Boundary controllability of the one-dimensional wave equation with rapidly oscillating density. C. Castro
Abstract.
We consider the one-dimensional wave equation with periodic density of period ! 0 in a bounded interval. By a counterexample due to Avellaneda, Bardos and Rauch we know that the exact controllability property does not hold uniformly as ! 0 when the control acts on one of the extremes of the interval. The reason is that the eigenfunctions with wavelength of the order of may have a singular behavior so that their total energy cannot be uniformly estimated by the energy observed on one of the extremes of the interval. We give partial controllability results for the projection of the solutions over the subspaces generated by the eigenfunctions with wavelength larger and shorter than . Both results are sharp. We use recent results on the asymptotic behavior of the spectrum with respect to the oscillation parameter , the theory of non-harmonic Fourier series and the Hilbert Uniqueness Method (HUM).
1 Introduction In this work we are interested in the boundary controllability of the one-dimensional wave equation with a very rapidly oscillating density. Wave propagation in highly heterogeneous media is a very complex issue. The problem under consideration is the simplest one in this context. Some recent results show that singular phenomena can appear when the wavelength of the solutions is of the order of the size of the microstructure. More precisely, as it was proved in [2], there exist stationary solutions which concentrate most of its energy in one part of the boundary (see also [1]). These solutions contain frequencies of the order of the inverse of the oscillation parameter. This type of phenomena constitutes an obstacle to the uniform boundary controllability when the microstructure becomes ner and ner. Later on, in [5] and [6], a rather complete description of the spectrum with respect to the oscillation parameter was given. Here we use the results in [5] and [6] to prove sharp partial controllability results. Let 2 L1 (IR) be a periodic function such that 0 < m (x) M < 1 a. e. x 2 IR. Given > 0, we set (x) = (x=) and consider the one-dimensional wave equation 8 > 0 < x < 1; 0 < t < T; < (x)utt ; uxx = 0; u(0; t) = 0; u(1; t) = f (t); 0 < t < T; (1) > : u(x; 0) = u (x); ut(x; 0) = u (x); 0 < x < 1: 0
1
Departamento de Matematica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain, (
[email protected]).
2 In (1) f 2 L (0; T ) is a boundary control function acting on the extreme x = 1 of the string and (u ; u ) the initial data. The function oscillates more and more as ! 0 and then (1) constitutes a good model for the wave propagation in a medium with very heterogeneous density. The following result states that system (1) is well-posed: Theorem 1 Assume that 2 W ;1(0; 1). System (1) is well-posed in the energy space L (0; 1) H ; (0; 1) in the sense that, given T > 0, if 2
0
1
1
2
1
(u ; u ) 2 L (0; 1) H ; (0; 1) and f 2 L (0; T ) 0
2
1
1
(2)
2
the solution u of (1) veri es
u(x; t) 2 C ([0; T ]; L (0; 1)) \ C ([0; T ]; H ; (0; 1)): 2
1
(3)
1
Furthermore, we can estimate the norm of the solution as follows
k(u ; ut)kL1
1=2 ;T ;L2(0;1)H ;1(0;1)) C k kW 1;1 (0;1) kf kL2 (0;T ) + k(u0; u1 )kL2 H ;1
(0
(4)
where C > 0 is a constant which only depends on m and M .
Remark 1 Estimate (4) is not uniform in due to the fact that kkW ;1 This leads to a bound for the solution which blows up as ! 0. 1
;
(0 1)
; . 1
Inequality (4) can be obtained using transposition and a suitable boundary regularity inequality for the adjoint uncontrolled system (see [10] and [11]). This last inequality can be proved using the multipliers method and it is easily extended to the n-dimensional case (see [10] and Appendix I at the end of the paper). Once stated the well-possedness of system (1) we consider the exact boundary controllability for which the following classical result is known: Theorem 2 Assume that 2 BV (0; 1). Then, for any T > 2pM ; 0 < < 1 and (u ; u ) 2 L (0; 1) H ; (0; 1) 0
1
2
(5)
1
there exists a control f 2 L2 (0; T ) such that the solution of (1) with f = f veri es
u(T ) ut(T ) 0:
(6)
Moreover, we can estimate the norm of the controls as follows:
kf kL
;T ) C2 exp (C3 TV (
2 (0
)) k(u
0
; u )kL2 1
; H ;1(0;1)
(0 1)
(7)
where C2 and C3 are positive constants which only depend on m , M and T .
Remark 2 Note that the BV -norm of is of the order of ; . This leads to a bound of the control (7) which blows up exponentially as ! 0. Remark 3 In the hypotheses of Theorem 2 there are many possible controls f 2 1
L (0; T ) which drive the system to the equilibrium. The so-called Hilbert Uniqueness Method (HUM) (see [10]) provides the one with minimal L -norm. The control obtained 2
2
by HUM satis es (7).
3 Inequality (7) is a consequence of a boundary observability inequality for the adjoint uncontrolled system and the HUM. This last inequality can be proved by sideways energy estimates which involve a change of variables between the space and time (see for instance the Appendix in [7] and the Appendix I below). This method is typically one-dimensional and cannot be extended to several space dimensions. It turns out that both inequalities (4) and (7) are sharp in what concerns their dependence on . The optimality of (4) means that a bounded sequence of data and L -controls can produce a sequence of solutions u which is not uniformly bounded in the class (3) as ! 0. The optimality of (7) means that there exist bounded sequences of initial data which cannot be controlled with a sequence of controls uniformly bounded in . More precisely, the following holds: 2
Theorem 3 There exist periodic functions 2 L1(IR) with m (x) M < 1 pM0 ,
2 exists a sequence j j ; j ! 0 and a bounded sequence of initial data (u ; u ) 2 L (0; 1) H (0; 1) for which 0
1
2
1
the controls fj of minimal L2-norm which drive the system (1) to the equilibrium verify
fj L2
(0
;T )
C eC =j ! 1 as j ! 0; 2
(8)
3
for suitable constants C2 ; C3 > 0.
Remark 4 Note that (8) holds for particular choices of the sequence j ! 0. Thus, in principle, an inequality of the form (7) may be uniform along other subsequences j ! 0 .
Remark R 5 When ! 0, the coecient (x) = (x=) converges weakly to its average
= (s)ds and then system (1) converges in some sense to the following one: 8 > 0 < x < 1; 0 < t < T; < utt ; uxx = 0; u(0; t) = 0; u(1; t) = f (t); 0 < t < T; (9) > : u(x; 0) = u (x); ut(x; 0) = u (x); 0 < x < 1: 1 0
0
1
For this limit system we can apply theorems 1 and 2 with = 1 and we deduce that the estimate for the solution (4) and the estimate for the control (7) hold. Theorem 3 establishes that we cannot deduce an estimate like (7) uniformly in , as a perturbation result of the limit one. This is due to the lack of a good enough convergence of the solutions of (1) to those of the limit system (9).
