u , x ∩ Tx P(0) ≠ ∅ } ,

P(

)

u , x ∩ Tx P( ) ≠ ∅ } ,…

):

u I P )) .

=

∈N

P( st (

P( ) , u IP ) )

(62)

,

One can also find an example of a sub-bundle

such that st ( u I P ) is not a limit ordinal number, and the sequence (62) stabilizes at the step st ( u I P ) - 1. We remark that if finite dimensional systems are under considerations then those constrained control systems are the subject of interest whose constitutive sub-bundles have finite values of their ∂ B - structural indices. Define s to be the set of all those controls u (⋅) : M ∋ x → u ( x) ∈ Ux , for which the following takes place: 1). f ( x, u ( x)) ∈ ( t.p.B ( u I P ))x , for each x ∈ B ( t.p.B ( u I P )) , and 2). for each u (⋅) ∈ s , the constrained differential equation (61) possesses a solution, for each initial state x(0) ∈ B ( t.p.B ( u I P )) . Assume that s ≠ ∅ . For each u (⋅) ∈ s , the subset B ( t.p.B ( u I P )) of P is the solution space of the constrained control system (61). The set s is the family of all controls u (⋅) , such that B ( t.p.B ( u I P )) is the sliding control subset of each of the constrained control systems (61), where u (⋅) ∈ s . In the applications, there is usually given a set a of available controls. In that case, if the set a of available controls is given, one should verify the conditions 1). and 2). by taking into considerations the controls u (⋅) ∈ a . If sa ≠ ∅, where a a ∩ s , then B ( t.p.B ( u I P )) is the sliding control set of the control system s = (61), for each u (⋅) ∈ sa . As an example, consider the following constrained control system u,

dx1 dx 2 dx = u1 , = ϕ ( x) + u 2 , 3 = ( x) , x3 = 0 , dt dt dt where: ϕ (⋅) :

u 2(⋅) :

3

3

∋ x → ϕ ( x) ∈ ,

3

(⋅) :

(63)

∋ x → ( x) ∈ , u1(⋅) ∈ C (

3

, ), and

∋ x → u 2( x) ∈ . Assume that ( x) ≠ 0, for x 2 ≠ 0, and

( x) = 0, for x 2 = 0.

(64)

The equation x3 = 0 defines the constrained subset P in the considered control system. The total space of the tangent sub-bundle u is given by u

3

= { ( x, x' ) ∈ T



6

: x'3 = ( x) },

(65)

and uIP

= { ( x, x' ) ∈ T

3

: x3 = 0 , and x'3 = ( x) }

- 32 -

(66)

is the total space of the constitutive bundle system u I P . By the assumption (64), one gets: u I P ) (1)

(

= { ( x, x' ) ∈T

3

u IP

=

uIP

of the constrained

: x2 = 0 , x3 = 0 , and x'3 = 0 }

and next, (

u I P ) (2)

= { ( x, x' ) ∈T

u I P ) (2) 3

Because (

sub-bundle in T

3

: x 2 = 0 , x3 = 0 , x'2 = 0 , and x'3 = 0 }.

(67)

is the tangent bundle to the x1 axis which is embedded as the u I P ) (2)

, then (

= t.p.B (

u I P ).

Thus, st (

u IP

) = 2.

By taking into account the constraint x'2 = 0 in (67), one obtains that the set

a s

of the controls is the set of all u (⋅) = (u1, u 2 )(⋅), such that: u1(⋅) ∈ C ( 3, ), and u 2 (⋅) = −ϕ (⋅). For each (u1, u 2 )(⋅) ∈ sa, the solution space u IP of the corresponding control system (63) is the x1 axis in equation

3

, and the differential algebraic

dx1 dx 2 dx = u1( x1, 0, 0) , = 0, 3 = 0, x 2 = 0, x3 = 0 dt dt dt is the consistent part of the d.a.e. (63) which in that case equals the integrable part of (63). Example 8. There is considered the following d.a.e. denoted by ,

dx1 dx 2 = x2 , = − x1 , dt dt 0, if x4 ≠ 0, dx3 = dt x3, if x 4 = 0, dx4 x , = 4 dt x21 + x 22 + x32 + x42 = r 2,

(68)

where r is a positive real number. The total space of the constitutive sub-bundle of the d.d.s.

=(

4

,

,

= {( x1, x 2, x3, x4 , 1, 1 = x2 ,

Because

2 = − x1,

) which corresponds to the d.a.e. 2, 3, 4 ) ∈T

3 = 0, if x4 ≠ 0,

4

is given by

: x21 + x 22 + x32 + x42 = r 2,

3 = x 3, if x4 = 0 , and

is not a tangent sub-bundle, then the d.d.s.

4 = x4 }.

(the d.a.e.

(69) ) is not

consistent. Let us use the projection-prolongation process in order to extract the consistent part . At the first step, we obtain , c of the d.d.s.

- 33 -

, (1)

= {( x1, x 2, x3, x4 , 1,

By noting that x3 , (1)

2 3 + x4

= 0, for ( x, ) ∈

= {( x1, x 2, x3, x4 , 1, x3 = 0, x4 = 0,

Because

, (1)

2, 3, 4 ) ∈

2

2 3 + x4

4

= − x1,

3

= 0 , and

4

(71)

= 0 }.

is the tangent sub-bundle being the vector field

1 = x2 ,

2

= − x1,

4

and 4 = 0 on the circle x21 + x 22 = r 2 embedded in = 1, process ends at the step number one. We have: st =

(70)

: x21 + x 22 + x32 + x 42 = r 2,

3 = 0,

,c

= 0 }.

, iff x3 = 0 and x4 = 0, we get

2, 3, 4 ) ∈ T

1 = x2 ,

: x3

, then the extraction g = 1, and

, (1) .

5. EQUIVALENCE OF DIFFERENTIAL DYNAMICAL SYSTEMS The following preliminary remarks are necessary in the considerations concerning equivalence of the d.d. systems. Let M 1 and M 2 be the differentiable Banach manifolds. Denote by ( T M 1, T M 2 ) the set of all maps G (⋅) : T M 1 ∋ → G ( ) ∈ T M 2 , and let ∂B (

T M 1,

TM 2 )

be the subset of all these maps G (⋅) ∈ (

T M 1,

TM 2 )

which

satisfy the commutation relation ∂ B G (⋅) = G ∂ B (⋅) .

The ∂ B - commutation of a map G (⋅) ∈

∂B(

T M 1,

(72) TM 2 )

denotes that G (⋅)



transforms the tangent part of each of the sub-bundles

TM1

onto the tangent

part of the corresponding image sub-bundle G ( ), [Szatkowski 2002b]. For a map G (⋅) ∈ ∂ B ( T M 1, T M 2 ) and an integer

∂B G (⋅) = G ∂B (⋅) ,

(73)

where ∂B (⋅) is the map ∂ B (⋅) composed times with itself. The set ∂ B ( T M 1, T M 2 ) contains the subset { df (⋅) : f (⋅) ∈ embd ( M 1, M 2 )} of maps, for embd ( M 1, M 2 ) being the set of all differentiable embeddings of M 1 into M2. Let G (⋅) ∈ ∂ B ( T M 1, T M 2 ) . Note that if st < for a sub-bundle



T M 1,

being the ordinal number of the set N of natural numbers, then st G (

) ≤ st

.

(74)

Let be an ordinal number, and M a differentiable Banach manifold. The map ∂ B (⋅) is defined in the following way,

∂ B (⋅) :

TM ∋

→ ∂B

- 34 -

=

( )

,

(75)

where the sub-bundle

( )

is given at the step number

of the ∂ B - iterations (19).

That is,

∂ B (⋅) = id (⋅), ∂ B1 (⋅) = ∂ B (⋅) , ∂ B +1(⋅) = ∂ B ∂ B (⋅) , o

(76)

and ∂ B (⋅) :

TM



→ ∂B

=

( ),

<

(77)

for

being a limit ordinal number. Let M 1 and M 2 be the differentiable Banach manifolds, and f (⋅) ∈ embd ( M 1, M 2 ) . Then, [Szatkowski 2001, 2002b], for the sub-bundles 1∈ T M 1 and 2 ∈ T M 2 such that 2

= df (

1)

(78)

2, ( )

,

there is satisfied the equation

df ( for each ordinal number

1, ( ) )

=

(79)

. Consequently, t.p.B

2

= df ( t.p.B

1)

(80)

and st

= st

2

1

.

(81)

The equation (81) ensures that the map df (⋅), for f (⋅) ∈ embd ( M 1, M 2 ) , preserves the structural index of a sub-bundle. Equivalence of the differential dynamical systems is considered according to the following definition. Definition 24. The d.d. systems a = ( M a , , a ) and a b

= (M b ,

a

b

of the d.d.s.

bijection of

a

,

b

a

) are equivalent iff there exists a map g (⋅) of the solution space

onto the solution space

onto

b

b

of the d.d.s.

b

which defines

by taking xb(⋅) = g xa (⋅), for all xa (⋅) ∈

a

.

It is easily verifiable that the given d.d. systems a and b are equivalent, if there exists a C 1 - embedding f (⋅) of the manifold M a into the manifold M b , such that b

= df (

a

).

(82)

If the equation (82) is satisfied for the d.d. systems a and b , for f (⋅) being a onto by C - embedding of M a into M b , then f (⋅) defines bijection of a b 1

taking xb(⋅) = f

xa (⋅), for all xa (⋅) ∈

a

. Consequently,

b

= f(

a

) . Next, by

taking into account the preliminary notes, one obtains the following relations:

- 35 -

b, (

for each ordinal number

)

= df (

b

= df (t.p.B

b

= df (

a,(

(83)

))

, t.p.B

a

),

(84)

and a

).

(85)

6. PROLONGATION ALGORITHM Another solution concerning the problem of extracting the consistent part of a differential dynamical system defined by a first-order implicit ordinary differential equation was developed in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995]. The algorithm which was described in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995] uses the differentiation procedure. In the successive steps of the algorithm there are performed prolongations on a given implicit differential equation. An implicit differential equation is differentiated until the step at which the obtained set of derivative array equations uniquely determines the derivative x' of the dependent variable x as a function of x. If the prolongation iterations are effective, then the smallest number d , such that d - th derivative array determines x' uniquely as a function of x is called the differentiation index of an implicit differential equation. Let us note that although this determinacy criterion is widely explored in the computing procedures, it is not the properly defined termination condition of the prolongation iterations, because the resulting vector field may be not a vector field which is tangent to its base space. Thus, an another criterion for termination of the prolongation algorithm should be formulated. The properly defined termination criterion for the prolongation process is proposed in this section. A better name for d would be determinacy index, as it gives the number of prolongations necessary to obtain a determined system [Seiler 1998]. The relations between the prolongation and the projection-prolongation algorithms are studied. The inequality will be proved which is satisfied by the geometric and the proper differentiation indices for a broad class of implicit differential equations. Also new conditions will be formulated which ensure that the prolongation process is effective. We shall prove that if the map which defines constraints for the constitutive subbundle of an implicit ordinary differential equation is sufficiently submersive, and additional solvability conditions are satisfied, then one can consider the prolongation algorithm in its formulation developed in [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995] as the calculation procedure which makes it possible to perform the successive steps of the projection-prolongation process, under assumption that the properly defined termination condition has been used for the prolongation iterations. In that case, if the geometric index of the equation is a finite number, then the proper differentiation index is well defined and it equals the geometric index of the equation being under considerations. There is considered an implicit ordinary differential equation

f ( t , y, y' ) = 0 ,

- 36 -

(86)

where: f (⋅) is a map of 2n +1 into j , t ∈ is the independent variable, y ∈ the dependent variable, and y' denotes derivative of a differentiable function

n

is

n

y (⋅) : I ∋ t → at a point t o∈ I , I being an open and nonempty interval in . We assume that the map f (⋅) is sufficiently differentiable, so that all needed differentiations can be performed. In the following we put (t , y ) for x, and we identify with t the independent variable which has been used in the definition of a solution of a d.d.s. In this section, there are used the notations of the references [Campbell 1993], [Campbell et al. 1994], [Campbell, Gear 1995], [Campbell, Moore 1995]. Write to denote the differential equation f ( t , y, y' ) = 0, t' - 1 = 0

(87)

being the equation (86) completed by the equation t' - 1 = 0 . Let n n +1 = ( + 1, , ) , where is the space of all points (t , y ) , and the sub-bundle of T

n +1

is

whose total space is given by n +1

= { ( x, x' ) = ( t , y, t' , y' ) ∈ T

: f ( t , y, y' ) = 0, and t' = 1 } .

(88)

In the following, the space T + 1 is considered as equivalent to 2 ( n + 1) . Put: ui = ( t , y , y' ,..., y ( i + 1 ) ) , and z i = ( t , t', y , y', y'', ..., y ( i + 1 ) ) , where i ∈ N , and wr = ( 1, y', y'', ..., y ( r + 1) ) , where r ∈ N +, N + being the set of positive natural numbers. By differentiating a given equation (86) k times with respect to t, one obtains the following system of (k + 1) ⋅ j differential equations n

f (t , y, y' ) = 0 , d f (t , y, y' ) = 0 , dt

(89)

d k f (t , y, y' ) = 0 , dt k where the variable y has been assumed to be a k + 1 times differentiable function of t. Next, [Campbell 1993], [Campbell, Gear 1995], [Campbell, Moore 1995], the system (89) of the equations, after completion by the equation t' - 1 = 0 , is transformed into the following system f (uo) ( Lw1 f )(u1) ( Lw2( Lw1 f ))(u 2) ( Lwk ( Lw k - 1 (...Lw1 f )...))(uk ) t' - 1 - 37 -

=0,

(90)

of the differential equations which is defined on the space T × T k +1 of all points zk . The symbol L denotes directional derivative. Where k ∈ N , let fk (⋅) be the map given by, fk (⋅) :

( k + 2) n + 1

∋ uk → fk (uk ) =

( f (uo), ( Lw1 f )(u1), ( Lw2( Lw1 f ))(u 2),..., ( Lwk ( Lw k - 1 (...Lw1 f )...))(uk )) ∈

n



( k + 2) n + 2

( k + 1) j

.

