Remark 1 Estimate (4) is not uniform in due to the fact that

Texto completo

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Boundary controllability of the one-dimensional wave

equation with rapidly oscillating density.

C. Castro

Abstract. We consider the one-dimensional wave equation with

peri-odic density of period ! 0 in a bounded interval. By a

counterex-ample due to Avellaneda, Bardos and Rauch we know that the exact controllability property does not hold uniformly as !0 when the

con-trol 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 L

1(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

>

<

>

:

(x)utt;uxx = 0; 0< x < 1; 0 < t < T;

u(0;t) = 0; u(1;t) = f(t); 0< t < T; u(x;0) = u0(x); ut(x;0) = u1(x); 0 < x < 1:

(1)

Departamento de Matematica Aplicada, Universidad Complutense de Madrid, 28040 Madrid,

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2

In (1) f 2L

2(0;T) is a boundary control function acting on the extreme x = 1 of the

string and (u0;u1) 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 2W

1;1(0;1). System (1) is well-posed in the energy space

L2(0;1) H

;1(0;1) in the sense that, given T > 0, if

(u0;u1) 2L

2(0;1) H

;1(0;1) and f 2L

2(0;T) (2)

the solution u of (1) veries

u(x;t)

2C([0;T];L 2

(0;1))\C 1

([0;T];H;1

(0;1)): (3)

Furthermore, we can estimate the norm of the solution as follows

k(u

;ut)

kL 1

(0;T;L 2

(0;1)H ;1

(0;1))

Ck

k 1=2

W1;1 (0;1)

kfkL 2

(0;T)+ k(u

0;u1) kL2

H ;1

(4)

where C > 0 is a constant which only depends onm and M.

Remark 1

Estimate (4) is not uniform in due to the fact that kkW 1;1

(0;1)

;1.

This leads to a bound for the solution which blows up as !0.

Inequality (4) can be obtained using transposition and a suitable boundary reg-ularity 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 con-trollability for which the following classical result is known:

Theorem 2

Assume that 2BV (0;1). Then, for any T > 2 p

M; 0 < < 1 and

(u0;

u

1) 2L

2(0;1) H

;1(0;1) (5)

there exists a control f 2L

2(0;T) such that the solution of (1) with f = f veries

u(T)ut(T)0: (6)

Moreover, we can estimate the norm of the controls as follows:

kfkL2 (0;T)

C

2exp(C3TV (

))k(u 0;u1)

kL2 (0;1)H

;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 ;1. 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

L2(0;T) which drive the system to the equilibrium. The so-called Hilbert Uniqueness

Method (HUM) (see [10]) provides the one with minimalL2-norm. The control obtained

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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 L2-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:

Theorem 3

There exist periodic functions 2L

1(IR) with0<

m (x)M <1

a.e. x 2 R such that the following holds: For any T > 2 p

M, there exists a sequence

j !0and a bounded sequence of initial data (u

j

0 ;u

j

1) 2L

2(0;1) H

;1(0;1) for which

the controlsfj of minimal L2-norm which drive the system (1) to the equilibrium verify

fj

L2 (0;T)

C 2e

C3=j

!1 as j !0; (8)

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 5

When !0, the coecient (x) = (x=) converges weakly to its average

=R 1

0 (s)ds and then system (1) converges in some sense to the following one: 8

>

<

>

:

utt;uxx = 0; 0< x < 1; 0 < t < T;

u(0;t) = 0; u(1;t) = f(t); 0< t < T; u(x;0) = u0(x); ut(x;0) = u1(x); 0 < x < 1:

(9)

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 2C

2(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 L

1(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 when2L

1(IR). However, when dealing with the high frequencies,

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4

wavelength greater than . Our results are derived from some uniform observability inequalities for the adjoint system. These inequalities concern the solutions which con-tain 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 L2(0;1) with the natural norm

kuk 2

L2 (0;1)=

Z

1

0

(x)

ju(x)j

2dx: (10)

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

kuk 2

L2 (0;1)=

Z

1

0

ju(x)j

2dx (11)

with uniform constants in. For this reason we will not make explicit the dependence on of this L2-norm.