The proof of the optimality of inequality (7) was given in [2] for 2 C (IR). Here, using the results in [5] and with almost the same argument as in [2] we prove the optimality of (4) and (7) when 2 L1 (IR). The main idea is to construct solutions of the adjoint system which concentrate most of its energy near one of the extremes as ! 0. These solutions contain frequencies with wavelength of the order of the oscillation parameter . The main goal of this paper is to prove that both estimates (4) and (7) can be done uniform in provided one removes from the solution u of (1) the Fourier components corresponding to the frequencies with wavelength of the order of . This can be done for the low frequencies when 2 L1(IR). However, when dealing with the high frequencies, more and more regularity of is needed to cover the whole range of frequencies with 2
4 wavelength greater than . Our results are derived from some uniform observability inequalities for the adjoint system. These inequalities concern the solutions which contain frequencies with wavelength larger and shorter than . To prove these inequalities we use the Ingham inequality (see [9]) and the following two spectral properties: The uniform observability for the eigenfunctions which shows that they do not exhibit any concentration of energy in the extremes as ! 0. The spectral gap between any two consecutive eigenvalues which is greater than a strictly positive constant independent of . Recent results show that both properties hold when the wavelength of the eigenfunctions is shorter and larger than but they are lost when we consider eigenfunctions with wavelength in the critical range (see [5] and [6]). These spectral results are known only for the one-dimensional problem. The rest of the paper is divided as follows: In Section 2 we give precise statements of our main results, i.e. the uniformity of inequalities (4) and (7) in when we remove from the solution the frequencies with wavelength in the critical range . In Section 3 we introduce the adjoint system and we give the statements of the regularity and observability properties which give rise to theorems 1, 2, 3, and the ones stated in Section 2 below with their proofs. In Section 4 we prove the control results stated in Section 2 from their associated observability results. The proof of theorems 1, 2 and 3 from their associated observability results is classical (see [10]). For the sake of completeness, in the Appendix I we give a proof of both the boundary regularity property which allows us to prove the estimate (4), and the classical observability inequality which gives rise to the control result stated in Theorem 2. Finally, in the Appendix II we prove a technical result. All along this paper we consider the space L (0; 1) with the natural norm Z kukL2 ; = (x)ju(x)j dx: (10) 2
1
2
(0 1)
2
0
Observe that this norm depends on the parameter . However, due to the fact that 0 < m (x) < M < 1 a.e. all these norms are equivalent to the usual norm Z kukL2 ; = ju(x)j dx (11) 1
2
(0 1)
2
0
with uniform constants in . For this reason we will not make explicit the dependence on of this L -norm. We also consider the subspace H (0; 1) with the norm Z kukH01 ; = ju0(x)j dx: 2
1 0
1
2
(0 1)
2
0
2 Statements of the main results Let us introduce a suitable decomposition of the space L (0; 1) in terms of the eigenfunctions of the eigenvalue problem associated to (1). Consider the eigenvalue problem ( 00 ' + (x)' = 0; x 2 (0; 1); (12) '(0) = '(1) = 0: 2
5 For each > 0, there exists a sequence of eigenvalues 0 < < < ::: < n < ::: ! 1 1
(13)
2
and a sequence of associated eigenfunctions ('n )n2N which can be chosen to constitute R an orthonormal basis in L (0; 1) with the norm jj'jjL2 ; = (x)j'(x)j dx: Observe that L (0; 1) = spanf('n)n2N g: (14) Multiplying the equation in (12) by any eigenfunction and integrating by parts we easily deduceR that the eigenfunctions q are also orthogonal in H (0; 1) with the norm 0 jj'jjH01 ; = j' j . In fact, ('n= n )n2N constitutes a orthonormal basis in H (0; 1). Given continuous increasing functions K (); M () such that K (); M () ! 1 as ! 0, we introduce the spaces 2
2
(0 1)
1 0
2
2
2
1 0
(0 1)
1 0
1 0
2
LK = spanf('n )nK g and H M = spanf('n)nM g ( )
( )
( )
( )
(15)
and its associated orthogonal projections K : L (0; 1) ! LK M : L (0; 1) ! H M \ L (0; 1): ( ) ( )
2
( )
2
( )
2
Observe that
L (0; 1) = LK spanf('n)K nM g H M \ L (0; 1) : 2
( )
( )
( )
( )
2
(16)
Note that LK() is a nite dimensional space generated by the low frequencies while M H () is the subspace spanned by the eigenfunctions associated to the high frequencies. The same decomposition is valid for all the spaces generated by the eigenfunctions. We have, in particular H (0; 1) = LK spanf('n)K nM g H M \ H (0; 1) : (17) 1 0
( )
( )
( )
( )
1 0
The negative result of Theorem 3 is due to the singular behavior of the eigenfunctions with wavelength of the order of , i.e. n ; . More precisely, the eigenfunctions which can exhibit a singular behavior are those for which its associated eigenvalues n verify q n 2 I (18) where I is the set of all so-called instability intervals of the associated Hill equation
'n
1
u00(y) + (y)u(y) = 0; y 2 R:
(19)
p ; ; : n (min x ) m x2I
(20)
q Using the Rayleigh formula it is easy to check that n=pM n n=pm for any 0 < < 1 and n 2 IN . On the other hand, as Iqis constituted by non-empty closed intervals we deduce that (18) only happens when n minx2I x > 0; i.e. 1
1
We refer to [8] for a detailed description of the instability intervals of the equation (19). The main results in this paper are the following two theorems:
6
Theorem 4 Let 2 L1 (IR) bepa periodic qR function such that 0 < m ;(x) M < 1 a.e. x 2 R; and consider T > 2 = 2 (s)ds. Then, if K () D with D small enough, for any (u ; u ) with (u ; u ) 2 L (0; 1) H ; (0; 1) and 0 < < 1 there exists f 2 L (0; T ) such that the solution u of (1) veri es 1 0
0
2
1
0
1
1
2
1
(K u(T ); K ut (T )) = 0: (21) Moreover, there exist constants C ; C > 0 independent of such that kf kL2 ;T C k(K u ; K u )kL2 ; H ;1 ; ; (22) k(K u ; K ut )kL1 ;T L2 ; H ;1 ; C kf kL2 ;T ; (23) for any (u ; u ) and 0 < < 1. Assume that there exists a subsequence, still denoted ! 0; such that u * u weakly in H ; (0; 1): (24) Then, the sequence of solutions (u; ut) and the sequence of controls f (t) verify (K u; K ut) * (u; ut) weakly in L1 (0; T ; L (0; 1) H ; (0; 1)) (25) f * f weakly in L (0; T ) (26) where (u; ut) and f (t) are the solution and the control respectively of the limit system 8 > 0 < x < 1; 0 < t < T; < utt ; uxx = 0; (27) u(0; t) = 0; u(1; t) = f (t); 0 < t < T; > : u(x; 0) = u (x); ut(x; 0) = u(x)=; 0 < x < 1: ( ) 1
(0
1
)
( )
0
( )
2
( )
( )
0
( )
(0
;
1
(0 1)
(0 1)
(0 1)
2
(0 1))
(0
)
1
1
( )
1
1
2
( )
1
2
0
1
Moreover, if u1 ! u1 strongly in H ;1 (0; 1); then
(K u; K ut ) ! (u; ut) strongly in L1 (0; T ; L (0; 1) H ; (0; 1)) (28) f ! f strongly in L (0; T ) (29) Remark 6 This result, roughly, establishes the continuity of the controls with respect to ! 0. The controls f and f are not unique as indicated in Remark 1 and the meaning of (26) has to be made precise. Even if the controls are not unique, HUM provides an unique choice of them. In (26) (29) we state the convergence of the controls obtained by HUM. Remark 7 Inequality (22) establishes, in particular, that the result of Theorem 3 does not hold when the initial data is generated by the low frequency eigenfunctions. Remark 8 In Theorem 4 we have assumed that the initial data (u ; u ) for system (1) are xed. However, our proof can be easily adapted to more general cases. In particular, if we consider (u ; u ) with ( u ; u ) * (u; u) weakly in L (0; 1) H ; (0; 1) then (21), (22) and (23) still hold. The convergence results (25) and (26) also hold and the limits (u; ut) and f are the solution and control of the limit system 8 > 0 < x < 1; 0 < t < T; < utt ; uxx = 0; u (0 ; t ) = 0 ; u (1 ; t ) = f ( t ) ; 0 < t < T; (30) > : u(x; 0) = u(x)=; ut(x; 0) = u(x)=; 0 < x < 1 ( )
2
( )
1
2
0
0
1
0
1
0
0
2
1
1
1
1
7 instead of (27). Moreover, if we assume that (u0; u1) ! (u0; u1) strongly in L2(0; 1) H ;1 (0; 1) then (28) and (29) hold with (u; ut) and f the solution and control of the limit system (27). Theorem 5 Let 2 W N +1;1(IR) (N 1) be a periodic R 1 qfunction such that 0 < m (x) M < 1 a.e. x 2 R; and consider T > 2 0 (x)dx. Then, if M () D;1;1=N with D large enough, for any (u0; u1) 2 L2(0; 1) H ;1 (0; 1) and 0 < < 1 there exists f 2 L2 (0; T ) such that the solution u of (1) veri es (M ()u(T ); M ()ut(T )) = 0: (31) Moreover, there exist constants C1; C2 > 0 independent on such that kf kL2 (0;T ) C1k(M ()u0; M ()u1)kL2(0;1)H ;1(0;1); (32) M () M () k( u ; ut )kL1(0;T ;L2(0;1)H ;1(0;1)) C2kf kL2 (0;T ); (33) for any (u0; u1) 2 L2(0; 1) H ;1 (0; 1) and 0 < < 1. Assume that there exists a subsequence, still denoted ! 0; such that (24) holds. Then, the sequence of solutions (u ; ut) and the sequence of controls f(t) verify (M ()u; M ()ut) * (0; 0) weakly in L1(0; T ; L2(0; 1) H ;1(0; 1)) (34) f * 0 weakly in L2(0; T ): (35) Moreover, if u1 ! u1 strongly in H ;1 (0; 1); then the convergence in (34) and (35) is strong. Remark 9 The time 2p needed to control the projection solution over the low R 1 qof(x)the dx needed for the higher eigenfrequencies (Theorem 4) is larger than the time 2 0 eigenfrequencies (Theorem 5). Remark 10 In Theorem 5 we assume that u1 2 H ;1(0; 1) instead of u1 2 H ;1 (0; 1) which is the assumption in Theorem 4. The reason is that both assumptions are equivalent when 2 H 1 which is the case in Theorem 5 where we have simpli ed the notation. Remark 11 The results of Theorems 4 and 5 are sharp in the following sense: We know that inequality (7) is sharp in what concerns the dependence on when solutions contain eigenfunctions 'n with n verifying (20). In Theorem 4 we prove that the constant in (7) can be chosen independent of when the solutions do not contain eigenfrequencies with n D;1 . At this moment we do not know if D coincides with the constant in (20) but we see that the Theorem 4 is sharp in what concerns the order in . Concerning Theorem 5, assume that 2 C 1 . Then it is well known that the set I in (18) is not bounded and then we cannot expect a constant independent of in (7) for solutions which contain eigenfunctions with n C;1 for any constant. On the other hand, Theorem 5 states that the constant can be done independent of if the solution contains eigenfunctions with n D; for any > 0. We see again that there is some kind of optimality in Theorem 5. The uniform bounds in Theorem 4 were announced in [4] where a sketch of the proof was given. Here we give a complete proof and complement this result on the low frequencies with the analysis of the limit as ! 0 and with Theorem 5, that establishes the high frequency counterpart.