(91)

The system (90) of the differential equations defines prolongation of the differential equation to T × T k +1 n . The equation prolonged to n k + 1 ( k ) T ×T is denoted by . The equation (90), considered at that place as algebraic equation, and called in that case the derivative array equation, defines a sub-bundle in T × T k +1 n which (k ) (k ) is denoted by . The base space of the sub-bundle is defined in the space n +1

(k )

of points x. The sub-bundle

(k )

,

(k)

(k )

Let

(k )

=

is the constitutive bundle of the equation

.

n+1

tangent bundle T

n +1

, where T

Next, define the d.d.s.

(k )

2 ( n + 1)



(k )

(k )

that

(k )

(k )



n+1

=(

of the d.d.s. which

,

(k )

).

(92)

is well defined iff the function f (⋅) is k times differentiable. Note , and

(k ) ⊆

, where:

(k )

is the infinitesimal generator for the

( k ) is the infinitesimal generator for the

solutions of the equation (87), and solutions of Denote by

n+1

,

(k )

,

on the

is the space of all points ( x, x' ) .

as the projection on T

corresponds to the prolonged equation

The d.d.s.

(k )

be a sub-bundle being the projection of the sub-bundle

. , k +1

the subset of all those solutions of the d.d.s.

k +1 times differentiable. Let

, k +1

, k +1

=

which are

be the infinitesimal generator for ~ Im D x (⋅) .

x (⋅) ∈

, k +1 ,

(93)

, k +1

The following lemma is necessary in the following. Lemma 1. Assume that the function f (⋅) is k times differentiable. Then the d.d.s. (k )

is well defined, and

, k +1

(k )



Thus, , k +1



which is equivalent to

(k )

Proof. Let ( x o, x' o ) = ( t o, y o, t' o, y' o ) ∈



. , k +1 .

, k +1



(k )

.

(94) This means that t' o = 1 , and that

there exists a solution y (⋅) of the equation (86) which is defined on an open interval

- 38 -

Dom y (⋅) containing t o , which is k +1 times differentiable and satisfies: y (t o ) = y o, and y ' (t o ) = y' o . Because f (t , y (t ) , y' (t ) ) = 0 , for all t ∈ Dom y (⋅) , then d i f (t , y (t ) , y' (t ) ) = 0 , dt i

(95)

for each i = 1,2,..., k and all t ∈ Dom y (⋅) . The equations (95), for i = 1, 2,..., k , assure that ( t o, t' o, y o, y' o, y (2) ( t o ) ,..., y ( k + 1) ( t o ) ) ∈

(k )

,

(96)

where t' o = 1 . By noting that (96) implies the inclusion ( t o, y o, t' o, y' o ) ∈ concludes that

, k +1



(k )

(k )

, one

.

Corollary. Let the function f (⋅) be k times differentiable. From (94) it follows that

, k +1 =

(k )

iff the solutions of the d.d.s.

(k )

are k +1 times differentiable.

Also note, that if the solutions of the differential equation (86) are k +1 times differentiable, then ( k ) = . Let us now formulate the idea of the prolongation algorithm in the following way. An implicit differential equation , which is given in (87), is prolonged in the successive steps of the algorithm. If there is a finite number k, such that the subbundle (k ) becomes a tangent sub-bundle, then the prolongations are stopped. In that case the prolongation algorithm is considered to be effective, and the smallest number k such that (k ) becomes a tangent sub-bundle is called the proper differentiation index of the differential equation . If the proper differentiation index is well defined for an implicit differential equation , then it is denoted by d . Consider the following example. Example 9. There is given the following implicit differential equation

( x'1 ) 2 + ( x'2 - x1 )2 + x 22 = 0

(97)

which is defined in T 2 . The constitutive sub-bundle of the differential equation (97) denoted by is shown in Figure 8. In result of the geometric analysis

Fig. 8. The constitutive sub-bundle

- 39 -

.

of the structure of the sub-bundle , one obtains that the total space of the subbundle t.p.B is the point ( x, x' ) = (0, 0), and that the projection-prolongation = 1 . The solution iterations terminate at the step number one. Hence, g = st space of the equation is the point x = 0, which is the equilibrium point of . By differentiating the differential equation (97) with respect to t, one obtains the following equation x'1 x'1' + ( x'2' - x1' )( x'2 - x1 ) + x2 x2' = 0 which is defined in T 2

2

. The equations (97) and (98) define the prolongation 2

to T 2

of the equation

. It is easy to see that the sub-bundle (1)

projection of the constitutive sub-bundle sub-bundle T

2

of T 2

(98)

2

is equal to

(1)

of the equation

(0)

(0)

, where

=

(1)

(1)

being the

on the tangent

. Thus the

prolongation process is not effective at the first step. Let us continue the prolongation iterations at the step number two. By differentiating the equation (98) with respect to t, one obtains the following equation

( x1'' )2 + x1' x'1'' + ( x'2'' - x1'' )( x'2 - x1) + ( x'2' - x1' ) 2 + ( x2' ) 2 + x2 x''2 = 0

(99)

which is defined in T 3 2. The equations (97), (98) and (99) define the prolongation (2) of the equation to T 3 2. Note that the total space of the sub-bundle constitutive sub-bundle of T 3

2

>

of the equation

d

(2)

(2)

is the point ( x, x' ) = (0, 0). Hence,

conclusion that d

(2)

(2)

being the projection of the

on the tangent sub-bundle T

= t.p.B

= 2. In that example, the indices

g

2

. One obtains in

and

d

satisfy the inequality

g.

Contrary to the projection-prolongation process, the prolongation algorithm may be not effective. This is shown in the following example. Example 10. Consider the implicit differential equation given by the constraints: x'1 = 0 , g ( x'2 - 1) = 0 , x1 = 0 , g ( x2 ) = 0 ,

(100)

where g (⋅) is the C ∞− function defined by,

g ( w) = exp(−1/ w2 ) , for w ∈ The total space 2

T ≅ that t.p.B

4.

\ {0}, and g (0) = 0.

of the constitutive bundle

of

(101)

is the point (0, 0, 0, 1 ) in

The constitutive bundle is not the tangent sub-bundle. One obtains = ∅ , and g = 1 for the equation .

Let us now use the prolongation algorithm in order to extract a consistent subsystem of the d.d.s. which in that example is necessarily equal to the consistent part

,c

of the system

. It is easily observed that no additional constraints arise

- 40 -

in result of the successive prolongations of the constraints (100) which implies that (k ) = , and ( k ) = , for each k ∈ N. Thus, ( k ), where k ∈ N , never becomes , c . In that example, the prolongation algorithm is not effective. We shall prove the basic relations between the projection-prolongation algorithm for extraction of the consistent part of a d.d.s. and the prolongation algorithm for extraction of a consistent sub-system of the d.d.s. corresponding to an implicit differential equation given in (87). For notational convenience let us write the differential equation (87) as

F ( x, x' ) = 0 ,

(102)

where

F ( x, x' ) = F ( t , y, t', y' ) = ( f ( t , y, y' ), t' - 1 ) .

(103)

Put: uˆ i = ( x , x',..., x ( i + 1 ) ) , where i ∈ N , and wˆ r = ( x' , x'', ..., x ( r + 1 ) ) , where r ∈ N + . For k ∈ N , let Fk (⋅) be the map defined by, Fk (⋅) : ( k + 2) (n + 1) ∋ uˆk → Fk (uˆk ) = ( F (uˆo), ( Lwˆ 1F )(uˆ1), ( Lwˆ 2( Lwˆ 1F ))(uˆ 2),..., ( Lwˆ k ( Lwˆ k - 1 (...Lwˆ 1F )...))(uˆk )) ∈ n +1

Note that ( k ) is the projection of ker Fk (⋅) on the space T uˆo = ( x, x' ). In the following, we write

(k )

for the sub-bundle of T k +1

space is given by ker Fk (⋅). The base space of the sub-bundle space

n +1

of the d.d.s.

of points x. Recall also that , (k )

, (k )

( k + 1) ( j +1)

. (104)

of all points n+1

(k )

whose total

is defined in the

denotes the constitutive sub-bundle

which one obtains at the step k of the projection-prolongation

reduction process (20) for extracting the consistent part

,c

of a d.d.s.

.

Let m be a natural number, and let the maps fo (⋅), f 1(⋅), f 2 (⋅),..., and fm -1 (⋅), for

m ≥ 1, be defined as in (91). The following theorem is true. Theorem 2. If f (⋅) is a C m − map, for m ≥ 1, and the maps fo (⋅), f 1(⋅), f 2 (⋅),..., and fm -1 (⋅), for m ≥ 1, are submersive on their kernels, then (k )



, (k ) ,

(105)

for each k = 0,1,2,..., m. Additionally, assume that for each i = 1,2,..., m, and for each point ( x o, x' o ) ∈

, (i ) ,

there exists a point uˆ io- 1 = ( x o, x' o,..., ( x ( i) )o ) ∈ T i

n +1

such that

the equation

( Dx ( i ) Fi - 1 )(x ( i + 1) ) = - ( D(x, x', ..., x ( i - 1 ) ) Fi - 1) ( x' o, x'' o,..., ( x ( i) )o ) ,

(106)

where the derivatives are calculated at uˆ i − 1 = uˆ io−1 , possesses a solution with respect to x ( i + 1) . - 41 -

Then (k )

=

, (k ) ,

(107)

for k = 0,1,2,..., m. Proof. Let us use the induction principle in the proof of the first part of the theorem. Because (o) = , (0) , then the thesis is true for k = 0. Next, assume that the inclusion (105) is satisfied for some k ∈{ 0, 1,..., m - 1 }. Let us derive from this assumption that ( k + 1) ( k + 1)

If ker Fk (⋅) = ∅ , then



, ( k + 1) (k )

=

.

(108)

= ∅ . The thesis is trivial.

Now suppose that ker Fk (⋅) ≠ ∅ . The assumption that the C m - k − map fk (⋅) is submersive on its kernel ensures that the C m - k − map Fk (⋅) is submersive on ker Fk (⋅). Thus, ker Fk (⋅) is a regular C m - k − submanifold of the space T k +1 all points uˆk . Let us write the equation Fk + 1 (uˆk + 1 ) = 0 as

Fk + 1 (uˆk + 1 ) = ( F (uˆo), ( Lwˆ k + 1 Fk )(uˆk + 1) ) .

n +1

of

(109)

It follows from (109), that if uˆ ko + 1 = ( x o, x' o,..., ( x ( k + 1 ) )o, ( x ( k + 2 ) )o ) is a point in ker Fk + 1 (⋅) , then wˆ ko+ 1 = ( x' o, x '' o,..., ( x ( k + 1 ) )o, ( x ( k + 2 ) )o ) is the tangent vector to the submanifold ker Fk (⋅) at uˆ ko = ( x o, x' o,..., ( x ( k + 1 ) )o ) . This assures that the coordinate x' o of wˆ ko+ 1 is the tangent vector to the projection of ker Fk (⋅) on the space n +1

of all points x at the point x o defined by the co-ordinate x of uˆ ko . By taking (k )

into account that is the projection of ker Fk (⋅) on the space T n + 1 of all points ( x, x' ), one obtains that the co-ordinate x' o of wˆ ko+ 1 is the tangent vector to B(

(k )

) at x o. By noting that ( x o, x' o ) ∈

( x o, x' o ) ∈T B(

(k )

)∩

.

From (105), we have that B (

( x o, x' o ) ∈ T B( By noting that

, ( k )) ∩ ( k + 1)

, one concludes that

(k )

) ⊆ B(

, where T B(

, ( k )) . , ( k )) ∩

Finally then,

=

, ( k +1 ) .

is the projection of ker Fk + 1 (⋅) on the space T

points ( x, x' ), we have proved that if ( x, x' ) ∈

( k + 1)

, then ( x, x' ) ∈

n +1

of all

, ( k + 1) .

In conclusion, because the inclusion (105) is valid for k = 0, and the inclusion (108) has been derived from (105) for some k ∈{ 0, 1,..., m - 1 }, then the induction principle assures that (105) is true for all k ∈{0,1,2,..., m }. Let us prove the second part of the theorem. Because (o) = , then the thesis is true for k = 0. , (0) =

- 42 -

Next, let k ∈{1,2,..., m }. By taking into account the first part of the theorem, it suffices to verify that there is satisfied the inclusion If

, (k )

= ∅ , then the inclusion

Now suppose that

, (k )

, (k )

, (k )

(k )



(k )



.

is trivially satisfied.

≠ ∅ , and let ( x o, x' o ) be a point in

, (k ).

The

solvability condition which has been assumed for the equation (106), with i = k , ensures that there exists a point uˆ ko - 1 = ( x o, x' o,..., ( x ( k ) )o ) ∈T k equation DFk − 1 ( x' o,..., ( x ( k ) )o, x ( k + 1 ) ) = 0 ,

n +1

, such that the

(110)

where the derivative is calculated at uˆk − 1 = uˆ ko − 1 , possesses a solution with respect to x ( k + 1 ) . Let ( x ( k + 1 ) )o be a solution of the equation (110). Then the vector wˆ ko = ( x' o,..., ( x ( k) )o, ( x ( k + 1 ) )o ) satisfies the equation ( Lwˆ k Fk - 1 )( x o, wˆ k ) = 0 . By noting that ( x o, x' o ) ∈

, ( k ),

where

, (k )

(111)



= ker F (⋅) , and by taking into

account the equation Fk ( uˆk ) = ( F (uˆo), ( Lwˆ k Fk - 1 )( uˆk ) )

one obtains that uˆ ko = ( x o, wˆ ko ) is a point in ( x o, x' o ) ∈

(k )

, (k )



= ker Fk (⋅). This implies that

.

Thus, we have proved that if ( x, x' ) ∈ that

(k )

(112)

(k )

, (k ) ,

then ( x, x' ) ∈

(k )

which assures

.

Let us note that if the submersivity condition in the part one of the Theorem 2 is to be fulfiled for m = g , g being the geometric index of the implicit differential equation (87), then there must be satisfied the inequality

g⋅ j

≤(

g

+ 1) ⋅ n + 1 , where j

is the dimension of the co-domain of the map f (⋅). Example 11. There is considered the d.a.e. (26) which has been analysed in the Example 4 with use of the projection-prolongation iterations. Let us extract the consistent part of the d.d.s. corresponding to the equation given in (26) by performing prolongations of . In this example, the k-th derivative array equation (90) is given by

- 43 -

y' - h(t , y ) g (t , y ) y'' - ( Lh h) (t , y ) ( L h g ) (t , y )

= 0 ,.