We also consider the subspace H1

0(0;1) with the norm

kuk 2

H1 0

(0;1)= Z

1

0 ju

0(x) j

2dx:

2 Statements of the main results

Let us introduce a suitable decomposition of the space L2(0;1) in terms of the

eigen-functions of the eigenvalue problem associated to (1). Consider the eigenvalue problem

(

'00+(x)' = 0; x

2(0;1);

(5)

5 For each > 0, there exists a sequence of eigenvalues

0<

1 <

2 < ::: < n < :::

!1 (13)

and a sequence of associated eigenfunctions ('n)n2N which can be chosen to constitute

an orthonormal basis inL2(0;1) with the norm jj'jj

2

L2 (0;1)=

R

1

0

(x)j'(x)j

2dx: Observe

that

L2(0;1) = span

f('n)n 2N

g: (14)

Multiplying the equation in (12) by any eigenfunction and integrating by parts we easily deduce that the eigenfunctions are also orthogonal in H1

0(0;1) with the norm jj'jj

2

H1 0

(0;1)= R

1

0 j'

0

j

2. In fact, ('n= q

n)n2N constitutes a orthonormal basis inH 1

0(0;1).

Given continuous increasing functions K();M() such that K();M() ! 1 as

!0, we introduce the spaces

LK() =span f('n)n

K()

g and H

M()=span f('n)n

M()

g (15)

and its associated orthogonal projections K() : L

2(0;1)

!LK ()

M() : L2(0;1)

!HM ()

\L 2(0;1):

Observe that

L2(0;1) = L

K()

spanf('n)K

()nM() g

HM() \L

2(0;1)

: (16)

Note thatLK() is a nite dimensional space generated by the low frequencies while

HM() 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

H1

0(0;1) = LK ()

spanf('n)K

()nM() g

HM() \H

1

0(0;1)

: (17)

The negative result of Theorem 3 is due to the singular behavior of the eigenfunctions 'n with wavelength of the order of , i.e. n

;1. More precisely, the eigenfunctions

which can exhibit a singular behavior are those for which its associated eigenvaluesn

verify

q

n 2I (18)

whereI is the set of all so-called instability intervals of the associated Hill equation u00(y) + (y)u(y) = 0; y

2R: (19)

Using the Rayleigh formula it is easy to check that n=p

M

q

n n= p

m for

any 0< < 1 and n2IN. On the other hand, as I is constituted by non-empty closed

intervals we deduce that (18) only happens when q

nminx

2Ix > 0; i.e.

n(min

x2I

x)p

m;1;1: (20)

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6

Theorem 4

Let2L

1(IR) be a periodic function such that0<

m (x)M <1

a.e. x2R; and consider T > 2 p

= 2q R

1

0 (s)ds. Then, if K()

D

;1 with D small

enough, for any(u0;u1) with(u0;

u

1) 2 L

2(0;1) H

;1(0;1) and0< < 1there exists

f 2L

2(0;T) such that the solution u of (1) veries

(K()u

(T);K

()ut(T)) = 0: (21)

Moreover, there exist constants C1;C2 > 0 independent of such that

kfkL2 (0;T)

C 1

k(K ()u0;

K

()u1) kL2

(0;1)H ;1

(0;1); (22) k(K

()u

;K

()ut) kL1

(0;T;L 2

(0;1)H ;1

(0;1)) C

2 kfkL2

(0;T); (23)

for any (u0;u1) and 0< < 1.

Assume that there exists a subsequence, still denoted !0; such that

u

1 * u

1 weakly in H

;1(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 L

1(0;T;L2(0;1) H

;1(0;1)) (25)

f * f weakly in L2

(0;T) (26)

where (u;ut) and f(t) are the solution and the control respectively of the limit system

8

>

<

>

:

utt;uxx = 0; 0< x < 1; 0 < t < T;

u(0;t) = 0; u(1;t) = f(t); 0< t < T; u(x;0) = u0(x); ut(x;0) = u

1(x)=; 0 < x < 1:

(27)

Moreover, if u

1 !u

1 strongly in H

;1(0;1); then

(K()u

;K

()ut)

!(u;ut) strongly in L

1(0;T;L2(0;1) H

;1(0;1)) (28)

f !f strongly in L

2(0;T) (29)

Remark 6

This result, roughly, establishes the continuity of the controls with respect to

!0. The controlsf 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(u0;u1) for system (1)

are xed. However, our proof can be easily adapted to more general cases. In particular, if we consider (u

0;u

1) with

(u

0;

u

1)* (u

0;u

1) weakly in L 2

(0;1)H ;1

(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

>

<

>

:

utt;uxx = 0; 0< x < 1; 0 < t < T;

u(0;t) = 0; u(1;t) = f(t); 0< t < T; u(x;0) = u

0(x)=; ut(x;0) = u

1(x)=; 0 < x < 1

(7)

7

instead of (27).