8
3 Observability of the adjoint system The transposition method and the Hilbert Uniqueness Method (HUM) provide an equivalence between the results of the previous sections and some suitable regularity and observability properties for the adjoint system. In this section we give precise statements of these properties and their proofs. In Section 3.1 we introduce the adjoint system and state the main results. In Section 3.2 we give some preliminaries where we state the two spectral results that we need: the spectral gap between any two consecutive eigenvalues of the adjoint system and a uniform observability property for the eigenfunctions. Both results are proved in [5] for the low frequency case and in [6] for the high frequency one. Finally, in Section 3.3 we prove Proposition 4 (the proof of Proposition 5 is similar). The method consists in three steps: in the rst one we introduce the Fourier decomposition of the solutions which allows us to write the inequality (41) in terms of the eigenvalues and eigenfunctions of (4). In the second step we use the uniform gap between two consecutive eigenvalues and a result on non-harmonic Fourier series to state a suitable inequality . Finally, in the third step we combine the result in Step 2 and the uniform observability of the eigenfunctions to obtain the observability inequality (41). 3.1 The statements
Let us introduce the adjoint system to (1): 8 > (x)vtt ; vxx = 0; 0 < x < 1; 0 < t < T; < v (0 ; t ) = v (1 ; t ) = 0 ; 0 < t < T; > : v(x; 0) = v (x); vt(x; 0) = v (x); 0 < x < 1: 0
(36)
1
The estimate (4) for the solutions of the controlled problem can be derived using the transposition method (see [10] and [11]) and the following boundary regularity property for system (36):
Proposition 1 Assume that (x) 2 W ;1(0; 1). There exists a constant C > 0 such 1
that
ZT
1
jvx (1; t)j dt C k(x=)kW ;1 ; k(v ; v )kH ; L ; for every (v ; v ) 2 H (0; 1) L (0; 1) with 0 < < 1 and v solution of (36). 2
0
1
1 0
1
1
0
(0 1)
0
1
2
1 0 (0 1)
2 (0 1)
(37)
2
The proof of Proposition 1 can be obtained by the multipliers method even for the n-dimensional case. This proof does not require the periodicity of . For the sake of completness we give the proof in the Appendix I below. The exact boundary controllability property of (1) stated in Theorem 2 and estimate (7) are derived from the HUM and the following clasical boundary observability property:
Proposition 2 Assume that (x) 2 pBV (0; 1) is periodic with 0 < m (x) M < 1 a.e. x 2 [0; 1]. Then, given T > 2 M ; we have ZT ;m TV p ; 2 minf ; 1g(T ; 2 )e k(v ; v )k jv (1; t)j dt (38) m
M
1
(
)
0
1
2
H01 (0;1)L2(0;1)
0
x
2
for every (v0 ; v1) 2 H01(0; 1) L2(0; 1) and 0 < < 1, with v solution of (36).
9
Remark 12 Inequality (38) provides an estimate of the total energy of the solutions
from the energy concentrated in one of the extremes of the interval (x = 1). This estimate is not uniform in because TV ( ) ;1 .
The proof of Proposition 2 is obtained by a method which involves a change of variables between the space and time (see the Appendix I). This method can be applied only in the one-dimensional case and it does not require the periodicity of . Observe that none of the estimates (37) and (38) is uniform in . As we said in the introduction, inequalities (4) and (7) are optimal in what concerns the dependence of the constants involved in the estimates with respect to . This result and Theorem 3 are derived from the following proposition which shows that both (37) and (38) are indeed sharp in what concerns its dependence on .
Proposition 3 Let 2 L1 (IR) be a periodic function such that 0 < m (x) M < 1 a.e. x 2 R;. Let T > 0. Then, at least one (and possibly both) of the following properties (a) or (b) holds: (a) There exists a sequence j ! 0 and a sequence of non-trivial solutions v j of (36) satisfying
ZT
jvxj (1; t)j dt C ;j k(vj ; vj )kH ; L ; : (39) (b) There exists a sequence j ! 0 and a sequence of solutions vj of (36) which 2
1
0
veri es
ZT
1
0
2
1
1 0 (0 1)
jvxj (1; t)j dt C e;C =j k(vj ; vj )kH
2 (0 1)
; L2(0;1):
(40) Furthermore, there exist non-constant periodic functions such that (a) holds. The same is true with (b). 0
2
2
3
0
1
2
1 0 (0 1)
Remark 13 Proposition 3 establishes the optimality of (37) and (38) for some periodic densities . Indeed, according to (39), for some periodic densities, there are solutions of (36) which concentrate most of its energy near the point x = 1 of the boundary where the energy is being observed or measured, as ! 0. On the other hand, equation (40) means that there are solutions of (36) which concentrate most of its energy away of the observability zone (the point x = 1 in the present case) as ! 0.
The partial controllability results stated in theorems 4 and 5 can be reduced to some partial observability results for system (36). To state them we recall the Fourier decomposition of the spaces L (0; 1) and H (0; 1) into the low and high frequency spaces given in (16) and (17). We have the following: 2
1 0
Proposition 4 Let 2 L1 (IR) be a periodic function such that 0 < m (x) p M < 1 a.e. x 2 R; and consider T > 2 . Then, if K () D; with D small enough, there exist constants C , C > 0 such that for any (v ; v ) 2 LK LK and 1
1
2
0
1
0 < < 1 the solution v of (36) veri es ZT C k(v ; v )kH01 ; L2 ; jvx (1; t)j dt C k(v ; v )kH01 1
0
1
2
(0 1)
(0 1)
2
0
2
0
1
2
( )
; L2(0;1):
(0 1)
( )
(41)
Proposition 5 Let 2 W N ;1(IR) with N 1 be a periodicR qfunction such that 0 < m (x) M < 1 a.e. x 2 R; and consider T > 2 (s)ds. Then, if +1
1 0
10
M () D; ; =N with D large enough, there exist constants C , C > 0 such that for any (v ; v ) 2 (H M \ H ) (H M \ L ) and 0 < < 1 the solution v of (36) veri es ZT C k(v ; v )kH01 ; L2 ; jvx (1; t)j dt C k(v ; v )kH01 ; L2 ; : (42) 1 1
0
( )
1
1
0
2
1
1 0
(0 1)
( )
1
2
2
(0 1)
2
0
0
1
2
2
(0 1)
(0 1)
Assume that 2 C 1. Propositions 4 and 5 state that both estimates (37) and (38) can be done uniform in provided one removes from the solution of (36) the Fourier components corresponding to frequencies with wavelength larger and shorter than . 3.2 Preliminaries to the proofs
In this section we recall some results about the behavior of eigenfunctions and eigenvalues of (36) with respect to . We consider three di erent cases: the low frequency case which correspond to eigenfunctions with wavelength larger than the periodicity of the density , the high frequency case which correspond to eigenfunctions with wavelength shorter than , and the critical case where the wavelength of the eigenfunctions is of the order of . For the rst two cases we state two spectral properties: the spectral gap (uniform in ) between two consecutive eigenvalues and a boundary observability property for the eigenfunctions. For the critical case, we state the existence of eigenfunctions which concentrates at one extreme. 3.2.1 Low frequencies Proposition 6 Assume that 2 L1 (IR) is a periodic function. Given > 0; there exists a constant C ( ) > 0 such that
q q n ; n p ; ;
(43)
+1
for all n and with n C ( ). Furthermore, there exist C1; C2 ; c > 0 such that the following estimates hold for the eigenfunctions 'n of (1) with n c;1:
C
Z
1
1 0
j('n)0(x)j j('n)0(1)j C 2
2
Z
2
1
0
j('n )0(x)j : 2
(44)
3.2.2 High frequencies Proposition 7 Let be a periodic function with 0 < m (x) M < 1. Assume that 2 W N +1;1(IR) for some N 1. Given > 0, there exists a constant C > 0 such that if n C;1;1=N we have
q q n ; n R q ; : (s=)ds +1
(45)
1 0
Furthermore, there exist C1; C2 ; c > 0 such that the following estimates hold for the eigenfunctions 'n with n c;1;1=N :
C
1
Z
1 0
j('n)0(x)j2 j('n)0(1)j2 C2
Z
0
1
j('n )0(x)j : 2
(46)
11 3.2.3 Critical case Proposition 8 Consider 2 L1 (IR) a non-constant periodic function with 0 < m (x) M < 1 a.e. x 2 IR. Then at least one of the following properties (a) or (b) hold: (a) There exist a sequence j ! 0, a sequence of eigenfunctions 'j of (12) and a positive constant C > 0 such that
R
j'0j (x)j dx (47) j'0j (1)j Cj : (b) There exist a sequence j ! 0, a sequence of eigenfunctions 'j of (12) and 1 0
2
2
positive constants C1 ; C2 > 0 such that
R
1 0
j'0j (x)j dx C =j j'0j (1)j C j e ; 2
1
2
(48)
2
Furthermore, there exist smooth non-constant periodic functions and j ! 0 such that (a) holds. The same is true for (b).
Remark 14 Proposition 8 guarantees that there are not constants C ; C > 0 such that C
Z
1
1 0
j('n)0(x)j dx j('n)0(1)j C 2
2
Z
2
1
1 0
j('n)0(x)j dx 2
2
(49)
for all n 2 IN and 0 < < 1. Indeed, according to Proposition 8, for any , at least one of the uniform estimates in (49) fails.
Remark 15 Note that the eigenfunctions 'j we have found in the proof of Proposition
8, both in what respects (47) and (48), correspond to eigenvalues that are of the order of ;2.
Remark 16 At this moment we do not know if for any non-constant periodic 2 L1(IR) one can always nd a sequence j ! 0 verifying (47). The same can be said about (48).
Remark 17 Note that (47) or (48) holds for particular choices of the sequence j ! 0. Thus, Proposition 8 is not an obstacle for (49) to hold uniformly along other sequences j ! 0.