(113)

k

y ( k + 1) - ( Lh h )(t , y ) ( Lhk g ) (t , y ) t' - 1 where the maps g (⋅) and h(⋅) have been assumed to be sufficiently differentiable. (k )

It follows from (113) that the total space of the sub-bundle the following family of equations: 2

is constrained by

k

g (t , y ) = 0, ( Lh g )(t , y ) = 0, ( Lh g )(t , y ) = 0,...., ( Lh g )(t , y ) = 0, t' = 1, and y' = h(t , y ) . (114) If the map g (⋅) is differentiable, where

times differentiable, and the map h(⋅) is

g g

, being the equation

prolonged

times, is well defined. Let us use now the

g

g.

It is easily verified that if g (⋅) is the C g - map, for C

g

-1

- map, for

g

times

has been assumed to be a finite number, then the equation

( g)

Theorem 2, with m =

g -1

g

≥ 1, and h(⋅) is the

≥ 2, and each of the maps g -1

g (⋅), ( g , L h g )(⋅), ( g , L h g , L h2 g )(⋅), ..., ( g , L h g , L h2 g ,..., L h g )(⋅) ,

(115)

+1 is submersive on its kernel, for the independent (t , y ) considered in , then the assumption of the part one of the Theorem 2 is satisfied with m = g . By noting that n

the assumption of the part two of the Theorem 2 is always satisfied for the d.a.e. being analysed, one concludes in that case that ( k ) = , ( k ) , for k = 0,1,2,..., g . Thus,

(k )

=

, (k ) ,

for k = 0,1,2,..., g , the proper differentiation index (

d

is well

d)

= ,c. The submersivity condition is the same as in the Example 4. Note that it requires the inequality g ⋅ j ≤ n + 1 to be satisfied. In this example, j is the dimension of the defined for the d.a.e. ,

d

=

g,

and

co-domain of the map g (⋅). The following Theorems 3 and 4 are the corollaries to the Theorem 2. Theorem 3. If the geometric index g of an implicit differential equation given in (87) is a finite number, and the assumptions contained in the part one and the part two of the Theorem 2 are satisfied for m = g , then the proper differentiation index

d

is well defined for the equation

- 44 -

,

d

=

g,

and

(

d)

=

,c .

Theorem 4. If the proper differentiation index d of an implicit differential equation given in (87) is well defined, and the assumptions in the part one and the part two of the Theorem 2 are satisfied for m = d , then g = d for the equation , (

d)

= , c. . For most implicit ordinary differential equations being the subject of interest, the tangent part to its base space of the constitutive sub-bundle of the equation is a subbundle which is locally vector continuous. In that case, the following theorem is valid. Theorem 5. Assume that the geometric index g of an implicit differential and

given in (87) is a finite number, and f (⋅) is a C g - map, for

equation

If the maps fo (⋅), f 1(⋅), f 2 (⋅),..., and f kernels, the sub-bundle t.p.B the differential equation are

(⋅), for

g

≥ 1.

≥ 1, are submersive on their

is locally vector continuous, and the solutions of g + 1 times differentiable, then ( g)

The index

d

equation

satisfy the inequality

=

.

is well defined for the equation ≤

d

Proof. Because the map f (⋅) is (

g -1

g

g

g.

(116)

, and the indices

g

and

d

of the

(117)

times differentiable, for

g

≥ 1, then the d.d.s.

g)

is well defined, and because the solutions of the differential equation are by assumption g + 1 times differentiable, then it follows from the Corollary to Lemma ( g)

1 that

=

( g)

. The equation

because f (⋅) is the C - map, for g

fo (⋅), f 1(⋅), f 2 (⋅),..., and f

g -1

g

(⋅), for

=

⊆ t.p.B

=

, where t.p.B

sub-bundle t.p.B In conclusion,



( g)

. Next,

≥ 1, and the maps g

≥ 1, have been assumed to be submersive on

their kernels, then the Theorem 2 assures that ( g)

implies that

( g)



, ( g) .

Thus,

by the assumed local vector continuity of the

.



( g)



, which yields (116).

Because the equation (116) assures that obtains that d ≤ g .

( g)

is a tangent sub-bundle, then one

The following Theorem 6 gives the conditions which ensure that if the sequence , (1),..., ( k ),... of the sub-bundles stabilizes at the step m of the prolongation

(o)

process, then (m ) is a tangent sub-bundle. In that case, the d.d.s. (m ) which one obtains in result of the prolongation iterations at the step m is a consistent sub-system which corresponds to the implicit differential equation (87). of the d.d.s Theorem 6. Let f (⋅) be a C m + 1 − map. Assume that the maps fo (⋅), f 1(⋅), f 2 (⋅),..., and f m (⋅) which are defined as in (91) are submersive on their kernels. - 45 -

m + 1)

m

Under these assumptions, if ( = ( ) , then (m ) is a tangent sub-bundle. Proof. Let the assumptions of the theorem be satisfied. Then it follows from the proof of the part one of the Theorem 2 that the fibres of ( k + 1 ) are tangent to B(

(k )

), for each k = 0,1,2,..., m. By taking into account that

( m + 1)

=

(m)

, one

n+1

(m )

obtains that is a tangent sub-bundle of T . Remark 3. 1). For each of the differential algebraic equations which have been considered in the Examples 2, 3, 5.1, and 5.2, the index d is well defined for the corresponding implicit differential equation obtained in the standard way, and its value equals the value of the index g . 2). After transforming the d.a.e. which describes the motion of the mass point in the Example 6 into the first-order d.a.e., one gets d = 3 for the obtained d.a.e. considered as implicit differential equation in the standard way. The map f (⋅) which defines constraints for the constitutive sub-bundle of the obtained implicit differential equation does not satisfy the submersivity assumption contained in the Theorem 5. As a consequence, d > g. The inequality (117) is not satisfied. 3). Note that the function f (⋅) : T

2

∋ ( x1, x 2 , x'1 , x'2 ) → ( x1' ) 2 + ( x'2 - x1 )2 + x 22 is

not submersive on its kernel. As a consequence,

d

>

g

for the d.a.e. (97).

Remark 4. Let Y and Z be the Banach spaces. Consider an implicit differential equation f ( t , y, y' ) = 0 , where f (⋅) is now a map of × Y × Y into Z, y is a point in Y, and y' is the derivative of a differentiable function y (⋅) : I ∋ t → Y at a point in an open and nonempty interval I in . The space n considered as the space of the dependent y in the Theorem 2 is now the space Y, and the image space j of the map f (⋅) in the Theorem 2 is now the space Z. The thesis of the Theorem 2 remains valid for the constitutive bundle of an implicit differential equation defined for the dependent y in the Banach space Y, and for the constraint map f (⋅) having values in the Banach space Z. However, one should take into account the definition of a submersive map which is used in the preimage theorem formulated for maps defined in Banach spaces [Nonlinear functional… 1988]. Further extensions are also available.

7. CONCLUSIONS The problem has been considered of reduction of a differential dynamical system (d.d.s.) to its equivalent sub-system which is as simple as possible. In the approach which has been developed, the reduced d.d.s. is called the consistent part of the given system. The constitutive bundle of the consistent part of a d.d.s. is equal to the tangent part to its base space of the constitutive bundle of the d.d.s. being under considerations. In general, the constitutive bundle of the integrable part of a d.d.s. is necessarily a tangent sub-bundle. For most systems which are the subject of interest, the constitutive bundle of the integrable part of a d.d.s. equals the tangent part to its base space of the constitutive bundle of the system. That is, the integrable part of a d.d.s. equals its consistent part. Hence, the algorithms for extraction of the tangent part to

- 46 -

its base space of a sub-bundle have been found to be of the basic importance when the problem of extraction of the integrable part of a d.d.s. was considered. Two basic algorithms of reduction of a d.d.s. to its consistent part have been discussed. These were the projection-prolongation and the prolongation algorithms. It has been shown that the projection-prolongation iterations are always convergent to the desired result while additional conditions imposed on a d.d.s. being analysed were necessary in order to ensure the effectiveness of the prolongation process.

SUPPLEMENT It has been proved that the projection-prolongation iterations define the basic process for extracting the tangent part to its base space of a sub-bundle of the tangent bundle to the manifold. They define also the basic reduction process of a d.d.s. to its consistent part. In the following, it will be shown that one can extend the projection-prolongation reduction iterations for a sub-bundle of the jet bundle [The geometry… 1989]. Next, one can use the generalized projection-prolongation algorithm to partial differential inclusions in order to complete the constraints by the hidden constraints. The projection-prolongation algorithm defines the basic process of extracting the hidden constraints in a partial differential inclusion. Let M and Z be the C k - Banach manifolds. Write x for a point in M. The space of k-jets of all k times differentiable maps of M into Z is denoted by J k( M , Z ), and J k( M , Z ), is the bundle

( J k( M , Z ), px (⋅), M ) ,

(S1)

px (⋅) being the projection map from J k( M , Z ) onto the base space M of J k( M , Z ).

Next, let

J k( M , Z )

be the set of all sub-bundles of J k( M , Z ). The empty subset

of J k( M , Z ) is included as the empty sub-bundle denoted by ∅ . The empty jet subbundle of J k( M , Z ) is considered as the empty sub-set of J k( M , Z ). A jet subbundle is denoted by . For a subset A of M, { A}M is the set of all C k - submanifolds of M. The empty set and all the zero dimensional submanifolds are included in { A}M . Let u (⋅) be a map defined on a subset of M and taking values in Z, such that for each C k - submanifold P ∈{Dom u (⋅)}M , including the empty set as the submanifold, u (⋅)IP is a k times differentiable map of P into Z. The map defined on a zero dimensional submanifold is considered as infinitely many times differentiable. Their partial derivatives are defined to be the zeros. The k-jet sub-bundle J k u is the sub-bundle of J k( M , Z ) whose total space J k u is given by J ku =

J k(u (⋅)IP) ,

P ∈{Dom u (⋅)}M

(S2)

where each of the k-jet sets J k(u (⋅)IP ), for P ∈{Dom u (⋅)}M , has been naturally embedded in J k( M , Z ). If Dom u (⋅) = ∅, then J k u = ∅ .

- 47 -

Definition S1. A partial differential inclusion (PDI) of the order k, denoted by , , ) , where: is a pair ( is a sub-bundle of the k-jet bundle J k( M , Z ), ∈ J k( M , Z ) , M, Z being the C k - Banach manifolds, and is a set of maps called the solutions to . is each map u (⋅) defined on a subset of M, such that for each A solution to submanifold P ∈{Dom u (⋅)}M , u (⋅)IP is a k times differentiable map of P into Z, and J ku ⊆

.

(S3)

The manifold M is the base space of , M × Z is called the ambient space of , is the constitutive bundle of . The base space denoted by of the and constitutive bundle of the inclusion is called the configuration space of . Define the map

k J (⋅)

in the following way, k JA

M ⊇A→

J k( P, Z ) ,

=

(S4)

P∈{ A}M

where each of the k-jet sets J k( P, Z ) has been naturally embedded in J k( M , Z ). The set Jk A is the total space of the sub-bundle denoted by Jk A of J k( M , Z ). The map ∂B :

J k( M , Z )

∋ ∂B

B( ∂B

→ ∂B =

∈ ∩

J k( M , Z )

k J B(

),

) being the base space of the sub-bundle is denoted by ∂B . A sub-bundle ∈ J k( M , Z ) , such that ∂B

is given in the following way, (S5)

. The total space of the sub-bundle =

is called the ∂B - invariant k-

is ∂B - invariant iff

jet sub-bundle. A k-jet sub-bundle



k J B(

).

The set of all ∂B - invariant sub-bundles of the jet bundle J k( M , Z ) is denoted by

inv . J k( M , Z )

Definition S2. The consistent part of a jet sub-bundle



J k( M , Z ) ,

denoted by

c.p.B , is the largest ∂B - invariant sub-bundle of J k( M , Z ) which is contained as a subset in , the largest in the sense of inclusion relation. There is considered a PDI = ( , ). Let be the sub-bundle of whose total space is the following subset of ,

=

u (⋅) ∈

J ku .

(S6)

The base space of the sub-bundle , denoted by , is called the entire domain space for the solutions of the inclusion . , ). Definition S3. There is given a PDI = ( =( , ) is called the integrable part of . The PDI It follows from the definitions that - 48 -

=

.

Definition S4. A PDI c.p.B

is relatively well-posed iff

is a proper sub-bundle of

. A PDI

= c.p.B

is well-posed iff

, and =

.

= One obtains that = B( c.p.B ) for a relatively well-posed PDI, and for a well-posed inclusion. For a relatively well-posed PDI , the problem of extracting the sub-bundle of reduces to the problem being the constitutive bundle of the integrable part of extracting the sub-bundle c.p.B of the constitutive bundle of . For a well-posed PDI , = . The basic algorithm which enables extraction of the sub-bundle c.p.B for a given PDI = ( , ) is defined with use of the projection-prolongation iterations. It is discussed in this Supplement. Definition S5. A PDI is consistent iff c.p.B = . In the other case a PDI is called an inconsistent inclusion. ), Definition S6. A PDI is essentially inconsistent iff the set B ( c.p.B

denoted by #, is a proper subset of . The set # is called the proper configuration space of the inclusion . \ # of the configuration space of Definition S7. The points in the subset , ) are called the impasse points of the an essentially inconsistent PDI = ( first kind of the inclusion (abbreviated 1-i.p.’s). Definition S8. There is given a PDI . The PDI c = ( c.p.B , c ) is called the consistent part of the inclusion . It follows from the definitions that

c

=

.