Moreover, if we assume that

(u

0;

u

1) !(u

0;u

1) strongly in L 2(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 WN

+1;1(IR) (N

1) be a periodic function such that 0 < m

(x) M < 1 a.e. x 2 R; and consider T > 2 R

1

0 q

(x)dx. Then, if M()

D;1;1=N with D large enough, for any (u 0;u1)

2 L 2(0;1)

H

;1(0;1) and 0 < < 1

there exists f 2L

2(0;T) such that the solution u of (1) veries

(M()u(T);M()u

t(T)) = 0: (31)

Moreover, there exist constants C1;C2 > 0 independent on such that

kfkL2 (0;T)

C 1

k(M ()u

0;

M()u 1)

kL2 (0;1)H

;1

(0;1); (32) k(M

()u;M()u

t)kL1 (0;T;L

2

(0;1)H ;1

(0;1)) C

2 kfkL2

(0;T); (33)

for any (u0;u1) 2L

2(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()u

t)* (0;0) weakly in L1(0;T;L2(0;1) H

;1(0;1)) (34)

f * 0 weakly in L2

(0;T): (35)

Moreover, ifu

1 !u

1 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 of the solution over the low eigenfrequencies (Theorem 4) is larger than the time 2R

1

0 q

(x)dxneeded for the higher eigenfrequencies (Theorem 5).

Remark 10

In Theorem 5 we assume that u1 2H

;1(0;1) instead of u 1

2H

;1(0;1)

which is the assumption in Theorem 4. The reason is that both assumptions are equiva-lent when2H

1 which is the case in Theorem 5 where we have simplied 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 solu-tions contain eigenfuncsolu-tions '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 2C

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 withn 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

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8

3 Observability of the adjoint system

The transposition method and the Hilbert Uniqueness Method (HUM) provide an equiv-alence between the results of the previous sections and some suitable regularity and ob-servability 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 decom-position 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 be-tween 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) = v0(x); vt(x;0) = v1(x); 0 < x < 1:

(36)

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)2W

1;1(0;1). There exists a constant C

1 > 0 such

that Z T

0

jvx(1;t)j 2dt

C 1

k(x=)kW 1;1

(0;1) k(v

0;v1) k

2

H1 0

(0;1)L 2

(0;1) (37)

for every (v0;v1) 2H

1

0(0;1) L

2(0;1) with 0< < 1 and v solution of (36).

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 esti-mate (7) are derived from the HUM and the following clasical boundary observability property:

Proposition 2

Assume that (x)2 BV (0;1) is periodic with 0< m (x)M < 1 a.e. x 2[0;1]. Then, given T > 2

p

M; we have

2minfm;1g(T ;2 p

M)e; ;1

mTV() k(v

0;v1) k

2

H1 0

(0;1)L 2

(0;1)

Z T 0

jvx(1;t)j

2dt (38)

for every (v0;v1) 2H

1

0(0;1) L

(9)

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 L

1(IR) be a periodic function such that 0 <

m (x)

M <1 a.e. x2R;. LetT > 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 vj of

(36) satisfying

Z T 0

jv

j

x(1;t)j 2dt

C 1

;1

j k(v

j

0 ;v

j

1 ) k

2

H1 0

(0;1)L 2

(0;1): (39)

(b) There exists a sequence j ! 0 and a sequence of solutions vj of (36) which

veries Z T

0

jvxj(1;t)j 2dt

C 2e

;C 3=j

k(v

j

0;v

j

1 ) k

2

H1 0

(0;1)L 2

(0;1): (40)

Furthermore, there exist non-constant periodic functions such that (a) holds. The same is true with (b).

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 pointx = 1of 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 spacesL2(0;1) and H1

0(0;1) into the low and high frequency spaces

given in (16) and (17). We have the following:

Proposition 4

Let 2 L

1(IR) be a periodic function such that 0 <

m (x)

M < 1 a.e. x 2 R; and consider T > 2 p

. Then, if K() D

;1 with D small

enough, there exist constants C1, C2 > 0 such that for any (v0;v1) 2LK

() LK

() and

0< < 1 the solution v of (36) veries

C1 k(v

0;v1) k

2

H1 0

(0;1)L 2

(0;1)

Z T 0

jvx(1;t)j 2dt

C 2

k(v 0;v1)

k 2

H1 0

(0;1)L 2

(0;1): (41)