Remark 18 Concerning the last statement of Proposition 8 much more can be said. In fact, for any as above there exists x0 such that ~(x) = (x + x0 ) satis es (a). The same can be said about (b).
Remark 19 The existence of eigenfunctions which concentrate most of its energy in the boundary is also known for the n-dimensional case where the oscillating coecient is in the principal part of the operator (see [1]).
12 3.3 Proof of Propositions 4 and 5.
The proofs of Propositions 4 and 5 are similar and we only prove the rst one. We proceed in three steps: Step 1: Fourier decomposition of the solutions. In this step we write the inequalities (41) in term of the eigenvalues and eigenfunctions of (12). When considering solutions of (36) in separated variables we are led to the eigenvalue problem (12). As we said above, for each > 0 there exists a sequence of eigenvalues fn gn2IN and a sequence of associated eigenfunctions f'ngn2IN which can be chosen to constitute an orthonormal basis in H (0; 1). De ne q q k = ; ;k and ';k = 'k ; k 2 ZZ . 1 0
Then, the solutions v of (36) can be represented as X ipkt v (x; t) = ak 'k (x)e k2ZZ
where fakgk2Z are complex Fourier coecients by the initial data. q determined Given > 0, we consider !k = ('k ; k 'k ). Then f!k gk2ZZ constitutes an orthonormal basis of the energy space H (0; 1) L (0; 1) with the norm Z h i = k(u; v)k = juxj + jvj dx : 1 0
2
1
2
1 2
2
0
So, any nite energy solution of (36) can be represented as X ipkt (v (x; t); vt (x; t)) = ak !k (x)e ; with coecients (ak ) 2 l ; 2
k2ZZ
and
k(v; vt)k = 2
X k2ZZ
jak j : 2
Inequalities (41) can be reduced to nd constants C ; C > 0 such that X Z T X ipk t 0 X C jakj ak e ('k ) (1) dt C jak j : jkjK e jkjK jkjK 1
2
2
2
1
( )
2
2
0
( )
(50)
( )
In the following two steps we prove two inequalities which combined give us (50). Step 2: A non-harmonic Fourier series result. Here we prove the following result which precises the uniform spectral gap stated in Proposition 6.
Proposition 9 Consider T > 2p. There exist constants C ; C > 0 such that X X Z T X ipkt jak j : (51) C jak j ak e dt C 1
2
2
1
2
jkjK ()
0
jkjK ()
2
2
jkjK ()
This result is a corollary of Proposition 6 and the following well-known result on nonharmonic analysis:
13
Theorem 6 (Ingham [9]) Let (k )k2ZZ be a sequence of real numbers such that k ; k > > 0. (52) +1
Then, for any T P > 2= there exist constants C1; C2 > 0, only depending on , such that for any f = k2ZZ ck eik t 2 L2 (0; T ) we have
Z T X X jck j : C jck j ck eik t dt C k2ZZ k2ZZ k2ZZ 1
X
2
2
(53)
2
2
0
The constants C1 ; C2 in (53) depend on T and but not in the sequence fk g satisfying (52).
Step 3: Observability of eigenfunctions. Using the uniform observability of the eigenfunctions stated in Proposition 6 we easily obtain the existence of constants C ; C > 0 such that Z T X Z p p T X C ak ei k t dt ak ei k t ('k )0 (1) dt jkjK jkjK Z T X p t i C ak e k dt (54) jkjK 1
2
2
1
0
2
0
( )
( )
2
2
0
( )
for any T > 0 and (ak ) 2 l . Combining Proposition 9 and (54) we obtain the inequalities in (50). In view of the Step 1 this conludes the proof of Proposition 4. 2
3.3.1 Proof of Proposition 2 The proof of Proposition 3 is a simple consequence of Proposition 8. We only prove the rst part because the second one is similar. By part (a) in Proposition 8 there exists a sequence j ! 0, a sequence of eigenfunctions 'j and a positive constant C > 0 such that R 1 j'0 (x)j2dx j 0 (55) j'0j (1)j2 Cj : Consider j the eigenvalue associated to 'j . Then
p
vj = cos( j t)'j (x) is a solution of (36) which veri es: p j ! ZT ZT p sin(2 1 jvxj (1; t)j dt = j'0j (1)j cos ( j t) = j'0j (1)j 2 + p j T ) : (56) 4 On the other hand Z Z k(vj (x; 0); vj (x; 0))kH01 ; L2 ; = j'0j (x)j dx + (x)j'j (x)j dx Z j'0j (x)j dx: (57) = 1 + 1j Combining (55) (56) and (57) we easily check that vj veri es (39). 2
2
0
2
2
0
2
1
(0 1)
(0 1)
1
2
0
2
0
1
0
2
14
4 Proofs of the main theorems In this section we prove Theorem 4. The proof of Theorem 5 is similar and we omit it. To clarify the exposition of this long proof we proceed in 8 steps. In the rst one we consider the exact boundary controllability that we obtain using the HUM. In the second step we obtain the bound (22) of the controls. In the third and fourth steps we prove the weak and strong convergence of the controls respectively. In the step 5 we use transposition to obtain the uniform bound (23) for the projections of the solutions. From this uniform bound we prove, in steps 6 and 7, the weak and strong convergence of the projections of the solutions respectively. Finally, a technical lemma is proved in the step 8. Step 1: Existence of the controls. In this step we adapt the HUM to obtain the exact controllability result. Let us assume that Proposition 4 holds. Given T > 2p we consider the following system 8 > 0 < x < 1; 0 < t < T; < (x)vtt ; vxx = 0; v(0; t) = v(1; t) = 0; 0 < t < T; (58) > : v(x; T ) = v (x); vt(x; T ) = v (x); 0 < x < 1: 0
1
Due to the time reversibility of the wave equation, system (58) is well-posed backwards in time. So, given (v ; v ) 2 H (0; 1) L (0; 1) there exists an unique solution v of (58) in the class v(x; t) 2 C (0; T ; H (0; 1)) \ C (0; T ; L (0; 1)): (59) Moreover, the energy of the system is conserved: Z Z E (t) = jvx (x; t)j dx + ( x )jv(x; t)j dx Z Z x = jv ;x(x)j dx + ( )jv (x)j dx = E (0); 8t 2 [0; T ]: Due to Proposition 4, there exist positive constants C ; C such that when (v ; v ) 2 LK LK we have ZT C k(v ; v )kH01 ; L2 ; jvx (1; t)j dt C k(v ; v )kH01 ; L2 ; (60) 0
1 0
1
2
1 0
1
1
2
0
1
0
2
2
0 1
2
0
1
2
1
0
1
( )
2
0
1
( )
1
0
1
2
(0 1)
(0 1)
2
2
0
0
1
2
(0 1)
(0 1)
for all 0 < < 1. Multiplying the equations (1) by v and integrating by parts we easily obtain the following identity: Z < (x)ut (x; T ); v (x) >H ;1;H01 ; (x)u(x; T )v (x)dx Z ZT ; < (x)u (x); v (x; 0) >H ;1;H01 + (x)u (x)vt (x; 0)dx + f (t)vx (1; t) = 0: (61) Here < ; >H ;1;H01 represents the duality product between H and its dual H ; . As the eigenfunctions are orthogonal in L (0; 1) and v 2 LK which is a subspace generated by the rst K () eigenfunctions we can simplify the second term in (61) as follows Z Z (x)u (x; T )v (x)dx = (x)K (u(x; T ))v (x)dx: (62) 1
0
1
0 1
0
0
0
0
1 0
2
1
0
1
1
( )
1
1
0
( )
1
15 In the same way, we can simplify the rst term in (61). To do this we observe that both ut(x; T ) and v can be spanned in terms of the eigenfunctions 'n and that Z < 'n; 'm >H ;1;H01 = (x)'n (x)'m(x)dx (63) 0
1
0
which is nonzero if and only if m = n. So, as v 2 LK we easily obtain 0
( )
< (x)ut(x; T ); v (x) >H ;1;H01 =< (x)K (ut(x; T )); v (x) >H ;1;H01 (64) Using (62) and (64) we can simplify (61) into Z < (x)K (ut(x; T )); v (x) >H ;1;H01 ; (x)K (u(x; T ))v (x)dx Z ZT ; < (x)u (x); v (x; 0) >H ;1;H01 + (x)u (x)vt (x; 0)dx + f (t)vx (1; t) = 0: (65) Now, observe that the condition (21) is equivalent to the following one: Z < (x)K (ut(x; T )); v (x) >H ;1;H01 ; (x)K (u (x; T ))v (x)dx = 0; 8(v ; v ) 2 LK : (66) From (65) and (66) we deduce that the rst statement in Theorem 4 is equivalent to the existence of f 2 L (0; T ) such that Z ZT ; < u (x); v(x; 0) >H ;1;H01 + (x)u (x)vt(x; 0)dx + f (t)vx (1; t)dt = 0 (67) 0