= c. Let us note that is a well-posed or a relatively well-posed PDI, iff , ) is complete iff Definition S9. A PDI = ( = . Definition S10. A PDI = ( , ) is incomplete iff the entire domain space for the solutions of the inclusion is a proper subset of the configuration space of . , ) is strongly incomplete iff the space of Definition S11. A PDI = ( the inclusion is a proper subset of the proper configuration space # of . In the other words, a PDI is incomplete or, respectively, a PDI is strongly or, respectively, the proper configuration incomplete iff the configuration space space # of the inclusion is not filled to the full extent by the domains of the solutions to . Thus, one can observe reduction of the configuration space of an incomplete PDI . If is a strongly incomplete PDI, then there is observed reduction of the proper configuration space # of the inclusion. Definition S12. A point in the subset #

#

\

of the proper configuration space

of the strongly incomplete PDI = ( , ) is called an impasse point of the second kind of the inclusion (abbreviated 2-i.p.). Definition S13. If the configuration space of a PDI = ( , ) is not a closed subset of the base space M of the inclusion , then a point in cl \ is called an impasse point of the third kind of (abbreviated 3-i.p.). Definition S14. A point in the topological closure cl of the configuration of a PDI = ( , ) which is an impasse point of the first kind, or an space

- 49 -

impasse point of the second kind, or an impasse point of the third kind is called an impasse point of the inclusion . By taking into account the definitions of the impasse points, one obtains the following decomposition

cl

= ( cl

\

)∪(

\

#

)∪(

#

\

)∪

,

(S7)

where: cl \ is the set of all 3-i.p.’s, \ # is the set of all 1-i.p.’s, # \ is the set of all 2-i.p.’s, and is the solution space of a given PDI . , ) is singular iff the entire domain space Definition S15. A PDI = ( for the solutions of the inclusion is a proper subset of the topological closure cl of in the base space M of . are called the Definition S16. For a singular PDI , the points in cl \ singularity points of . The relations between the classes of consistent, inconsistent, incomplete and singular PDIs are given in the same diagram as it has been drawn for ordinary differential inclusions in Figure 4, Section 2. The consistent part c.p.B of the constitutive bundle of a PDI defines the constitutive bundle of the consistent part c of the inclusion . Next, because for a broad class of inconsistent PDIs, which are the subject of interest in the applications, the consistent part c of a PDI is equal to the integrable part of , then the problem of extraction of the consistent part of a jet sub-bundle is of the basic importance when integrability conditions are considered for PDIs with hidden constraints. Let us formulate the iterative projection-prolongation reduction process, called also the ∂B - reduction process, for extracting the consistent part of a jet sub-bundle. The formulation of the projection-prolongation iterations which is proposed for the jet sub-bundles in this section is the extension of the iterations defined for the subbundles of the tangent bundles to the manifolds. With use of the ∂B - reduction algorithm, it is possible to find the sub-bundle and next the space for a relatively well-posed PDI . In general, the ∂B - reduction iterations do not stabilize after a finite number of steps which implies that transfinite number of iterations is to be considered. Let ∈ J k( M , Z ) . The successive steps of the ∂B - reduction process for extracting the consistent part of a jet sub-bundle

- 50 -

are given by

=

(0) (1) (2)

= ∂B

= ∂B

( m + 1)

=

(0)



=

( m)

=

( )

k J B( ) , k J B ( (1) ) ,



=

(1)

= ∂B

,

k J B(



()

∈N

(m ) ) ,

(S8)

,

( + 1)

= ∂B

( )

=



k J B(

( )) ,

( + 1)

= ∂B

( )

=



k J B(

( )) ,

( ).

The ∂B - reduction iterations terminate at the step , where is the smallest ordinal number such that ( ) becomes a ∂B - invariant jet sub-bundle of . It has been taken into account in (S8) that ( + 1)

=

( )



k J B(

( ))

for each ordinal number . Theorem S1. The following equation

=

= c.p.B

( )

( + 1) ,

Proof. Note that the sub-bundle

k J B(



( )) ,

(S9)

is true.

which one obtains at the successive step

+ 1 of the iterations (S8), contains all these jets of the sub-bundle contained in ( ),

k J B(

Next, for

k J B(

ordinal number

( ) ),

for

which are contained in each of the

< . It follows then that c.p.B

. By taking into account that

, one finally obtains that

( )

which are

being a limit ordinal number the sub-bundle

≠ 0, contains all these jets in

for

sub-bundles of

( )).

( )

= c.p.B

( )



( ),

for each

is the ∂B - invariant sub-bundle

.

Definition S17. The ordinal number of steps of the ∂B - reduction process of a jet subwhich are necessary in order to extract the consistent part c.p.B ∈ J k( M , Z ) is called the ∂B - structural index of the sub-bundle . bundle The ∂B - structural index of a sub-bundle



J k( M , Z )

is denoted by st

.

The ∂B - reduction iterations (S8) have been called the projection-prolongation iterations, because at each successive step + 1 one first projects ( ) on M, next

B(

( ))

( + 1)

=

is prolonged to ( )



k J B(

k J B(

( )),

and one finally defines

( ) ).

Generally, the ∂B - structural index of a jet sub-bundle may be not a finite number. Hence, transfinite ∂B - reduction iterations have been considered. However,

- 51 -

we note that the values of the ∂B - structural index are finite numbers for the constitutive jet bundles of the PDIs which are the subject of interest in applications. Transfinite ∂B - iterations may be of interest in algebraic-topological studies of the jet bundles. , ) be a PDI. We assign the sequence Let = ( ( )

= ( M,

, ( ),

= 0,1, 2,... st

,( ) ) ,

,

(S10)

of the PDIs to the inclusion . The constitutive jet bundle , ( ) of the inclusion of the reduction iterations (S8), where one should set ( ) is given at the step for

. Note that

,( )

=

, for each

= 0,1, 2,... st

, and

(st

)

=

c,

c=

( c.p.B , ) being the consistent part of the inclusion . This process of generating the sequence (0) , (1) , ( 2) ,... (st ) is called the reduction of a given inclusion to a consistent inclusion, where the reduction process corresponds to the projection-prolongation iterations (S8) for extracting the consistent part of a subbundle being here the constitutive jet bundle of the given inclusion . The consistent part c of a PDI is the end element of the sequence (S10). Let us note that the described projection-prolongation process is always effective. The structural index st of the constitutive bundle of a PDI defines the geometric index of the inclusion . , ) is defined as equal Definition S18. The geometric index of a PDI = ( to the ∂B - structural index of the constitutive jet bundle of the inclusion , and it is denoted by g . With use of the geometric index

g

of a PDI, one can characterize consistent,

inconsistent, and essentially inconsistent PDIs in the following way. A PDI = ( , ) is a consistent inclusion iff g = 0 . A PDI inconsistent inclusion iff

g

> 0 . If

g > 1,

then the PDI

is an

is essentially inconsistent.

The projection-prolongation reduction process is the basic procedure which makes it possible to construct all hidden constraints in a PDI. If is a relatively well-posed PDI, then the projection-prolongation process yields the integrable part of the inclusion which in that case is the end element of the sequence (S10). For a well-posed inclusion , = In the following considerations, the prolongation algorithm, and the Theorems 2, 3, 4, and 6 given in Section 6 should be extended to implicit partial differential equations.

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ON GEOMETRIC FORMULATION OF DYNAMICAL SYSTEMS WITH HIDDEN CONSTRAINTS. PART II – DYNAMICAL ELECTRICAL NETWORKS

Andrzej Szatkowski*

In the part two, the geometric theory of electrical networks is discussed. Electrical networks are considered as the interesting examples of general dynamical systems which are possibly essentially inconsistent.

Keywords: dynamical electrical networks; geometric approach; state spaces; algebraic independent state variables; higher index networks; reduction algorithms

*Andrzej Szatkowski, Department of Mechatronics, High School of Computer Science and Management in Olsztyn, Artyleryjska Str. 3f, 10-165 Olsztyn, Poland. Email: [email protected]

- 56 -

1. INTRODUCTION Electrical networks are the interesting and important examples of dynamical systems. Connections between the theory of electrical networks and the theoretical dynamics date from the time when the methods of ordinary differential equations, and also the differential difference and integral equations have been applied to the analysis of the behaviour of electrical networks. Manifolds and sub-bundles provide a natural framework for the study of the trajectory problems of general nonlinear electrical networks. In the geometric approach to electrical networks, the ‘response’ of a dynamical network starting from a given operating point is considered as motion occurring in a subset of a differentiable manifold being the solution space of the network. The solution space of an electrical network is usually called its dynamic space. The main advantages of the geometric approach to electrical networks are the following. a). It is co-ordinate free: the results obtained by a geometric method do not depend on the particular choice of a tree, a loop matrix, the algebraic independent state variables etc. Therefore, the geometric method is suitable for studying intrinsic properties of networks. b). The restrictions on nonlinearities in a network are very weak. Thus, a very broad class of networks is covered. c). The concepts of the state space and the state equation appear in a natural way and are given in a rigorous formulation. d). Generally, the geometric approach is suitable for studying more or less qualitative or global properties. e). The concept of the tangent bundle connects in a natural way the theories of nonlinear and linear networks with use of the linearization operation at a point. From the point of view of nonlinear networks theory, the notion of the tangent space to a subset of a differentiable manifold has the following interpretation: if a of subset of a differentiable manifold is the solution space (the dynamic space) an electrical network , then for electrical networks which are of practical interest Tx is the solution space (the dynamic space) of the ‘small-signal’ electrical network which has been obtained by linearization of the constitutive relation of the given nonlinear network at its ‘operating point’ [Desoer, Wu 1972], [Szatkowski 1982]. The theorems will be proved which characterize the solution spaces (the dynamic spaces), the infinitesimal generators, and the algebraic independent state variables of dynamical electrical networks. One of the main results concerning the dynamic spaces and the sets of the solutions of electrical networks is the theorem which says that if the networks contain the same numbers of capacitive, inductive, and resistive branches, respectively, and the ambient spaces of the networks have been considered as the same spaces, then these networks possess the same sets of the solutions iff they have the same dynamic spaces. For electrical networks which are considered most often, the dynamic space of the network equals its entire configuration space. But for example for networks containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current sources one usually identifies the dynamic space of the network as a proper subset of its configuration space. For

- 57 -

these networks, the structural index of the constitutive bundle of the network is usually equal two. For higher index networks it is necessary to employ the reduction algorithms in order to extract the infinitesimal generator and the dynamic space of a network. Both, the projection-prolongation and the prolongation processes will be used here to reduce the d.d.s. corresponding to the higher index network to its consistent part. For a broad class of dynamical electrical networks, which are usually considered, the behaviour of a network is given by the trajectories of a vector field defined on the dynamic space of the network. Most theorems which will be proved concern this class of networks.

2. DYNAMIC SPACES AND EQUATIONS OF MOTION We begin considerations with the definitions of a network and of an electrical network. Definition 1. A network, in the sense of graph theory, is an ordered triple is a nonempty and finite set called the set of nodes, = ( , , (⋅)), where : is a finite set which is disjoint with and is called the set of branches, and & , where & = {{ ' , '' } : ' ∈ and '' ∈ }. map of into Definition 2. Let = ( , , (⋅)) be a network. The network if 1 ⊆ , 1 ⊆ and 1= ( 1 , 1 , 1 (⋅)) is a subnetwork of 1(⋅) = (⋅) I 1 .

(⋅) is a

One assumes in the following that: i). a network = ( , , (⋅)) does not contain isolated nodes (i.e., for each node ' ∈ there are a node '' ∈ and a branch ∈ ≠ ∅ such that { ', '' } = ( ) ; and ii). the network does not contain self-loops (i.e., the set Im (⋅) does not contain elements of the type { , }, where ∈ ). Definition 3. A network = ( , , (⋅)) is connected if there do not exist the subnetworks ( 1, 1, 1(⋅)) and ( 2 , 2, 2 (⋅)) of such that = 1 ∪ 2 and 1∩

= ∅. Definition 4 (electrical network). Let = ( , , (⋅)) be a network satisfying the assumptions (i) and (ii) given above. Assume that the elements of the sets and have been ordered, and let = ( C , L , R ), where C , L , and R are the disjoint and ordered subsets of . Write mC , ( mL , and mR , respectively), for the number of branches in C , (in L and R , respectively). Set m = mC + mL . The successive elements of the set C are denoted by C1, C1,..., CmC , the successive elements of the set L are denoted by L1, L 2,..., LmL , and the successive elements of the set R are denoted by R 1, R 2,..., RmR . 2

To each Cj ∈

C,

j = 1, 2,..., mC , there is assigned a triple ( qCj , vC j , iC j ) ∈

variables, to each Lk ∈ (

Lk , vLk , iLk

)∈

3

L,

k = 1, 2,..., mL , there is assigned a triple

of variables, and to each R l ∈

- 58 -

R,

l = 1, 2,..., mR , there is

3

of

assigned a pair (vR l , iR l ) ∈

2

of variables. The elements of the set

C

are called the

capacitive branches, the elements of the set L are called the inductive branches, and the elements of the set R are called the resistive branches. The variables qCj , vC j , iC j , where j ∈ {1, 2,..., mC } , are called, respectively, the charge, the voltage, and the current associated with the capacitive branch Cj , the variables

Lk , vLk , iLk ,

where k ∈ {1, 2,..., mL } , are called, respectively, the flux, the voltage and the current associated with the inductive branch Lk , and the variables vR l , iR l , where

l ∈ {1, 2,..., mR } , are called, respectively, the voltage and the current associated with the resistive branch R l . Set m = mC + mL + mR , and let n = m + 2m , ( m = mC + mL ). Denote by n an n-dimensional space of points y : n ∋ y = (qC , L, vC , vL, vR , iC , iL, iR ) = ( qC , L, v, i ), qC = ( qC ,1 ,..., qC , m ) = (qC ,..., qC m ) ,..., and iR = ( iR ,1,..., iR, m ) = ( iR ,..., iR ) , and 1 C

let

R

C

1

mR

n +1

be the space of all points x = (t , y ), where t ∈ R denotes time. A quartet =( , , (⋅), W ), where ( , , (⋅)) = ( , , (⋅)),

and W

is a subset of the space

n +1

of points x, is called an electrical network. n +1

The set W is called the constitutive relation of , and the space of points x is the space of the network . a dynamical network if it is explicitly considered as a differential We call dynamical system. Remark 1. j). In the following, we identify the terms: an R (resp., L, C ) branch and an R (resp., L, C ) two-terminal element of an electrical network. jj). Note that in the definition of an electrical network, which has been proposed, ‘couplings’ among the elements of the same kind, as well as ‘couplings’ among the elements of different kinds are allowed. Remark 2. We will assume in the following that for =( , W ) being an electrical network, =( , , (⋅)) is the connected and directed network. has assigned direction, which denotes that (⋅) is the map of Each branch of into × . Definition 5. The Kirchhoff space of an electrical network , denoted by K , is the space of all points x = ( t , qC , L, v, i ) whose voltage and current co-ordinates v and i, respectively, satisfy the Kirchhoff equations. Given a directed and connected electrical network =( , W ) , pick any tree for (for ). Let be the associated co-tree. Denote by B and D the fundamental loop and cut set matrices associated with and , respectively, [Brayton, Moser 1964], [Graph theory… 1997]. With use of the matrices B and D , the Kirchhoff space K of an electrical network is given by, K

= { x = ( t , qC ,

L , v, i ) ∈

n +1

: B ⋅ vT = 0 and D ⋅ iT = 0 } .