0
( )
1
0
( )
1
( )
0
1
0
0
0
0
1
( )
0
0
1
( )
0
0
( )
2
1
1
0
0
0
for all (v ; v ) 2 LK . Let us introduce the following quadratic functional: Z ZT J(v ; v ) = ; < u (x); v(x; 0) >H ;1;H01 + (x)u (x)vt(x; 0)dx+ 21 jvx (1; t)j dt: (68) Thanks to (60), J is a continuous, strictly convex and coercive functional in LK . So, there exists an unique minimizer (w ; w ) 2 LK which can be characterized by the formula Z ZT ; < u (x); v (x; 0) >H ;1;H01 + (x)u (x)vt (x; 0)dx + wx (1; t)vx (1; t)dt = 0; 8(v ; v ) 2 LK : (69) 0
1
( )
1
0
1
1
2
0
0
0
( )
0
( )
1
1
1
0
0
0
1
0
( )
We set f (t) = wx (1; t) where w solves (58) with the minimizer (w ; w ) as data. Then f veri es (67) and it is the control we were looking for. Step 2: Uniform bound of the controls. Here we prove (22). Observe that, due to the fact that (w ; w ) is a minimizer of J, we have 0
0
1
1
J(w ; w ) J(0; 0) = 0: 0
Then,
ZT 0
jf (t)j2dt < u
(70)
1
1 (x); w (x; 0) >H ;1 ;H 1 0
Z
; (x)u (x)wt(x; 0)dx: 0
1
0
(71)
16 Using the Fourier decomposition of u and u , the fact that w ; w 2 LK and the orthogonality of the eigenfunctions we can write the second term in (71) as Z < K (u (x)); w (x; 0) >H ;1;H01 ; (x)K (u (x))wt(x; 0)dx = < (K u ; K u ); (;wt(x; 0); w(x; 0)) >L2H ;1 ;L2H01
(K u ; K u )
L2 ; H ;1 ; k(w ; w )kH01 ; L2 ; : (72) 0
1
0
( )
1
1
1
( )
( )
( )
( )
0
0
( )
0
( )
0
1
1
(0 1)
0
(0 1)
(0 1)
1
(0 1)
Now, in view of (60) we can estimate the last term in (72) by != ZT
pC (K u ; K u ) 2 jf (t)j dt : 1 L ; H ;1 ; 1 2
1
( )
0
( )
2
1
(0 1)
(0 1)
0
(73)
Combining (71), (72) and (73) we easily complete the proof of (22). Step 3: Weak convergence of the controls. Thanks to the hypothesis (24) and the bound (22) the controls f are uniformly bounded in L (0; T ) and therefore there exists a subsequence, still denoted by f, such that 2
f * g weakly in L (0; T ):
(74)
2
Let us see that g = f where f is the control of the limit system (27) provided by HUM. The control f (t) is given by f (t) = wx(1; t) where w(x; t) is the solution of ( vtt ; vxx = 0; 0 < x < 1; 0 < t < T; (75) v(0; t) = v(1; t) = 0; 0 < t < T; with nal data (w ; w ), i.e. 0
1
v(x; T ) = w (x); vt(x; T ) = w (x); 0
0 < x < 1:
1
(76)
Here, (w ; w ) 2 H L (0; 1) is the minimizer of the limit quadratic functional Z 1 J (v ; v ) = ; < u (x); v(x; 0) >H ;1;H0 + u (x)vt(x; 0)dx ZT 1 + 2 jvx(1; t)j dt (77) where v(x; t) is the solution of (75) with nal data (v ; v ): Hence, the control f (t) can be characterized by the following two properties: 0
1 0
1
2
1
0
1
1
0
0
2
0
0
1
1. f (t) = wx(1; t) where w is a solution of (75). 2. f (t) veri es Z ZT ; < u (x); v(x; 0) >H ;1;H01 + u (x)vt(x; 0)dx + f (t)vx(1; t)dt = 0 8(v ; v ) 2 H L (0; 1): (78) 1
1
0
1
0
1 0
0
0
2
where v is the solution of (75) with nal data
v(x; T ) = v (x); vt(x; T ) = v (x); 0 < x < 1: 0
1
(79)
17 We are going to see that g veri es this two properties. We start with the rst one. By the boundness of the controls and the rst estimate in (60) we deduce that (w ; w ) is uniformly bounded in H (0; 1) L (0; 1). So, we can extract a subsequence, still denoted (w ; w ), such that (w ; w ) * (w ; w ) weakly in H (0; 1) L (0; 1): De ne w the weak limit in L (0; 1) of w (x=). The theory of Homogenization (see [3], Theorem 3.2) gives us that the solution w of (58) with nal data (w ; w ) converges to the solution w of (75) with nal data (w ; w =) in the following sense w * w weak-* in L1 (0; T ; H (0; 1)) \ W ;1(0; T ; L (0; 1)): (80) Furthermore, for any element v 2 L (0; 1), Z Z (x)wt(x; t)v(x)dx ! wt(x; t)v(x) strongly in C ([0; T ]): (81) 0
1 0
1
0
1
0
0
1
1 0
1
2
1
0
0
2
2
1
1
1
1 0
1
2
2
1
1
0
0
0
Multiplying the equations (58) by tests functions and integrating we easily obtain the following identity for w : ZTZ Z TZ 0= (x)wt (x; t)v(x)lt(t)dxdt ; wx (x; t)vx(x)l(t)dxdt ZT + wx (1; t)v(1)l(t)dt; 8v(x) 2 C ((0; 1]); l(t) 2 C (0; T ): (82) 1
0
1
0
0
0
1 0
0
1 0
We can pass to the limit in (82) thanks to (74), (80) and (81), and then w veri es ZTZ ZT ZTZ 0= wt(x; t)v(x)lt(t)dxdt ; wx(x; t)vx(x)l(t)dxdt + g(t)v(1)l(t)dt 8v(x) 2 C ((0; 1]); l(t) 2 C (0; T ): (83) On the other hand, as w is a solution of the limit system (75), it also veri es ZTZ ZTZ 0= wt(x; t)v(x)lt(t)dxdt ; wx(x; t)vx(x)l(t)dxdt ZT + wx(1; t)v(1)l(t)dt; 8v(x) 2 C ((0; 1]); l(t) 2 C (0; T ): (84) 1
1
0
0
0
1 0
0
0
1 0
1
0
1
0
0
0
1 0
0
From (83) and (84) we nally deduce ZT ZT g(t)l(t)dt = wx(1; t)l(t)dt; 0
0
1 0
8l(t) 2 C (0; T ) 1 0
and then g = wx(1; t) with w a solution of (75) as we wanted to prove. Now, we check that g also veri es the second property above. Assume that the following Lemma holds: Lemma 1 Given (v ; v ) 2 H L (0; 1); there exists a sequence (v ; v ) 2 LK such that (v ; v ) ! (v ; v ) in H L (0; 1): (85) Furthermore, if v is the solution of (58) with initial data (v ; v ) and v is the solution of (75) with initial data (v ; v ) the following holds: v ! v in L1 ([0; T ]; H (0; 1)) \ W ;1([0; T ]; L (0; 1)); (86) vx(1; t) ! vx(1; t) in L (0; T ): (87) 0
1 0
1
0
2
0
0
1
1
1 0
2
0
0
1
1
1 0
1
2
2
1
( )
18 We prove this Lemma at the end of the section (Step 8). We consider (v ; v ) 2 H L (0; 1): By Lemma 1 there exists a sequence (v ; v ) 2 LK such that (85) holds. Furthermore, from formula (86) we deduce in particular that (v(x; 0); vt(x; 0)) ! (v(x; 0); vt(x; 0)) strongly in H (0; 1) L (0; 1) (88) where v is the solution of (58) with initial data (v ; v ) and v is the solution of (75) with initial data (v ; v ): Passing to the limit, as ! 0, in formula (69) with (v ; v ) = (v ; v ) and taking into account (24), (87) and (88) we obtain that g satis es Z ZT ; < u (x); v(x; 0) >H ;1;H01 + u (x)vt(x; 0)dx + g(t)vx(1; t)dt = 0 8(v ; v ) 2 H L (0; 1): (89) So, g veri es the second property above. Step 4: Strong convergence of the controls. We consider formulas (69) and (78) with (v ; v ) = (w ; w ) and (v ; v ) = (w ; w ) respectively: Z ZT ; < u (x); w (x; 0) >H ;1;H01 + (x)u (x)wt (x; 0)dx + jwx (1; t)j dt = 0; (90) Z ZT ; < u (x); w(x; 0) >H ;1;H01 + u (x)wt(x; 0)dx + jwx(1; t)j dt = 0: (91) 0
1 0
1
2
0
1
( )
1 0
1
0
0
2
1
0
1
0
1
1
1
0
0
0
0
1
1 0
1
0
1
0
2
0
1
0
1
1
1
2
0
0
0
1
1
0
2
0
0
Due to the hypothesis on the strong convergence u ! u in H ; and (81), the rst two terms in (90) converge to the rst two terms in (91) and therefore ZT ZT jwx(1; t)j dt ! jwx(1; t)j dt; as ! 0: (92) 1
2
0
1
1
2
0
The strong convergence of the controls in L (0; T ) is a consequence of the weak convergence stated in step 4 and the convergence of the norms stated in (92). Step 5: Uniform bound of the projections of the solutions. We now prove (23) using transposition and (60). To introduce the concept of transposition we consider (x; t) 2 L (0; T ; LK ) and the following system: 8 > < (x) tt ; xx = (x)(x; t); 0 < x < 1; 0 < t < T; (0; t) = (1; t) = 0; 0 < t < T; (93) > : (x; 0) = t(x; 0) = 0; 0 < x < 1: 2
1
( )
Multiplying the equations of (1) by the solution of (93) and integrating by parts we easily obtain the following identity: ZTZ ZT udxdt = ; f (t) x (1; t)dt; 8(x; t) 2 L (0; T ; LK ): (94) 1
0
1
0
( )
0
Here we have used the fact that u(x; T ) = ut(x; T ) = 0 because f is a control. Taking into account the Fourier decomposition of the space L (0; 1) given in (16) and the fact that (:; t) 2 LK we can write (94) as ZTZ ZT K (u )dxdt = ; f (t) x (1; t)dt; 8(x; t) 2 L (0; T ; LK ): (95) 2
( )
1
0
0
( )
1
0
( )
Assume that the following result, which is an extension of the second inequality in (41), holds:
19
Proposition 10 Let 2 L1 (IR) be a periodic function such that 0 < m (x) M < 1 a.e. x 2 R; and consider T > 0. Then, for all ((x; t); (x; t)) 2 C (0; T ; LK ) 1
there exists C2 > 0 such that for any 0 < < 1 the solution
ZT 0
j x (1; t)j dt C kkL 2
2
2
of (93) veri es
;T ;L2(0;1)):
( )
(96)
1 (0
Moreover, the solution of 8 > < (x)tt ; xx = (x)t(x; t); 0 < x < 1; 0 < t < T; (0; t) = (1; t) = 0; 0 < t < T; > : (x; 0) = t(x; 0) = 0; 0 < x < 1: satis es ZT jx (1; t)j2dt C2k k2L1(0;T ;H01(0;1)):
(97) (98)
0
The proof of Proposition 10 from Proposition (4) is rather standard (see [10]). However, for the sake of completeness, we give a proof in the Appendix II below. By Proposition 10 the right hand side of (95) is bounded by Z T f (t) (1; t)dt C kf k 2 kk 1 2 : (99) x L ;T L ;T L ; (0
0
)
(0
;
(0 1))
Then, the map
ZT ! ; f (t) x (1; t)dt is linear and continuous from L (0; T ; LK ) ! IR and therefore K u is indeed characterized by (95). Furthermore, from (95) and (99) we have
K u
C kf k 2 : 0
1
( )
L1 (0;T ;L2(0;1))
( )
( )
L (0;T )
In a similar way we deduce the bound for ut: Consider the system 8 > < (x)tt ; xx = (x)t(x; t); 0 < x < 1; 0 < t < T; (0; t) = (1; t) = 0; 0 < t < T; > : (x; 0) = t(x; 0) = 0; 0 < x < 1:
(100)
Multiplying the equations of (1) by the solution of (100) and integrating by parts we formally obtain ZTZ ZT utdxdt = ; f (t)x (1; t)dt; 8 2 C (0; T ; H (0; 1)): (101) 1
0
1
0
1 0
0
Hence, ZT ZT < ut ; >H ;1;H01 dt = f (t)x (1; t)dt; 8 2 C (0; T ; H (0; 1)): 1
0
1 0
0
(102)
Taking into account the Fourier decomposition of the space H ; (0; 1) and the fact that (:; t) 2 LK we can simplify (102) into ZT ZT < K ut ; >H ;1;H01 dt = f (t)x (1; t)dt; 8 2 C (0; T ; H (0; 1)): (103) 1
( )
0
( )
1
0
1 0
20 By Proposition 10 the right hand side of (103) can be bounded by Z T f (t) (1; t)dt C kf k 2 k k 1 x L ;T L ;T H01 ; : (0
0
Then, the map
)
(0
;
(104)
(0 1))
ZT
! f (t)x (1; t)dt is linear and continuous from L (0; T ; LK ) ! IR and then K ut can be indeed characterized by formula (103). Moreover, from (103) and (104) we deduce that
K u
C kf k 2 : 0
1
( )
( )
( )
t L1 (0;T ;H ;1(0;1))
L (0;T )
Step 6: Weak convergence of the projections of the solutions. By (23) and the uniform bound of the controls we deduce that the projections (K u ; K ut) are uniformly bounded in L1(0; T ; L H ; (0; 1)). 2
( )
1
( )
So, there exists a subsequence, still denoted by (K u; K ut) such that (K u; K ut) * (p; q) weakly-* in L1 (0; T ; L H ; (0; 1)): (105) Let us see that (p; q) = (u; ut=) where (u; ut) is the solution of the limit system (27). Observe that, by the transposition argument of the step 5, the projection K u can be characterized by the formula ZTZ ZT K (u)dxdt = ; f (t) x (1; t)dt; 8 2 L (0; T ; L (0; 1)) (106) ( )
( )
( )
2
( )
1
( )
1
0
1
( )
0
2
0
where is the solutions of (93). In a similar way u can be characterized by ZT Z TZ udxdt = ; f (t) x(1; t)dt; 8 2 L (0; T ; L (0; 1)) 1
0
1
2
0
0
where is the solutions of the following system: 8 > < (x) tt ; xx = (x; t); 0 < x < 1; 0 < t < T; (0; t) = (1; t) = 0; 0 < t < T; > : (x; 0) = t(x; 0) = 0; 0 < x < 1:
(107)
(108)
At this point, we need the following Lemma:
Lemma 2 Given (; ) 2 C (0; T ; L (0; 1) H (0; 1)); there exists a sequence (; ) 2 C (0; T ; LK LK ) such that ( ; ) ! (; ) in C (0; T ; L (0; 1) H (0; 1)): (109) 1
1
( )
2
1 0
( )
1
2
1 0
Furthermore, if is the solution of (93) with nonhomogeneous term = and the solution of (108) the following holds: x(1; t) ! x(1; t)
in L2 (0; T ):
is
(110) Analogously, if is the solutions of (93) with nonhomogeneous term = t and is the solution of (108) with nonhomogeneous term = t , we have x (1; t) ! x(1; t) in L (0; T ): (111) 2
21 The proof of this lemma is similar to the proof of Lemma 1 and we omit it. We consider 2 C (0; T ; L (0; 1)): By Lemma 2 there exists 2 C (0; T ; LK ) such that (109) holds. Then, formula (106) gives, in particular ZTZ ZT K (u ) dxdt = ; f (t) x (1; t)dt: (112) 1
2
1
( )
1
0
( )
0
0
Passing to the limit, as ! 0, in formula (112) and taking into account (105), (110) and the weak convergence of the controls, we obtain that p satis es ZTZ ZT pdxdt = ; f (t) x(1; t)dt; 8 2 L (0; T ; L (0; 1)): (113) 1
0
1
0
2
0
On the other hand, as u is characterized by (107) we deduce that p = u. Let us see now that ut = q=. The projection K ut can be characterized by the formula ZT ZT < K ut; >H ;1;H01 dt = f (t)x (1; t)dt; 8 2 C (0; T ; H (0; 1)) (114) ( )
0
where
1
( )
1 0
0
is the solutions of (100). On the other hand, ut can be characterized by ZT ZT < ut; >H ;1;H01 dt = f (t)x(1; t)dt; 8 2 C (0; T ; H (0; 1)) (115) 1
0
1 0
0
where is the solutions of the following system: 8 > < (x)tt ; xx = t(x; t); 0 < x < 1; 0 < t < T; (0; t) = (1; t) = 0; 0 < t < T; > : (x; 0) = t(x; 0) = 0; 0 < x < 1:
(116)
We consider 2 C (0; T ; H (0; 1)): By Lemma 2 there is 2 C (0; T ; LK ) such that (109) holds. Then, formula (114) gives, in particular ZT ZT 1 < K ut ; >H ;1;H0 dt = ; f (t)x (1; t)dt: (117) 1
1 0
1
( )
0
( )
0
Passing to the limit, as ! 0, in formula (117) and taking into account (105), (111) and the weak convergence of the controls, we obtain that q satis es ZT ZT < q; >H ;1;H01 dt = f (t)x(1; t)dt; 8 2 C (0; T ; H (0; 1)): (118) 1
0
1 0
0
On the other hand, as ut is characterized by (115) we deduce that q = ut=. Step 7: Strong convergence of the projections of the solutions. We start with the convergence of K u. It suces to prove that for any sequence 2 L (0; T ; L (0; 1)) with (119) * weakly in L (0; T ; L (0; 1)) the following holds ZTZ ZTZ udxdt: (120) K u dxdt ! 1
( )
1
2
1
0
1
( )
0
2
0
0
Here, can be spanned in terms of the eigenfunctions and then we can write the rst term in (120) as ZTZ ZTZ K u dxdt = K u K dxdt: (121) 1
0
0
1
( )
0
0
( )
( )
22 By formula (106) we have in particular that ZTZ ZT K uK dxdt = ; f (t) x (1; t)dt; 1
0
( )
0
( )
0
where is the solution of (93) with = K . On the other hand, formula (107) give us ZTZ ZT udxdt = ; f (t) x(1; t)dt;
(122)
( )
1
0
(123)
0
0
where is the solution of (108). Let us see that the right hand term of (122) converges to the right hand term of (123).