- 59 -

(1)

It follows from (1) that the Kirchhoff space of a network is an n +1 (m + m + 1) - dimensional linear subspace of the space of , [Matsumoto et al. 1981]. Define = K ∩W .

(2)

The set is called the configuration space of the network . be a function defined Definition 6. Let x(⋅) : t → ( t , qC (t ), L (t ), v(t ), i (t )) ∈ on an open and nonempty interval in . The function x(⋅) is the solution of the network if: k). it is differentiable on Dom x(⋅), and kk). d ( q , )(t ) = (iC , vL )(t ) (3) dt C L for all t ∈ Dom x(⋅) . For the first time the geometric point of view has been presented in the theory of dynamical electrical networks in [Brayton, Moser 1964]. Next, the geometric theory of electrical networks was developed in connection with the theory of differential dynamical systems in the references [Beitrag… 1990], [Desoer, Wu 1972], [Differential-algebraic… 2008], [Haggman, Bryant 1984], [Matsumoto et al. 1981], [Reich 1990, 1991], [Smale 1972], [Szatkowski 1982, 1983, 1988, 1989, 1990, 2002]. The basic idea is the following. The constitutive relation of an electrical n +1 is given as a subset W of the space whose co-ordinates are network defined by the time t variable and the electrical variables associated with the elements of the network. Next, the Kirchhoff equations define a linear subspace K n +1

n +1

of . The intersection = K ∩ W is a subset of the space of the of the network , called also the dynamic space of network. The solution space , is a subset of . For a large class of electrical networks = , but for example for networks containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current . sources (see Fig. 2) one usually obtains as a proper subset of Let us now consider an electrical network , W ) as a d.d.s. =( n +1

=(

,

),

,

(4)

where the variable t has been taken as the independent in . The space of the constitutive sub-bundle of the d.d.s. - of the constitutive sub-bundle of the is given by network = {( x, ) ∈T

n +1



2( n + 1)

for j = 1, 2,..., mC , and

: x∈

k + mC +1

,

= 1,

= xm + 1 + m

The base space of the constitutive sub-bundle )= . of the network , B (

- 60 -

1

j +1

= x m + m +1 + j ,

, C+k

for k = 1, 2,..., mL }.

(5)

equals the configuration space

Note that the set of the solutions of an electrical network and the set of which corresponds to the network are the same sets the solutions to the d.d.s. of functions. The infinitesimal generator for the solutions of an electrical network is defined by the same formulae as the infinitesimal generator for the solutions of the d.d.s. which corresponds to the network . The following theorem is true. Theorem 1. Let and ' be the electrical networks which contain the same numbers of the elements: mC' = mC , m'L = mL , and mR' = mR , and let the ambient iff spaces of the networks be considered as the same spaces. Then, ' = . '= Proof. It suffices to prove that if , then . '= ' = implies that ⊆ , then The assumption '= ' . Hence, if x(⋅) ∈ Im x(⋅) ⊆ ⊆ and ' . By taking into account that the spaces of the networks ' are the same spaces, and the function x(⋅) satisfies the conditions (k) and (kk) of the Definition 6, one gets that x(⋅) is the solution of the network '. In the same way, one obtains that . ' ⊆ For each solution x(⋅) of the dynamical network , the velocities qC' (t ) and 'L(t ) are given, respectively, by the values of currents through the capacitive branches, and the values of voltages across the inductive branches of . If the equations qC' (t ) = iC (t ) , and 'L(t ) = vL (t ) , the equation t' = 1, and the constraints for T define a vector field = (x ) on a subset S of , and the vector field = (x ) defines a flow on S, then S is the dynamic space of the network , and the set of the solutions of is equal to the set of the solutions of the differential dx equation = (x ). dt The following Theorem 2 gives the conditions which ensure that the solutions of an electrical network are given by the solutions of a differential equation defined on the dynamic space of the network [Szatkowski 1982, 1983]. Theorem 2. Assume that the dynamic space of a network is an

(1 ≤ l ≤ m + 1) - dimensional C 1- submanifold of n + 1 , and has a C 1 - atlas {(U , f -1(⋅))} ∈ , – being a set of indices, such that f -1(⋅) = ( p t, z ( ) U )(⋅), where: z ( ) = (qC , j ( ) ,..., qC , j 1

r( )(

)

,

,..., L , k 1( )

), L , ku( )( )

1 ≤ j1( ) < j 2 ( ) < ... < jr ( )( ) ≤ mC , 1 ≤ k1( ) < k 2 ( ) < ... < ku ( )( ) ≤ mL , and

r ( ) + u ( ) = l - 1;

U

(⋅) is the inclusion map,

p t, z ( ) (⋅) denotes the natural projection from

U (⋅) : U n +1

∋ x → x∈

onto the space

n +1

, and

l

of all points

( t, z ( )). If the above is satisfied, then the solutions of the network are given by the dx solutions of a differential equation = (x ) defined on dynamic space of dt .

- 61 -

Set w( ) = ( iC , j

1(

)

,..., iC ,

jr ( ) ( )

, vL , k

1(

where ∈ . Then, in

) ,..., vL , k u ( ) ( ) ) ,

,

the local co-ordinates on (⋅) : U ∋ x →

( x ) = ( D f ) f - 1( x ) ⋅ ( 1, pw( ) ( x) )T ∈ Tx

,

(6)

for each ∈ . Proof. At first, let us prove that the family U ∋ x → ( D f ) f -1( x ) ⋅ ( 1, pw( ) ( x) )T ∈Tx

(7)

of maps, which is parameterized by the index ∈ , defines a vector field on For the proof, it suffices to verify the relation ( Df ' ) f - 1 ( x ) ⋅ (1, pw( ' )( x))T = ( Df '' ) f - 1 ( x ) ⋅ (1, pw( '

''

T '' )( x ))

.

(8)

which would be satisfied for each point x ∈ and for all indices ' and '' , such that x ∈ U ' ∩ U '' . Let x o = (t o, y o ) be a point contained in the dynamic space of the network satisfying the assumptions of the Theorem 2, and x(⋅) a solution of such that o o o x(t ) = x . For each pair ' and '' such that x ∈ U ' ∩ U '' , the equation (f

'

pt , z (

')

)( x(t )) = ( f

''

pt , z (

)( x(t ))

(9)

'' )

is well-defined for all these t in a nonempty and open subset of Dom x(⋅), for which x(t )∈U ' ∩ U '' . By differentiating the equation (9) one gets the following equation

d (f dt

'

pt , z (

x )(t ) = d ( f dt

')

pt , z (

''

'' )

x )(t )

(10)

which is equivalent to

( Df ') f - 1( x (t )) ⋅ '

d (p dt t , z (

'

)

x )T(t ) = ( Df '' ) f - 1 ( x (t )) ⋅ ''

d (p dt t , z (

''

x )T(t ) .

)

(11)

The equations (10) and (11) are defined for all t ∈ Dom x(⋅), such that x(t )∈U ' ∩ U ' ' . Next, by taking into account the definition of the solution of an electrical network, one transforms (11) into

( Df ') f - 1( x (t )) ⋅ (1, pw( '

'

T ) ( x (t )))

= ( Df '' ) f - 1 ( x (t )) ⋅ (1, pw( ''

''

T )( x (t ))) .

(12)

By setting t = t o in (12), one obtains the equation

( Df ') f - 1( xo ) ⋅ (1, pw( '

')

( x o ))T = ( Df '') f - 1 ( xo ) ⋅(1, pw( ''

( x o ))T .

'' )

The proof that the equation (8) is satisfied for each point x ∈ and for all indices ' and '' , such that x ∈U ' ∩ U ' ' has been completed. Thus, the family (7) of maps, which is parameterized by the index ∈ , defines the vector field on

which is the C o - vector field. In result - 62 -

dx = dt

(x )

(13)

is the well-defined ordinary differential equation on whose solution curves fill . the entire space Let us prove now that the solutions of an electrical network which is considered in the Theorem 2 satisfy the differential equation (13). For x(⋅) being a solution of the network , let o ∈ be a value of the index such that Im x(⋅) ∩ U o ≠ ∅. Then the set Dom o x(⋅) = { t ∈ Dom x(⋅) : x(t )∈U o } is a nonempty and open subset of Dom x(⋅). By taking into account the definition of the solution of an electrical network and the C 1 - atlas which has been assumed for one obtains that

d T d x (t ) = ( Df o) f -o1(x (t )) ⋅ ( pt , z ( dt dt

o)

x )T(t ) = ( Df o) f -o1(x (t )) ⋅ (1, pw(

o)

,

( x(t )))T (14)

for all t ∈ Dom o x(⋅). This completes the proof. Next, let x(⋅) be a solution of the differential equation (13). The function x(⋅) is the solution of the network , iff it satisfies the equation

d (p dt qC ,

L

x)(t ) = ( piC , vL x)(t )

(15)

for all t ∈ Dom x(⋅). In order to verify that each solution of the differential equation (13) satisfies the equation (15), let us first prove that the following relation ( piC , vL ( x))T = ( D ( pqC ,

L

f )) f - 1( x ) ⋅ ( 1, pw( ) ( x) )T

(16)

is true for each ∈ and for all x ∈U . In order to prove this relation, let ∈ , and x o = (t o, y o) ∈U . Let x(⋅) be a solution of the network , such that x(t o) = x o. One obtains

d (p dt qC , = D( pqC ,

L

L

x)T( t o ) =

f ) f - 1( x o ) ⋅

d (p dt t , z (

d (p dt qC , )

L

f

pt , z (

x )T( t o ) = D( pqC ,

)

L

x )T( t o ) = f ) f - 1( x o) ⋅ ( 1, pw( ) ( x o) )T . (17)

By taking into account that

d (p dt qC ,

L

x)( t o) = piC , vL ( x o),

(18)

one verifies that the relation (16) is satisfied for any chosen ∈ and a point x o ∈U , and hence, for each ∈ and all x∈U . For x(⋅) as a solution of the differential equation (13), let o ∈ be a value of the index such that Im x(⋅) ∩ U o ≠ ∅. One obtains that - 63 -

d (p dt qC ,

L

x)T ( t ) = D( pqC ,

L

f o ) f -o1(x (t )) ⋅ (1, pw(

o)

( x(t )))T .

(19)

for all t ∈ Dom o x(⋅). By taking into account the relation (16), which is satisfied for each ∈ and all x∈U , one obtains with use of (19) that the solution x(⋅) verifies the equation (15), for all t ∈ Dom o x(⋅). Finally, because the equation (15) does not depend on the index , then it is satisfied by the solution x(⋅) of the equation (13) for all t ∈ Dom x(⋅). The proof that the solutions of the differential equation (13) are the solutions of the network being under considerations has been completed. The thesis of the Theorem 2 may be summarized in the following way: if the dynamic space of a dynamical network is an (1 ≤ l ≤ m + 1) - dimensional n +1

C 1- submanifold of the space of the network, and if for each point x ∈ one can define the local co-ordinates in an open neighbourhood of x in in such a way, that they are the time t variable, and (l - 1) charge and flux variables chosen from the set of all charge and flux variables associated, respectively, with the capacitive and the inductive branches of , then the solutions of the network are dx given by the solutions of a differential equation = (x ) defined on dynamic dt space of . The assumptions of the Theorem 2 are satisfied by most dynamical electrical networks which are of practical interest. However, there are the networks which are important in the applications, where relaxation oscillators are the examples, for which the analysis of the solutions in the area of the dynamic space is not sufficient [Szatkowski 2002, 2003]. Here the solutions are considered in the sense of the Definition 6. The d.d. systems which correspond to these networks are the singular systems, in the sense of the Definition 17 in part I. For these networks, their dynamic spaces are not the closed subsets of their ambient spaces. ⊆ for an electrical network which has been In the general case, considered in the Theorem 2. The following Theorems 3 and 4, concern the class of = , [Szatkowski 1982, 1988]. networks for which Theorem 3. 3.1. Assume that the configuration space of a dynamical electrical network n +1

is an (m + 1) - dimensional C 1- submanifold of the space

, and that there exists a indices, such that

C 1- atlas

{(U ,

f -1(⋅) = ( pt, qC , for each ∈ . Then

f -1(⋅))}

L

U



)(⋅) ,

,

on

of the network

being a set of

(20)

, and the solutions of the dx network are given by the solutions of a differential equation = (x ) defined dt on . In the local co-ordinates on , (⋅) : U ∋ x →

is the dynamic space of

( x ) = (D f ) f - 1( x ) ⋅ (1, piC , vL ( x ))T∈ Tx

- 64 -

,

(21)

for each ∈ . be an (1 ≤ l ' ≤ m + 1) - dimensional C 1- submanifold of the space

3.2. Let n +1

of the network

has a C 1- atlas {(U , f -1(⋅))} ∈ such

. Assume that

that f -1(⋅) = ( p t, z ( ) U )(⋅), for each ∈ , where: z ( ) = (qC , j ( ) ,..., qC , jr ( ) ( ) , L , k ( ) ,..., L , k ( ) ) , 1

u( )

1

1 ≤ j1( ) < j 2 ( ) < ... < jr ( )( ) ≤ mC , 1 ≤ k1( ) < k 2 ( ) < ... < ku ( )( ) ≤ mL , and r ( ) + u ( ) = l ' - 1 . Then is the dynamic space of if, and only if, the relation

( pw ( ) ( x))T = ( D( pz (

)

f )) f -1 ( x ) ⋅ ( 1, pw( ) ( x) )T

(22)

is satisfied for each ∈ , and for all x ∈ U , where: z ( ) is the complementary subsequence of the co-ordinates of ( qC , L ) with respect to z ( ), w( ) = ( iC , j ( ) ,..., iC , j ( ) , vL , k ( ) ,..., vL , k ( ) ) , and w ( ) is the complementary r( )

1

u( )

1

subsequence of the co-ordinates of (iC , vL ) with respect to w( ). 3.3. If is the submanifold satisfying the conditions given in 3.2, and the is the dynamic relation (22) is satisfied for each ∈ , and all x ∈ U , then space of the network , and the solutions of are given by the solutions of a dx . In the local co-ordinates on differential equation = (x ) defined on dt (⋅) : U ∋ x →

( x ) = (D f ) f - 1( x ) ⋅ ( 1, pw( ) ( x) )T∈Tx

,

,

(23)

for each ∈ . Proof. 3.1. At first, we shall prove that the family

U ∋ x → (D f ) f - 1( x ) ⋅ (1, piC , vL ( x ))T∈ Tx

(24)

of maps, which is parameterized by the index ∈ , defines a vector field on For the proof, it suffices to verify the relation

( Df ' ) f -' 1( x ) ⋅ (1, piC , vL ( x ))T = ( Df '' ) f -'1' ( x ) ⋅ (1, piC , vL ( x ))T ,

.