Lemma 3 Consider a sequence 2 L (0; T ; L (0; 1)) which veri es (119). Then, the 1
solution
of (93) with = K () converges to the solution *
Moreover,
2
of (108) in the class
weakly in C (0; T ; H01(0; 1)) \ C 1(0; T ; L2(0; 1)): x (1; t) * x (1; t)
Proof: Observe that the solution (x; t) =
weakly in L2 (0; T ):
(124) (125)
of (93) with = K can be written as
Zt 0
( )
(x; t ; s; s)ds
(126)
where is the solution of 8 > 0 < x < 1; 0 < t; s < T; < (x) tt ; xx = 0; (0; t; s) = (1; t; s) = 0; 0 < t; s < T; > : (x; 0; s) = 0; t(x; 0; s) = K (x; s); 0 < x < 1; 0 < s < T:
(127)
( )
From (119) it is easy to see that
K *
(128)
( )
(The boundness of K is an obvious consequence of the boundness of while for the identi cation of the limit we can use Lemma 1 as in Step 2) A classical result in homogenization (see [3]) shows that ( )
(x; t; s) ! (x; t; s) weakly in C (0; T ; H (0; 1)) \ C (0; T ; L (0; 1)) 1 0
1
2
for all s 2 [0; T ] where is the solution of 8 > 0 < x < 1; 0 < t; s < T; < tt ; xx = 0; (0 ; t; s ) = (1 ; t; s ) = 0 ; 0 < t; s < T; > : (x; 0; s) = 0; t(x; 0; s) = ; 0 < x < 1; 0 < s < T: Furthermore, for any l 2 L (0; 1) and s 2 [0; T ]; Z Z (x) t (x; t; s)l(x)dx ! t(x; t; s)l(x)dx strongly in C ([0; T ]):
(129)
(130)
2
1
0
1
0
(131)
23 From (126) and (129) we obviously deduce (124). Now we check (125). Observe that by Proposition 4 x(1; t; s) is uniformly bounded in L (0; T ) for all s. Then x(1; t; s) * p(t; s) weakly in L (0; T ): (132) Let us see that p(t; s) = x(1; t; s) Multiplying (127) by the test function x (t) and integrating by parts we obtain the following expression for : Z TZ ZTZ ZT (x) t t(t)dxdt ; x (t)dxdt = x(1; t; s) (t)dt; 8 2 C (0; T ): (133) Analogously, multiplying the equations in (130) by x (t) we have ZTZ ZTZ ZT t t(t)dxdt ; x (t)dxdt = x(1; t; s) (t)dt; 8 2 C (0; T ): (134) Taking into account (129) and (131) the rst two terms in (133) converge to the rst two terms in (134). Then ZT ZT x(1; t; s) (t)dt = x(1; t; s) (t)dt; 8 2 C (0; T ) (135) 2
2
1
0
1
0
0
0
1
0
1 0
0
1
0
0
0
1 0
0
0
1 0
0
and we deduce that p(t; s) = x(1; t; s). Finally observe that for any 2 L (0; T ) 2
ZT 0
x (1; t; s) (t)dt
=
! =
ZTZt ZTZT 0
0
s
ZT 0
x(1; t ; s; s)ds (t)dt = x(1; t ; s; s) (t)dtds =
ZTZT s
ZTZt 0
0
0
x(1; t ; s; s) (t)dtds x(1; t ; s; s)ds (t)dt
x (1; t; s) (t)dt:
0
This concludes the proof of Lemma 3. By Lemma 3 and the strong convergence of the controls we deduce that right hand term of (122) converges to the right hand term of (123) and then (120) holds. We study now the convergence of K ut. Once again it is enough to prove that for any sequence 2 C (0; T ; H (0; 1)) with * weakly in L (0; T ; H (0; 1)) (136) the following holds ZT ZT < K ut ; >H ;1;H01 dt ! < ut; > dt: (137) 1
( )
1 0
1
( )
0
1 0
0
Observe that ZT ZT < K ut ; >H ;1;H01 dt = < K ut; K >H ;1;H01 dt: ( )
0
0
( )
( )
By formula (114) we have in particular that ZT ZT < K ut ; K >H ;1;H01 dt = f (t)x (1; t)dt; 0
( )
( )
0
(138)
24 where is the solution of (100) with = K . On the other hand, formula (115) give us ZT ZT < ut; >H ;1;H01 dt = f (t)x(1; t)dt; ( )
0
(139)
0
where is the solution of (116). Now we need the following lemma:
Lemma 4 Consider a sequence 2 C (0; T ; H (0; 1)) which veri es (136). Then, the 1
1 0
solution of (100) with t = K ()t converges to the solution of (116) in the class
! weakly in C (0; T ; H (0; 1)) \ C (0; T ; L (0; 1)):: 1 0
Moreover,
1
2
x (1; t) ! x(1; t) weakly in L (0; T ): 2
Proof: The solution of (100) with t = K t can be written as ( )
Z tZ t @ (x; r ; s; s)drds (140) @x is the derivative with respect to the third component and is the solution (x; t) =
0
0
3
where @x@ 3 of 8 > 0 < x < 1; 0 < t; s < T; < (x)tt ; xx = 0; (0; t; s) = (1; t; s) = 0; 0 < t; s < T; > : (x; 0; s) = K (x; s); t(x; 0; s) = 0; 0 < x < 1; 0 < s < T:
(141)
( )
This system is analogous to system (127) and from now on, the proof is similar to the proof of Lemma 3. By Lemma 4 and the strong convergence of the controls we deduce that left hand term of (137) converges to the left hand term of (138) and then (136) holds. Step 8: Proof of Lemma 1. Consider (v ; v ) 2 H (0p; 1) L (0; 1): We recall that the eigenfunctions 'n (x) = sin(nx= ) of pthe limit system (75) constitute an orthonormal basis of H (0; 1) and !n (x) = ('n; n 'n) constitute an orthonormal basis of H (0; 1) L (0; 1) with the norm Z Z k(v ; v )kH01L2 ; = jv0 (x)j dx + jv (x)j dx: 0
1 0
0
1 0
2
1 0
2
1
2
1
1
(0 1)
0
0
1
2
0
1
2
So, we can expand (v ; v ) in Fourier series as follows: X (v ; v ) = an!n (x): 0
1
0
1
n2IN
Recall also that the eigenfunctions 'n (x)qof system (58) constitute an orthonormal basis of H (0; 1) and then !n (x) = ('n; n 'n ) constitute an orthonormal basis of H (0; 1) L (0; 1) with the norm Z Z 0 k(v ; v )kH01L2 ; = jv (x)j dx + jv (x)j dx: 1 0
1 0
2
0
1
1
2
(0 1)
0
0
1
2
0
1
2
25 De ne
(v ; v ) = 0
where r() = ;1=2.
1
It is clear that
r X
( )
an!n (x)
n=1
(v ; v ) 2 LK LK : 0
( )
1
( )
We now prove the convergence stated in (85). To simplify the notation we only prove the convergence of the rst component. Observe that
r X X
kv ; v kH01 ; =
an'n (x) ; an'n(x)
n n2IN H01 ;
r
1
X
X
an ['n(x) ; 'n(x)]
+
an'n(x)
n r
H 1 ; n H01 ; 0r 1 = 0r 1 = 00 1 1= X X X @ janj A @ k'n (x) ; 'n (x)kH01 ; A + @ janj A : (142) ( )
0
0
(0 1)
=1
(0 1)
( )
=1
= ( )
(0 1)
1 2
( )
( )
1 2
2
2
n=1
(0 1)
1 2
2
(0 1)
n=1
As the last series is convergent, we deduce that 0 1 1= X @ janj A ! 0 1 2
n=r()
as ! 0:
2
n=r()
(143)
Concerning the rst term in (142) we use the following result on the convergence of eigenfunctions which is proved in [5]:
Lemma 5 Let 2 L1(IR) be a periodic function such that 0 < m < (x) < M < 1
a.e. in IR. Then, there exist constants C; c > 0 such that the eigenvalues n and eigenfunctions 'n of (58) with n C;1 verify
q q n ; n cn ; k'n ; 'n kW 1;1 ; cn ;
(144) (145)
3 2
3 2
(0 1)
where n and 'n are the eigenvalues and eigenfunctions of the limit system (75).
The rst term in (142) can be bounded by 0r 1= 0 r 1= X @ janj A @c X n A 1 2
( )
2
1 2
( )
2 4
n=1
6
n=1
kv kH kv kH 0
+ kv kL2
;
+ kv kL2
1 0 (0 1)
kv kH 0
;
1 0 (0 1)
0
;
+ kv kL2
1 0 (0 1)
1
1
1
;
(0 1)
;
(0 1)
;
(0 1)
c 1 +
Zr
( )+1
!=
1 2
s ds != ( r ( ) + 1) c 1 + 7 ; = + 1) ! = ( c 1 + 7 2
1
1 2
(146)
1 2
7
2
2
6
7
1 2
(147)
26 which converges to zero as ! 0. Combining (142), (143) and (147) we deduce that kv ; v kH01 ; ! 0; as ! 0 as we wanted to prove. For the proofs of (86) and (87) we can proceed in a similar way taking into account that the solution v(x; t) of (58) and vx(1; t) can be also represented in Fourier series of the eigenfunctions of (58), and the good convergence of the spectra stated in Lemma 5. 0
0
(0 1)
5 Appendix I In this section we prove the regularity result stated in Proposition 1 and the observability one stated in Proposition 2. In fact we prove more general results that do not require the periodicity of the density. Let us consider the wave equation 8 > 0 < x < 1; 0 < t < T; < (x)vtt ; vxx = 0; v(0; t) = v(1; t) = 0 0 < t < T; (148) > : v(x; 0) = v (x); vt(x; 0) = v (x) 0 < x < 1: 0
1
Note that (148) corresponds to (36) with = 1. The following holds: Proposition 1 Assume that (x) 2 W ;1(0; 1) with 0 < m (x) M < 1 a.e. x 2 [0; 1]. Then, for any T > 0 we have ZT h i jvx(1; t)j dt kkW 1;1 ; + 1 T + kkL1 ; (;m + 1)k(v ; v )kH01 ; L2 ; : 1
2
(0 1)
0
1
(0 1)
0
1
2
(0 1)
(0 1)
Proposition 2 Assume that (x) 2 BV (0; 1) with 0 < m (x) M < 1 a.e. x 2 [0; 1]. Then, for any T > 2pM we have ZT ;m1 TV p ; 2 minfm; 1g(T ; 2 M )e k(v ; v )kH01 ; L2 ; jvx (1; t)j dt (149) ( )
0
1
2
(0 1)
2
(0 1)
for every (v0 ; v1) 2 H01 (0; 1) L2 (0; 1) with v solution of (148).
0
Proof of Proposition 1: Multiplying the equation in (148) by xvx and integrating
by parts we easily obtain ZTZ ZTZ 0 = (x)vttxvxdxdt ; vxxxvxdxdt Z T Z T Z x d ZTZ jv j dxdt = ; (x)vtxvxtdxdt + (x)vtxvxdx ; 2 dx x Z T Z T Z (x)x d = ; j v j dxdt + ( x ) v xv dx t x 2 dx t Z T Z jvxj Z T jvx(1; t)j + dxdt ; 2 2 dt Z T Z T jvx(1; t)j ZTZ h i 1 0 = 2 ((x)x) jvtj + jvxj dxdt + (x)vtxvxdx ; 2 dt: 1
0
1
0
0
0
1
0
1
0
0
1
0
0
2
2
0
0
2
0
0
1
0
0
2
1
0
1
0
0
1
2
2
1
2
0
0
0
0
Hence, we deduce that ZT Z T Z h i 0 jvx(1; t)j dt k(x) kL1 ; + 1 jvtj + jvxj dxdt Z Z + (x)vt(x; T )xvx(x; T )dx ; (x)vt(x; 0)xvx(x; 0)dx: 2
0
1
0
1
(0 1)
0 1
0
2
2
0
(150)
27 We observe now that Z i kxkL1 ; Z h j v (x)vt(x; T )xvx(x; T )dx t(x; T )j + jvx(x; T )j dx 2 Z h i kkL1 ; ; 2 (m + 1) (x)jvt(x; T )j + jvx(x; T )j dx Z h i kk 1 = L2 ; (;m + 1) (x)jvt(x; 0)j + jvx(x; 0)j dx kk 1 = L2 ; (;m + 1)k(v ; v )kH01 ; L2 ; (151) where we have used the conservation of the energy valid for the system (148). In a similar way, Z kk 1 (; + 1) (x)vt(x; 0)xvx(x; 0)dx L ; 2 m k(v ; v )kH01 ; L2 ; : (152) Combining (150), (151) and (152) we obtain ZT h i jvx(1; t)j dt kkW 1;1 ; + 1 T + kkL1 ; (;m + 1)k(v ; v )kH01 ; L2 ; : 1
1
(0 1)
2
2
0
0
(0 1)
1
1
2
2
0
(0 1)
1
1
2
2
0
(0 1)
1
0
2
1
1
(0 1)
(0 1)
1
(0 1)
0
0
2
(0 1)
0
2
1
1
(0 1)
(0 1)
0
1
(0 1)
2
(0 1)
(0 1)
Proof of Proposition 2: Let us introduce the following functional Z T ; F (x) = 12
(1
;x) h
(1;x)
i (x)jvt(x; t)j + jvx(x; t)j dt 2
2
where = pM , and T > 2pM : Observe that ZT 1 F (1) = 2 jvx(1; t)j dt: (153) On the other hand the derivative of F is given by # Z T ; ;x " 0(x) 0 F (x) = (x)vt(x; t)vxt(x; t) + vx(x; t)vxx(x; t) + 2 jvt(x; t)j dt ;x h i ; 2 (x)jvt(x; ; x)j + jvx(x; ; x)j h i ; 2 (x)jvt(x; T ; + x)j + jvx(x; T ; + x)j : Integrating by parts we get Z T ; ;x ; x 0(x)jvt(x; t)j dt + (x)vt(x; t)vx(x; t)jtt T ; x F 0(x) = 21 ;x h i ; 2 (x)jvt(x; ; x)j + jvx(x; ; x)j h i ; 2 (x)jvt(x; T ; + x)j + jvx(x; T ; + x)j : Hence, by our choice of and the fact that 0 < m (x); we have Z T ; ;x 0 jF 0(x)j 21 j0(x)j ;x jvt(x; t)j dt j(x)j F (x): m 2
0
(1
(1
)
2
)
2
2
2
(1
(1
)
2
= + =
2
)
2
2
2
(1
(1
)
)
2
2
28 Using Gronwall's inequality we now deduce that ;m1 R 1 j0 s jds ;m1 R 1 j0 s jds x F (x) e F (1) e 0 F (1) ( )
( )
for each x 2 (0; 1). Thus, the energy of the system E (t); given by Z h i E (t) = (x)jvt(x; t)j + jvx(x; t)j dx; 1
2
2
0
veri es (T ; 2 )E (t) (T ; 2 )E (t ; 2 ) Z Z t; h i 1 2 (x)jvt(x; t)j + jvx(x; t)j dtdx Z R1 F (x)dx e;m1 0 j0 s jdsF (1): 1
2
2
0
1
( )
0
From (153), we nally deduce that ZT R1 jvx(1; t)j dt 2F (1) 2(T ; 2 )e;;m1 0 j0 s jdsE (t) ;1 R 1 = 2(T ; 2 )e;m 0 j0 s jdsE (0) R1 2 minfm; 1g(T ; 2 )e;;m1 0 j0 s jdsk(v ; v )kH01 ( )
2
0
( )
( )
0
1
2
; L2 (0;1):
(0 1)
In the proof we have used the Rfact that 2 W ; (0; 1). The same result holds when 2 BV (0; 1). We then replace j0(s)jds by TV (). 11
1 0
6 Appendix II In this section we give a proof of Proposition 10. Let us recall its statement. Proposition 10 Let 2 L1(IR) be a periodic function such that 0 < m (x) M < 1 a.e. x 2 R; and consider T > 0. Then, for all ((x; t); (x; t)) 2 C (0; T ; LK ) there exists C > 0 such that for any 0 < < 1 the solution of (93) veri es ZT j x(1; t)j dt C kkL1 ;T L2 ; : (154) Moreover, the solution of (97) satis es ZT jx(1; t)j dt C k kL1 ;T H01 ; : (155) 1
( )
2
2
2
0
2
2
0
Proof: Observe that
2
2
(0
(0
;
;
(0 1))
(0 1))
Zt
(x; t) = (x; t ; s; s)ds where (x; t; s) is the solution of 8 > 0 < x < 1; 0 < t < T; < (x) tt ; xx = 0; (0 ; t; s ) = (1 ; t; s ) = 0 ; 0 < t < T; > : (x; 0; s) = 0; t(x; 0; s) = (x; s); 0 < x < 1: 0
(156)
29 By Proposition 5 we have k x(1; t; s)kL2 ;T C k(x; s)kL2 ; : (157) On the other hand, using Minkowski inequality
Z T
Z t
k x(1; t)kL2 ;T =
x(1; t ; s; s)ds
2 =
;t (s) x(1; t ; s; s)ds
L ;T L2 ;T Z T
;t x(1; t ; s; s) L2 ;T ds (158) where ;t (s) is the characteristic function of the interval (0; t). Hence, the above expression is equal to ZT ZT ZT k x(1; t ; s; s)kL2 s;T ds k x(1; r; s)kL2 ;T ds C k(x; s)kL2 ; ds = C kkL1 ;T L2 ; (159) and we have proved (154). Now, we prove (155). Observe that Zt (x; t) = (x; t ; s; s)ds where 8is the solution of > 0 < x < 1; 0 < t < T; < (x)tt ; xx = 0; (0; t; s) = (1; t; s) = 0; 0 < t < T; (160) > : (x; 0; s) = 0; t(x; 0; s) = s(x; s); 0 < x < 1: On the other hand, (x; t; s) = R t @s@ W (r; s)dr where W is solution of 8 > 0 < x < 1; 0 < t < T; < (x)Wtt ; Wxx = 0; W (0 ; t; s ) = W (1 ; t; s ) = 0 ; 0 < t < T; (161) > : W (x; 0; s) = (x; s); Wt(x; 0; s) = 0; 0 < x < 1: By Proposition 5 we have kWx(1; t; s)kL2 ;T C k (x; s)kH01 ; : (162) Now, observe that Z t Z t;s @ Z t d Z t;s (x; t) = W (x; r; s)drds = ds W (x; r; s)dr ds @s Zt + W (x; t ; s; s)ds Z t;s s t Z t Zt = W (x; r; s)dr + W (x; t ; s; s)ds = W (x; t ; s; s)ds s and then, taking into account (162) and proceeding as in (158) and (159) we easily deduce (155). (0
(0
)
2
)
0
(0
(0 )
0
)
(0 )
0
(0 1)
(0
(0
)
)
(0 )
(
0
)
(0
0
2
(0
;
2
)
0
(0 1)
(0 1))
0
0
(0
0
2
)
0
0
(0 1)
0
0
=
0
=0
0
0
Acknowledgments. The author wishes to express his gratitude to E. Zuazua for
suggesting the problem and for many stimulating conversations. This work was part of the Ph.D. Thesis of the author in \Universidad Complutense de Madrid" in 1996. The author acknowledges the doctoral grant received from \Universidad Complutense". This work was written while the author was visiting the CEA (France) with the support of the DGES (Spain). The hospitality and support of CEA is acknowledged. The author was also partially supported by grants PB96-0663 of the DGES (Spain) and CHRX-CT94-0471 of the European Union.
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