(25)

which should be satisfied for each point x∈ and for all indices ' and '', such that x ∈ U ' ∩ U ''. Let x o be a point contained in the configuration space of the network satisfying the assumptions of the Theorem 3.1. For each pair ' and '' such that x o ∈U ' ∩ U '', one obtains

( Df '' ) f -''1( x o) ⋅ (1, piC , vL ( x o))T = D( f = D( f

''

pt, qC ,

L

''

pt, qC ,

L

f ' ) f -'1( x o) ⋅ (1, piC , vL ( x o))T

)x o ⋅ ( Df ' ) f -'1(x o) ⋅ (1, piC , vL (x o ))T = ( Df ' ) f -'1(x o) ⋅ (1, piC , vL (x o))T , (26)

- 65 -

where it has been taken into account in (26) that D( f for each ∈

p t , qC ,

L

)x is the identity map,

and all x∈ U . Hence,

( Df ' ) f -1(x o) ⋅ (1, piC , vL (x o))T = ( Df '' ) f -1( x o) ⋅ (1, piC , vL ( x o))T . '

''

The proof that the relation (25) is satisfied for each point x∈ and for all indices ' and '' such that x ∈U ' ∩ U '' has been completed. Thus, the family (24) of maps, which is parameterized by the index ∈ , defines which is the C o - vector field. In result,

the vector field on

dx = dt

(x )

(27)

is the well-defined ordinary differential equation on whose solution curves fill the entire space . Let x(⋅) be a solution of the differential equation (27). We shall prove that x(⋅) is the solution of the network . Let o ∈ be a value of the index , such that Im x(⋅) ∩ U o ≠ ∅. Then Dom o x(⋅) = { t ∈ Dom x(⋅) : x(t )∈U o } is a nonempty and open subset of Dom x(⋅). One obtains that

d (p dt t , qC , = D( p t, qC ,

x )T ( t ) = (

L

,1,...,

T , m + 1 ) ( x (t ))

(28) L

f o ) f -o1( x (t )) ⋅ (1, piC , vL ( x (t )))T = (1, piC , vL ( x (t )))T

for all t ∈Dom o x(⋅), where it has been taken into account in (28), that D( p t, qC , L f ) f -1(x )(⋅) is the identity map, for each ∈ and for all x∈ U . It follows from (28) that the solution x(⋅) satisfies the conditions of the Definition 6, for all t ∈Dom o x(⋅). Next, because the equation

d (p dt t , qC ,

L

x )( t ) = (1, piC , vL ( x (t )))

which has been derived in (28) does not depend on the index , then the conditions which ensure that x(⋅) is the solution of the considered network are fulfilled for all t ∈ Dom x(⋅). By taking into account that the solution curves of the differential equation (27) fill the entire space , we have also proved that the configuration space of the considered network is the dynamic space of . Let us now prove that the solutions of an electrical network , which is considered in the Theorem 3.1 satisfy the differential equation (27). For x(⋅) being a solution of the network , let o ∈ be a value of the index , such that Im x(⋅) ∩ U o ≠ ∅. By taking into account the definition of the solution of an electrical network and the C 1- atlas which has been assumed for that - 66 -

, one obtains

d T d x (t ) = ( f dt dt

p t , qC ,

o

x )T ( t ) = ( D f o) f -o1(x (t )) ⋅

L

= ( D f o) f

-1 o

( x (t )) ⋅ ( 1,

piC , vL

d (p dt t, qC ,

L

x )T ( t ) (29)

( x (t )) )T

for all t ∈Dom o x(⋅). This completes the proof. 3.2 and 3.3. Let be an electrical network whose configuration space satisfies the assumptions of the Theorem 3.3. At first, we shall prove that the family U ∋ x → (D f ) f - 1( x ) ⋅ ( 1, pw( ) ( x) )T∈Tx

(30)

of maps, which is parameterized by the index ∈ , defines a vector field on For the proof, it suffices to verify the relation ( Df ' ) f - 1( x ) ⋅ ( 1, pw( ' ) ( x) )T = ( Df '' ) f - 1( x ) ⋅ ( 1, pw( '

''

T '' ) ( x ) ) ,

.

(31)

which should be satisfied for each point x∈ and for all indices ' and '', such that x ∈ U ' ∩ U '' . of the network Let x o be a point contained in the configuration space satisfying the assumptions of the Theorem 3.3. The relation (22), which has been assumed to be valid for each index ∈ and all x∈U assures that ( 1, pw(

'' ) ( x

o))T

= D( pt, z (

'' )

f ' ) f - 1(x o) ⋅ ( 1, pw( ' ) ( x o))T '

(32)

for all indices ' and '', such that x ∈ U ' ∩ U '' . With use of (32) one obtains that ( Df '' ) f - 1( x o) ⋅ ( 1, pw( ''

= D( f

''

pt, z (

'' ) ( x '' )

o))T

= ( Df '' ) f - 1( x o) ⋅ D( pt, z ( ''

'' )

f ' ) f - 1(x o) ⋅ ( 1, pw( ' ) ( x o))T '

f ' ) f - 1( x o) ⋅ ( 1, pw( ' ) ( x o))T = ( Df ' ) f - 1(x o) ⋅ ( 1, pw( ' ) ( x o))T , ' ' (33)

for all ' and '', such that x o ∈U ' ∩ U '' . From (33) it follows that ( Df ' ) f - 1(x o) ⋅ ( 1, pw( ' ) ( x o))T = ( Df ) - 1 o ⋅ ( 1, pw( ' '' f ( x ) ''

'' ) ( x

o))T.

(34)

The proof that the equation (31) is satisfied for each point x∈ and for all indices ' and '', such that x ∈ U ' ∩ U '' has been completed. Thus, the family (30) of maps, which is parameterized by the index ∈ , defines the vector field on

which is the C o - vector field. In result,

dx = dt

(x )

(35)

is the well-defined ordinary differential equation on . the entire space

- 67 -

whose solution curves fill

Let x(⋅) be a solution of the differential equation (35). We shall prove that x(⋅) is the solution of the network . Let o ∈ be a value of the index such that Im x(⋅) ∩ U o ≠ ∅. Note that the relation (22), which has been assumed to be valid for each ∈ assures that

D ( pqC ,

L

f o) f -o1( x ) ⋅ ( 1, pw(

o)

( x ))T = ( piC , vL ( x ))T

(36)

for all x ∈U o . One obtains that d (p x )T ( t ) = ( ,1,..., , m + 1 )T( x(t )) dt t , qC , L = D( p t , qC , L f o ) f -o1( x (t )) ⋅ ( 1, pw( o) ( x (t )))T,

(37)

for all t ∈Dom o x(⋅). From (36) and (37) it follows that

d (p dt t , qC ,

x )( t ) = (1, piC , vL ( x (t ))),

L

(38)

where t ∈Dom o x(⋅), which implies that the solution x(⋅) satisfies the conditions of the Definition 6, for t ∈Dom o x(⋅). Next, because the equation (38) does not depend on the index , then the conditions which ensure that x(⋅) is the solution of the considered network are fulfilled for all t ∈ Dom x(⋅). By taking into account that the solution curves of the differential equation (35) fill the entire space , we have also proved that the configuration space of the considered network is the dynamic space of . whose Let us now prove that the solutions of an electrical network configuration space fulfills the assumptions of the Theorem 3.3 satisfy the differential equation (35). For x(⋅) being a solution of the network , let o ∈ be a value of the index , such that Im x(⋅) ∩ U o ≠ ∅. By taking into account the definition of the solution of an electrical network and the C 1- atlas which has been assumed for that d T d x (t ) = ( f dt dt

p t, z (

o

o)

x )T ( t ) = ( D f o) f -o1(x (t )) ⋅

= ( D f o) f -o1( x (t )) ⋅ ( 1, pw(

o)

d (p dt t, z (

o)

, one obtains

x )T ( t ) (39)

( x (t )))T,

for all t ∈Dom o x(⋅). This completes the proof of the Theorem 3.3. The proof of the Theorem 3.2 is contained implicitly in the proof of the Theorem 3.3. Definition 7. The space of an electrical network is defined as the following (2(n + 1) − m − 1) − dimensional affine subspace of T = {( x, ) ∈T and

n +1



k + mC +1

2( n + 1)

:

1

= xm + 1 + m

= 1,

C+k

j +1



2( n + 1)

= x m + m +1 + j , for j = 1, 2,..., mC ,

, for k = 1, 2,..., mL }.

- 68 -

n +1

, (40)

The sub-bundle of T

n +1

Definition 8. The space

whose total space equals

is denoted by

of an electrical network

(n + 1) - dimensional affine subspace of T

n +1

= TK ∩



2( n + 1)

.

(41)

.

is the following ,

n +1

. The sub-bundle of T whose total space equals is denoted by be an electrical network which has been considered in the Theorem 2, and Let n +1 the sub-bundle of T , such that the space of the sub-bundle is the graph of the function (⋅) given by the family (6) of maps. Then, for an electrical network satisfying the assumptions of the Theorem 2,

=

∩T

.

(42)

=

The dynamical electrical networks, such that:

, and the configuration n +1

, space of the network is an (m + 1) - dimensional C 1- submanifold of see the Theorem 3.1, are considered most often in electrical networks theory. The following Theorem 4 concerns this class of networks (see e.g.: [Desoer, Wu 1972], [Matsumoto 1976], [Matsumoto et al. 1981], [Smale 1972], [Szatkowski 1982, 1983, 1988]). Theorem 4. Assume that the constitutive relation W of an electrical network n +1

, the submanifolds K is an (m + m + 1) - dimensional C 1- submanifold of W are transversal, and K ∩ W ≠ ∅ . Then the sub-bundle ∩ TW is a vector field on

and

(43)

(i.e., each of the fibres of the sub-bundle (43) is a vector) if,

and only if, the submanifold is locally C 1- parameterized by the variables t , qC , and L , which means that for each point x ∈ there is an open neighbourhood U of x in

, such that pt , qC ,

L

U (⋅)

is the C 1- embedding.

If the assumptions of the theorem are satisfied, and the sub-bundle ∩ TW is a vector field on then the solutions of the network are the solutions of an ordinary differential equation which is defined on the configuration space . In that considered case the infinitesimal generator for the solutions of the d.d.s. (of the network ) is equal to the sub-bundle ∩ T W , and the solution space of the d.d.s. (the dynamic space of the network ) is equal to the configuration space of . Proof. Let be an electrical network satisfying the assumptions of Theorem 4. Because the submanifolds K and W are transversal, and K ∩ W ≠ ∅, then the configuration space of the network is an (m + 1) - dimensional

C 1- submanifold of n + 1. Let the sub-bundle ∩ TW is locally

C 1- parameterized

be a vector field on

by the variables t , qC , and

- 69 -

. We shall prove that L.

Let x o be a point in

2m

and a C 1- submersion g (⋅) : ∋ x → is given by the constraint

, such that the submanifold

g ( x) = 0.

∩ TW =

v

=(

1,

qC

=(

m + 2 ,...,

2

,...,

mC + 1 ) ,

m + m +1 ),

L

and ≅

For ( x, ) in the open subset T constraints for the subset

and W

∩ TK ∩ T W =

∩T

i

=(

=(

n +1

.

(45)

be partitioned, and next labeled

mC + 2 ,...,

m+m+2

n +1

×

of T

restricted to

are transversal, one

∩T

n +1

Let the co-ordinates of a tangent vector ∈ according to: =

n +1

(44)

By taking into account that the submanifolds K obtains that

t

of x o in

. There exist an open neighbourhood

of T

,..., n +1

m +1 ), n +1 ) .



2( n + 1)

restricted to T

(46) , the

are given by:

g ( x) = 0, ( Dg ) x ⋅ T = 0, t = 1, qC L

(47)

= iC ,

= vL .

For each fixed x ∈ , the constraints (47) yield the following system of n +1 linear equations with respect to ∈ Tx

n +1



,

( Dg ) x 1 0 0 0 0 0 1 0 0 0



0 0 1 0 0

t T qC T L

T v T i

0 =

1 iCT vLT

.

(48)

Because it has been assumed that ∩ T W is a vector field on , then the system (48) of the equations possesses unique solution with respect to , for each x∈ ∩ . This takes place iff det ( D (v, i ) g ) x ≠ 0, for all x ∈ ∩ , which that is implies that there is such an open neighbourhood of x o in 1 C - parameterized by the variables t , qC , and L . Next, the Theorem 3.1 assures that the dynamic space of the considered network is equal to the configuration space of , and the solutions of . By taking into are the solutions of an ordinary differential equation defined on account the expression (21), one also verifies that the sub-bundle - 70 -

is the infinitesimal generator for the solutions of the ∩ TW = ∩T network being under considerations. Now assume that the submanifold is locally C 1- parameterized by t , qC , and L. Because the submanifolds K and W are transversal, then

∩ TW =

. Let x o = ( t o, qCo ,

∩T

o o o L,v , i )

assumption, there exists an open neighbourhood U of

pt , qC ,

L

U (⋅)

be a point in

xo

in

is the C 1- embedding. Define f (⋅) = ( pt , qC , = ( t,

Let us use the labeling given in (46). A vector

∩T

contained in the fibre (

t

qC ,

, such that -1 U ) (⋅) . L

,

v, i )

is

)x o , iff:

= ( Df ) f -1(x o) ⋅ ( t , = 1,

L

. By

qC

qC ,

= iCo , and

From (49) it follows that the fibre ( ( Df ) f -1(x o) ⋅ (1, iCo , vLo )T.

L L

∩T

)T ,

= vLo .

(49)

)x o is the vector

Theorem 4 explains why the capacitor charges, inductor fluxes and time are the basic quantities in describing dynamics of electrical networks. For the class of dynamical networks which are the subject of the Theorem 4, and are considered most , and the constraints for the space often in the applications, the constraints for T define a vector field on the configuration space of the network , if, and only if the submanifold is locally C 1- parameterized by the variables t , qC , and In that case the sub-bundle ∩ T W is the infinitesimal generator for the solutions of the network . Under assumptions of the Theorem 4, the sub-bundle = . ∩ T W is the continuous vector field on L.

3. HIGHER INDEX NETWORKS Let us consider the general case, when the dynamic space of an electrical network is possibly a proper subset of the configuration space of . That is, the d.d.s. which corresponds to the network is possibly incomplete. be a dynamical electrical network. If Let t.p.B

= l.v.c.

,

(50)

then = t.p.B

.

(51)

If the relation (50) is satisfied for the constitutive sub-bundle of the d.d.s. which corresponds to an electrical network , and is not a tangent subbundle, then the d.d.s. is relatively well-posed. In that case, one can use the projection-prolongation iterations (19), in the part one, in order to extract the and the solution space of the network - of the infinitesimal generator

- 71 -

d.d.s. . In the applications, usually there are considered the regular networks (a dynamical electrical network is called the regular network, if the corresponding d.d.s. is regular – see the Definition 6, part one). The dynamical networks, such that st = 0 for an electrical network are a = 1. However, there are the very special class of networks. For most networks st networks which are important in the applications, such that the geometric index of is defined the network equals two. The geometric index of an electrical network as equal to the structural index st of its constitutive bundle . The projection-prolongation iterations for extracting the infinitesimal generator and the dynamic space of an electrical network are illustrated in Figure 1. It has been assumed that g = 2 for the network , and that the d.d.s. which corresponds to

is relatively well-posed.

- 72 -

Fig. 1. The sub-bundle solutions of the network

∩ T B(

, (1) )

, and B (

is the infinitesimal generator

∩T

) is the dynamic space

for the of

.

For an electrical network containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current sources one usually obtains the dynamic space of the network as a proper subset of the configuration space of .

- 73 -

Fig. 2. (a) Electrical network containing a loop consisting of capacitors and independent voltage sources. (b) Network containing a cut set consisting of inductors and independent current sources. In the following example, we use the projection-prolongation iterations (19) in the part one in order to extract the infinitesimal generator and the dynamic space of an electrical network containing capacitive-only loop. The d.d.s. which corresponds to the network is essentially inconsistent (see the Definition 8, part one). Example 1. Consider an autonomous electrical network shown in Figure 3. It consists of a capacitive loop and a resistor connected in parallel. The space of the is defined to be the space 8 of all points network

x = (qC , qC , vC , vC , vR , iC , iC , iR ) . 1

2

1

2

1

2

(52)

Because the network is autonomous, the variable t has not been taken as one of the co-ordinates of the space of the network.

Fig. 3. Electrical network containing the capacitive loop. Let the constitutive relation W

of the network

be given by the constraints

vC = g C (qC ), vC = g C (qC ), and iR = gR ( vR ) , 1

1

1

2

2

2

(53)

where g C (⋅), g C (⋅), and gR (⋅) are the C 2 - functions defined on . Thus, W 1

2

is the

5-dimensional C 2 - submanifold having the following C 2 - parametric representation 5

∋ (qC , qC , vR , iC , iC ) → x = (qC , qC , g C (qC ), g C (qC ), vR , iC , iC , gR ( vR ) ) . 1

2

1

2

1

2

1

1

2

2

2

1

We assume in the following that g C (⋅) is the C - diffeomorphism of 2

and

- 74 -

2

,

D( g C- 1 g C1 ) qC 1 ≠ −1 ,

(54)

2

for all qC ∈ . 1

The Kirchhoff equations:

vC = vC = vR ,

(55)

iC + iC + iR = 0

(56)

1

and

2

1

define the Kirchhoff space K

2

. The Kirchhoff space K

of the network

is the

8

5-dimensional linear subspace of the space of the network . By taking into account (53), (55), and (56), it is easily verified that the submanifolds K and W are transversal, and the configuration space 8

network has the structure of a 2-dimensional submanifold of following C 2 - parameterization 2

of the

having the

∋ (qC1, iC1 ) → x = ( qC1, ( g C- 1 g C1 )(qC1), g C1(qC1 ), g C1(qC1 ), g C1(qC1 ), iC1,

(57)

2

- ( gR g C1 )(qC1 ) - iC1 , ( gR g C1 )(qC1 ) ) .

The space of the constitutive sub-bundle network ) is given by

=(

×

8

)∩

,

where the constraints which define the space 1

1

= iC , and 1

2

1

2

(of the

(58)

are given by the equations:

= iC , and

)= , and the fibre Note that B( given by the constraints:

of the d.d.s.

,x

= iC . 2

above the point x ∈

of

= -( gR g C1)( qC1) - iC1 .

is

(59)

The sub-bundle is not a tangent sub-bundle of T 8. Because the infinitesimal generator for the solutions of the network is a tangent sub-bundle of the tangent of the constitutive bundle , let us first use the part to its base space t.p.B projection-prolongation iterations (19), part one, in order to extract the sub-bundle t.p.B . At the step number one of the projection-prolongation process, one obtains , (1)

=

∩ TB(

)=

The constraints which define the subspace

- 75 -

∩T , (1)

= of T

∩T 8



. 16

(60)

are given by:

vC − g C (qC ) = 0 ,

constraints for

,

x = (qC , qC , vC , vC , vR , iC , iC , iR ) ∈ 1

2

1

2

1

1

1

1

2

2

2

vC − g C (qC ) = 0 , iR − gR ( vR ) = 0 ,

8

,

2

(61)

vC − vC = 0 , 1

2

vC − vR = 0 , 2

iC + iC + iR = 0 , 1

2

and 3 - ( D g C1 )qC 1 ⋅ 1

constraints for Tx , x = (qC , qC , vC , vC , vR , iC , iC , iR ) ∈ 1

2

1

= ( 1,

2

2

,...,

8

1

8

)∈

2

= 0,

4 - ( D g C 2 )qC 2⋅ 2

= 0, 8 - ( D g R )vR ⋅ 5 = 0 , 3- 4= 0, 4- 5= 0, 6+ 7 + 8= 0,

,

,

(62)

and

x∈

8

,



iC −

,

constraints for

1

iC −

8

,

2

1

= 0,

= 0. 2

(63)

Note that by the assumption (54) being satisfied, the relations (61)-(63) yield the , following additional constraint for the co-ordinates of the points in iC = − 1

Thus, B(

, (1))

( gR g C )( qC ) 1

1+

D( g C- 1 2

is a proper subset of

of the 1-dimensional C 1- submanifold of C 1- parameterization,

- 76 -

1

g C 1 ) qC 1

.

(64)

. The subset B( 8

, (1))

has the structure

which has the following

qC = qC , 1

qC = 2

( g C- 1 2

1

g C )(qC ) , 1

vC = g C (qC ), 1

1

1

1

1

1

vC = g C (qC ), 2 1 1 vR = g C (qC ), x = F (qC ) : ( qC ∈ 1

1

iC = −

)

1

iC = − 2

( gR g C )(qC ) 1

1+

D( g C- 1 2

g C )qC 1

D( g C- 1 g C )qC 1 1 2 1 + D( g C- 1 g C )qC 1

2

(65)

,

1

1

⋅ ( gR g C1 )( qC1 ) , 1

iR = ( gR g C1 )( qC1 ) . One obtains the following expression for the subset , (1)

= x ∈B(

( x, E x + a ( x )) ,

, (1)

of T

8

16



,

(66)

, (1) )

where a( x) ∈Tx , and E x is the vector subspace of Tx given by the constraints = 2 = 3 = 4 = 5 = 8 = 0, and 7 = − 6 (see Figure 4). The co-ordinates of the 1 vector a(x) are given by:

a1( x) = iC1 , a 2( x) = iC 2 , a3( x) = −( D g C1 )qC 1⋅ iC1, a4 ( x) = − (D g C 2 )qC 2⋅ iC 2 ,

a5( x) = − (D g C 2 )qC 2⋅ iC 2 , a6 ( x) = 0, a 7( x) = − ( D g R )vR ⋅ (D g C 2 )qC 2⋅ iC 2 , and a8( x) = ( D gR )vR ⋅ (D g C 2 )qC ⋅ iC . 2

Thus,

, (1)

2

is the C 1- affine sub-bundle of T

point x∈ B( The sub-bundle

, (1))

. The fibre of

, (1)

at the

is a 1-dimensional affine subspace of the tangent space Tx , (1)

is not a tangent sub-bundle.

- 77 -

.

Fig. 4. For the considered network

C 1- submanifold, and

get

, (1)

, B(

, (1))

is the 1-dimensional

is the affine sub-bundle of T

over B(

, (1)) .

Let us continue the projection-prolongation iterations at the step number two. We , ( 2)

=

∩ TB(

, (1))

=

∩ T B(

The constraints which define the subspace

T

x = F ( qC 1 ) , = (DF ) qC 1⋅ 1 = iC 1 , 2 = iC 2 .

, ( 2)

1,

, (1)) . 8

of T

16



(67) are given by:

(68)

The constraints (68) define a tangent C o - vector field

(⋅) on B(

, (1))

which is

given explicitly by (⋅) : B( Thus,

, ( 2)

, (1))

∋ x → − (DF ) qC 1 ⋅

( gR g C )(qC ) 1

1

1 + D( g C- 1 g C )qC 1 1 2

is the proper sub-bundle of the sub-bundle

step number one of the projection-prolongation iterations, and The tangent sub-bundle

, ( 2)

, (1)

.

(69)

obtained at the , ( 2)

= t.p.B

.

which has been obtained at the step number two

of the projection-prolongation iterations is l.v.c. Hence, = l.v.c. which assures that , ( 2) = t.p.B , ( 2) is the infinitesimal generator for the solutions of the considered network B( . , (1)) is the dynamic space of the network

. The submanifold

We have proved, that the ∂ B - structural index st of the constitutive bundle of the network equals two. The d.d.s. which corresponds to the network is relatively well-posed, and its geometric index g equals two. The solutions of the network

are the solutions of the differential equation

- 78 -

dx = dt defined on the submanifold B(

, (1))

(70)

( x)

being the dynamic space

of

.

Let us consider a dynamical electrical network whose constitutive relation n +1 W is a subset of given by the constraint equation ( t , qC ,

= 0,

L , v, i )

(71)

n +1

where into m , m being the number of branches of the (⋅) is a map of network. The Kirchhoff space K of the network is an (m + m + 1) - dimensional n +1 linear subspace of the space of the network. For being any tree for , and the associated co-tree , let B and D be the fundamental loop and cut set matrices associated with and , respectively. With use of the matrices B and D , the constraints for K are given by the equations

B ⋅ vT = 0 , and D ⋅ i T = 0 .

Let g (⋅) be the following map of g ( t , qC ,

L , v, i )

=(

n +1

( t , qC ,

into

(72)

2m

L , v, i ) ,

,

v ⋅ B T, i ⋅DT ) .

(73)

The equation g ( t , qC ,

L , v, i )

=0

defines the constraint for the configuration space

(74) of the network =(

space of the constitutive sub-bundle of the d.d.s. which corresponds to the network is given by = {( x, ) = ( t , qC ,

L , v,

g ( x) = 0,

i, t

t , qC ,

= 1,

qC

L

,

n +1

v , i ) ∈T

= iC , and

L



= vL },

. The total n +1

,

2( n + 1)

,

:

)

(75)

where there has been used the notation (46) for the co-ordinates of a vector . The are the solutions of the d.a.e. solutions of the network dqC = iC , dt d L v = L, dt g ( t , qC , L , v, i ) = 0 .

(76)

The following two cases are analysed. The case 1 concerns the class of dynamical electrical networks which are considered most often in the applications. Case 1. Let (⋅) be a C 1- map, and ker g (⋅) ≠ ∅, and let the Jacobi matrix g I(v, i )( t , qC , L , v, i ) be nonsingular for all ( t , qC , L , v, i ) ∈ ker g (⋅) . Then the - 79 -

submanifolds K

and W

are transversal, and the configuration space

of the

n +1

network is an (m + 1) - dimensional C 1- submanifold of which is locally C 1- parameterized by the variables t , qC , and L . In view of the Theorem 3.1, the configuration space of the network is the dynamic space of , and the solutions of are the solutions of an ordinary differential equation defined on . The total space , (1) of the constitutive sub-bundle , (1) of the d.d.s. , (1)

obtained at the step number one of the projection-prolongation reduction

process for extracting the consistent part , (1)

,c

of the d.d.s.

= {( x, ) = ( t , qC , L , v, i, t , qC , , v , i ) ∈T L g ( x) = 0, t = 1, qC = iC , = vL , and T I qc ( x ) ⋅ i C +

g I t ( x) + g

g I

( x) ⋅ vLT + L

L

g Iv ( x ) ⋅

T v

is given by, n +1



2( n + 1)

: (77)

+ g I i ( x) ⋅

T i

= 0}.

It follows from (77), by taking into account nonsingularity of the Jacobi matrix g I( v, i )( x), for all x∈ , that , (1) is the sub-bundle of vectors which are tangent to

. For a point x∈

, the fibre (

, (1) )x

is the tangent vector to

at x whose co-ordinates are given by, ( t , qC , , v , i )( x ) L = (1, piC ( x), p vL( x) , - ( g - 1 (v, i ) ⋅ ( g I t + g I( qc , I

L)

piTC , vL ))T( x )) .

(78)

In the expression (78), g I( qc , L ) is considered as the linear map defined at x. Thus, the geometric index g of the d.d.s. (of the network ) equals one. The d.d.s.

, (1)

. The trajectories of the network

is the consistent part of

. The vector field under , (1) . Case 2. For electrical networks containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current sources, the dynamic space of the network is usually a proper subset of its configuration space. In view of the Theorem 3.1, if the configuration space of a 1 network is a C - submanifold, and the dynamic space of is a proper subset of , then the configuration space of the network is not locally C 1- parameterizable by the variables t , qC , and L . Thus, for electrical networks containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current sources, the configuration space of the network is usually not a differentiable submanifold which is locally C 1- parameterizable by the variables t , qC , and L . Let be a dynamical electrical network, such that the Jacobi matrix g I(v, i )( t , qC , L , v, i ) is singular for all ( t , qC , L , v, i ) ∈ ker g (⋅) . are the trajectories of the vector field defined on consideration corresponds to the sub-bundle

However, let there exist the subsequences: qC of the sequence

qC = (qC 1 ,..., qC m ), C

L

of the sequence

L

=(

L 1 ,...,

L mL

) , v L of

vL = ( vL 1 ,..., vLm ), vR of vR = ( vR 1 ,..., vR m ), i C of iC = ( iC 1 ,..., iC m ), and i R L

R

- 80 -

C

of iR = ( iR ,..., iR ), such that the length of the sequence 1

x = ( qC ,

mR

L , vC , vL , vR , i C , iL, i R

) equals 2m, and

det g Ix ( x) ≠ 0 ,

(79)

for all x ∈ ker g (⋅) . In the following, we assume that ker g (⋅) ≠ ∅ . Thus,

≠ ∅. Let x = ( t , qC , L , vL , vR , i C , iR ) be the complementary subsequence of the co-

ordinates of x with respect to x. By the assumption (79), the configuration space n +1

is an (m + 1) - dimensional C 1- submanifold of

of the network

which is

C 1- parameterized

locally The total space , (1)

by the variable x . , (1) of the constitutive sub-bundle

, (1)

of the d.d.s.

is given by,

, (1) = {( x,

) = ( t, qC ,

g (x) = 0,

L, v, i, t, qC ,

n +1

, v, i ) ∈T



2(n + 1)

:

qC = iC , L = vL , and g It (x) + g + g Iq (x)⋅ iCT + g I (x) ⋅vLT + g I (x) ⋅vLT C L L T +g T +g T +g T +g T ( x ) ⋅ ( x ) ⋅ ( x ) ⋅ ( x ) ⋅ vC vL vR vL IvC IvL IvL IvR IvR (x) ⋅ vR T T T T T IiC (x)⋅ iC + g IiC (x)⋅ iC + g IiL (x)⋅ iL + g IiR (x)⋅ iR + g IiR (x)⋅ iR = 0 } . t

= 1,

L

T I qC (x)⋅iC

+g +g If

, (1)

(80)

is a tangent sub-bundle, then the projection-prolongation iterations

terminate at the step number one. In the other case, one should continue the extraction process at the step number two. By the assumption (79) being satisfied, note that ( x, ) ∈ , (1) iff: g ( x ) = 0, t

= 1,

qC

= iC ,

L

= vL , and

iCT vLT

T vC T vL T vR T iC T iL T iR

= − [ g I( qC , L ) g I ( vC , v L , vR , iC , iL , iR ) ]I x ⋅ ( g I t ( x) + g I( qC , L )( x) ⋅ ( i C , vL )T + g I( vL , vR , iC , iR ) ( x) ⋅ ( vL , vR , i C , iR )T ) . -1

, we partition the matrix [ g I( qC , denoted by E (x), according to For x∈

E ( x) =

E1( x) E 2( x )

- 81 -

(81)

g I ( vC , v L , vR , iC , iL , iR ) ]I x , L) -1

}r }u

,

where r denotes the length of the sequence ( iC , v L ), and u is the length of the sequence ( vC , vL , vR , iC , iL , iR ) . Let u be the length of the sequence ( vL , vR , i C , iR ) . It is easily observed that the base space of the sub-bundle

, (1)

is

given by,

B( (

, (1) )

n +1

= { x∈

vL , vR , i C , iR ) ∈

+ g I( qC ,

: g ( x) = 0, and there exists a vector

u

, such that p i C , v L ( x) = - ( E1( x) ⋅ ( g I t ( x) ( x)⋅( i C , vL )T + g I( vL , vR , iC , iR ) ( x) ⋅ ( vL , vR , i C , iR )T ) )T }. L)

Next, the total space

of the sub-bundle

, ( 2)

, ( 2)

(82)

which one obtains at the

step number two of the projection-prolongation algorithm for extraction of the subis defined by, bundle t.p.B

If

, ( 2)

= {( x, ) ∈TB(

, ( 2)

is a tangent sub-bundle, then the projection-prolongation iterations

, (1)) : t

= 1,

= piC ( x) , and

qC

L

= pvL( x) }.

terminate at the step number two. In that case, the geometric index

g

(83)

of the d.d.s.

(of the network ) equals two. Let us now use the prolongation process in order to extract the dynamic space and the infinitesimal generator for the solutions of an electrical network which is possibly essentially inconsistent. There is considered a dynamical electrical network whose constitutive relation W is a subset given by the constraint equation ( t , qC ,

= 0,

L , v, i )

(84)

n +1 n +1 a in the space of , into which is sufficiently (⋅) being a map of differentiable. The solutions of the network are the solutions of the following differential algebraic equation denoted by ,

q'C - iC = 0,

'

L - vL

( t , qC ,

B ⋅ vT = 0, Where y = ( qC , given by,

f ( t , qC ,

L, v, i ),

L, v, i, q'C ,

'

L , v, i ) D ⋅ iT

let f (⋅) :

L , v ', i'

= 0, t' - 1 = 0, = 0,

2 n +1

(85)

= 0.

∋ ( t , y, y' ) →

) = ( q'C - iC , L' - vL ,

m +m +a

be the map

L , v, i ) , v ⋅ B

( t , qC ,

T

, i ⋅D T ) . (86)

The map F (⋅) :

F ( t , qC ,

2 ( n + 1)

L, v, i, t', q'C ,

∋ ( x, x' ) → '

L , v ', i '

m +m + a +1

) = ( f ( t , qC ,

, where x = (t , y ), is defined by, L, v, i, q'C ,

'

L , v ', i '

) , t' - 1 ) .

With use of the maps f (⋅) and F (⋅), respectively, one can write the equation as - 82 -

(87)

f ( t , y, y' ) = 0, t' - 1 = 0 ,

(88)

and as

F ( x, x' ) = 0,

(89)

respectively. Note also that the total space of the constitutive sub-bundle of the network equals ker F (⋅). In the following, there are used the notations of Section 6, part one. The (k ) prolongation of the implicit differential equation to T ×T k +1 n is given by

f (uo) ( Lw1 f )(u1) ( Lw2( Lw1 f ))(u 2)

=0,

(90)

( Lwk ( Lw k - 1 (...Lw1 f )...))(uk )

t' - 1 where: ul = ( t , qC ,

L, v, i, q'C ,

(l + 1), ' L , v ', i',..., qC

l = 0, 1, 2,..., k , and wr = ( 1, q'C ,

(l + 1), v (l + 1), i (l + 1) ) , for L ( r + 1), ( r + 1), v ( r + 1), i ( r + 1) ) , for ' L , v ', i',..., qC L n k + 1 ( k + 2) n + 2 T ×T ≅ on which the prolonged

r = 1, 2,..., k . A point in the space equation is defined is denoted by zk , zk = ( t , t', qC , L, v, i, q'C , L' , v ', i' ,..., qC( k + 1),

( k + 1), v ( k + 1), i ( k + 1) ) . L

The equation

(90), considered as algebraic equation, is the k - th derivative array equation associated with the constitutive implicit differential equation (88) of the network The map f , k (⋅) is defined by

f , k (⋅) : ( k + 2) n + 1 ∋ uk → f , k (uk ) = ( f (uo), ( Lw1 f )(u1), ( Lw2( Lw1 f ))(u 2),..., ( Lwk ( Lw k - 1 (...Lw1 f )...))(uk )) ∈

.

( k +1) j

, (91)

where k ∈ N , and j = m + m + a . Set wo = ( 1, q'C , L' , v ', i' ). One obtains the following expression for Lwl ( L wl - 1 (...Lw1 f )...)(ul ) ,

Lwl ( L wl - 1 (...Lw1 f )...)(u l) = ( q ( l + 1) - iC(l ), C

( L w l - 1 ( L w l - 2 (...L wo

)...))(u l - 1 ) , (k )

The total space of the sub-bundle the prolonged equation (k )

= { zk ∈

(k )

( l + 1) - v (l ), L L v (l ) ⋅ B T , i (l ) ⋅ D T

(92)

).

being the constitutive bundle

is given by,

( k + 2) n + 2

: f

, k (uk )

- 83 -

= 0, and t' - 1 = 0 }.

(93)

(k )

of

(k )

The sub-bundle

(k )

is the projection of the sub-bundle

n 1

bundle T + , where T 6 in the part one. The geometric index

n +1

g



2 ( n + 1)

on the tangent

is the space of all points ( x, x' ), see Section

of a dynamical electrical network has been defined as

equal to the structural index of the constitutive bundle of the network. There is also defined the proper differentiation index d for an electrical network whose constitutive relation is given by a constraint equation. The index d of a dynamical electrical network equals the proper differentiation index of the constitutive implicit differential equation of - see also [Differential-algebraic… 2008] and [Schwarz, Tischendorf 2000]. However, the index d of an electrical network depends on the description, given by the constraint equation (84), of the constitutive relation W of the network. Thus, it is not defined uniquely for . Assume that the map g (⋅),

g ( t , qC ,

L , v, i )

=(

( t , qC ,

L , v, i ) ,

v ⋅ B T , i ⋅ D T ) , is a C 1- submersion on its

kernel. Then the total space of the sub-bundle

, (1)

which one obtains at the first

step of the projection-prolongation algorithm is given by n +1 ≅ , (1) = { ( x, x' ) ∈ T B ⋅ v' T = 0 , and

2 ( n +1)

: F ( x, x' ) = 0 , ( D D ⋅ i' T = 0 }.

It is easily verifiable that, in that case, the sub-bundle (1)

the sub-bundle tangent bundle T

(1)

)x ( x' ) = 0,

(94)

being the projection of

obtained at the first step of the prolongation algorithm on the

n +1

equals the sub-bundle

, (1) .

Let us now use the Theorem 5 in Section 6 in the part one, in order to give the conditions under which the prolongation process enables extraction of the dynamic space and the infinitesimal generator for the solutions of an electrical network whose dynamic behaviour is given by the trajectories of the d.a.e. (85). Let be a dynamical electrical network whose dynamics is given by the solutions of the equation (85). Theorem 5. Assume that the geometric index g of the network is a finite number, and

(⋅) is a C g - map, for

If the map g (⋅), for g

g

g

≥ 1.

≥ 1, and the maps f

,1(⋅),

f

,2(⋅),..., and

≥ 2, are the submersions on their kernels, the sub-bundle t.p.B

vector continuous, and the solutions of the network

are

g

f

,

g

- 1(⋅) ,

for

is locally

+ 1 times

differentiable, then ( g)

=

.

(95)

The proper differentiation index d is well defined for the constitutive differential algebraic equation (85) of the network , and the indices g and d of satisfy the inequality

- 84 -

d

For

g

= 1,

d



g.

(96)

= 1.

Proof. We use the Theorem 5 in the part one. The implicit differential equation (87), part one, is now considered to be the differential algebraic equation (85) – the equation (88). By taking into account that the map f (⋅) is g times differentiable iff the map

(⋅) is

g

times differentiable, one notices that the constitutive

differential algebraic equation of the network being under considerations fulfills the assumptions of the Theorem 5 in the part one. The proof is complete.

4. CONCLUSIONS Dynamical electrical networks have been discussed from the point of view of the theory of differential dynamical systems in its geometric formulation. Electrical networks are the interesting and important examples of general differential dynamical systems which are possibly essentially inconsistent. The theorems have been proved which characterize the solution spaces (the dynamic spaces), the infinitesimal generators, and the algebraic independent state variables of dynamical electrical networks. For electrical networks which are considered most often, the dynamic space of the network equals its entire configuration space. But for example for networks containing loops consisting of capacitors and independent voltage sources and/or cut sets consisting of inductors and independent current sources one usually identifies the dynamic space of the network as a proper subset of its configuration space. For these networks, the structural index of the constitutive bundle of the network is usually equal two. For higher index networks it was necessary to employ the reduction algorithms in order to extract their infinitesimal generators and dynamic spaces. Both, the projection-prolongation and the prolongation processes have been used to reduce the differential dynamical system corresponding to the higher index network to its consistent part.

REFERENCES Beitrag zur theorie der algebrodifferentialgleichungen. Reich S. T.U. Dresden, 1990 (Ph.D. diss.). Brayton R.K., Moser J.K. A theory of nonlinear RLC networks. "Qt. Appl. Math." 1964, no 1 and no 2. Desoer C.A., Wu F.F. Trajectories of nonlinear RLC networks: a geometric approach. "IEEE Trans. Circuit Theory" 1972, no 6.

Differential-algebraic systems: analytical aspects and circuit applications. Riaza R. World Scientific, NJ and London 2008. Graph theory and its engineering applications. Chen W.-K. Word Scientific, London 1997. Haggman B., Bryant P.R. Solutions of singular constrained differential equations: a generalization of circuits containing capacitor-only loops and inductor-only cutsets. "IEEE Trans. Circuits Syst." 1984, no 12. - 85 -

Matsumoto T. On several geometric aspects of nonlinear networks. "J. Franklin Inst." 1976, no 1-2. Matsumoto T., Chua L. O., Kawakami H., and Ichiraku S. Geometric properties of dynamic nonlinear networks: transversality, local-solvability and eventual passivity. "IEEE Trans. Circuits Syst." 1981, no 5. Reich S. On a geometrical interpretation of differential-algebraic equations. "Circ. Syst. Sign. Proc." 1990, no 4. Reich S. On an existence and uniqueness theory for nonlinear differential algebraic equations. "Circ. Syst. Sign. Proc." 1991, no 3. Schwarz D.E., Tischendorf C. Structural analysis of electric circuits and consequences for MNA. "Int. J. Circ. Theor. Appl." 2000, no 2. Smale S. On the mathematical foundations of electrical circuit theory. "J. Diff. Geom." 1972, no 1-2. Szatkowski A. On the dynamic spaces and on the equations of motion of non-linear RLC networks. "Int. J. Cir. Theor. Appl." 1982, no 2. Szatkowski A. On the dynamics of non-linear RLC networks from a geometric point of view. "Int. J. Cir. Theor. Appl." 1983, no 2. Szatkowski A. Geometric methods in the description of non-linear RLC circuits: Dynamic equations and dynamic spaces of RLC systems. "Electrotechnical Dissertations", ("Rozprawy Elektrotechniczne") 1988, no 2. Szatkowski A. A general dynamical system model arising from electrical networks. European Conference on Circuit Theory and Design, Brighton 1989. Szatkowski A. On generalised dynamical systems – differentiable dynamic complexes and differential dynamic systems. "Int. J. Sys. Sci." 1990, no 8. Szatkowski A. Geometric methods in dynamical systems modelling: electrical, mechanical and control systems. "Sys. Anal. Model. Simul." 2002, no. 11. Szatkowski A. Non-regular, incomplete and singular dynamical systems: impasse points in dynamical systems. "Sys. Anal. Model. Simul." 2003, no 8.

- 86 -

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