Proposition 5

Let 2 WN

+1;1(IR) with N

1 be a periodic function such that

0 < m (x) M < 1 a.e. x 2 R; and consider T > 2 R

1

0 q

(10)

10

M()D

;1;1=N with D large enough, there exist constants C

1, C2 > 0 such that for

any(v0;v1)

2(HM ()

\H 1

0)

(HM ()

\L

2)and 0< < 1the solution v of (36) veries

C1 k(v

0;v1) k

2

H1 0

(0;1)L 2

(0;1)

Z T 0

jvx(1;t)j 2dt

C 2

k(v 0;v1)

k 2

H1 0

(0;1)L 2

(0;1): (42)

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 eigenval-ues of (36) with respect to . We consider three dierent 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 prop-erty 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 L

1(IR) is a periodic function. Given > 0; there

exists a constant C() > 0 such that

q

n+1 ;

q

n

p

;; (43)

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 nc ;1:

C1 Z

1

0 j('n)

0(x) j

2

j('n) 0(1)

j 2

C 2

Z

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

that2WN

+1;1(IR) for some N

1. Given > 0, there exists a constant C > 0 such

that if nC

;1;1=N we have q

n+1 ;

q

n

R

1

0 q

(s=)ds ;: (45)

Furthermore, there exist C1;C2;c > 0 such that the following estimates hold for the

eigenfunctions 'n with nc

;1;1=N:

C1 Z

1

0 j('n)

0

(x)j 2

j('n) 0(1)

j 2

C 2

Z

1

0 j('n)

0

(x)j 2

(11)

11

3.2.3 Critical case

Proposition 8

Consider 2 L

1(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

1

0 j'

0

j(x)j 2dx

j' 0

j(1)j 2

Cj: (47)

(b)

There exist a sequence j ! 0, a sequence of eigenfunctions 'j of (12) and

positive constants C1;C2 > 0 such that R

1

0 j'

0

j(x)j 2dx

j' 0

j(1)j 2

C 1je

C2=j; (48)

Furthermore, there exist smooth non-constant periodic functionsand j !0such that

(a) holds. The same is true for (b).

Remark 14

Proposition 8 guarantees that there are not constantsC1;C2 > 0such that

C1 Z

1

0 j('n)

0(x) j

2dx

j('n) 0(1)

j 2

C 2

Z

1

0 j('n)

0(x) j

2dx (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 + x~ 0) satises (a). The

same can be said about (b).

(12)

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 eigenvaluesfngn

2IN and a sequence of

associated eigenfunctions f'ngn

2IN which can be chosen to constitute an orthonormal

basis inH1

0(0;1). Dene q

k =; q

;k and '

;k ='k; k 2ZZ.

Then, the solutions v of (36) can be represented as

v(x;t) = X

k2ZZ

ak'k(x)eip

kt

wherefakgk

2Z are complex Fourier coecients determined by the initial data.

Given > 0, we consider !k = ('k;q

k'k). Thenf!kgk

2ZZ constitutes an

orthonor-mal basis of the energy spaceH1 0(0;1)

L

2(0;1) with the norm

k(u;v)k= Z

1

0 h

juxj 2+

jvj 2

i

dx 1=2

: So, any nite energy solution of (36) can be represented as

(v(x;t);vt(x;t)) = X

k2ZZ

ak!k(x)eip

kt; with coecients (ak)

2l 2;

and

k(v

;vt)

k 2

=

X

k2ZZ jakj

2:

Inequalities (41) can be reduced to nd constants C1;C2 > 0 such that

C1 X

jkjK(e) jakj

2

Z T

0

X

jkjK()

akeip

kt('k)0

(1)

2

dtC 2

X

jkjK() jakj

2

: (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 C1;C2 > 0 such that

C1 X

jkjK() jakj

2

Z T

0

X

jkjK()

akeip

kt

2

dtC 2

X

jkjK() jakj

2: (51)

(13)

13

Theorem 6

(Ingham [9]) Let (k)k2ZZ be a sequence of real numbers such that

k+1

;k > > 0. (52)

Then, for any T > 2= there exist constants C1;C2 > 0, only depending on , such

that for any f =P

k2ZZcke

ikt 2L

2(0;T) we have

C1 X

k2ZZ jckj

2

Z T

0

X

k2ZZ

ckeikt

2

dtC 2

X

k2ZZ jckj

2

: (53)

The constants C1; C2 in (53) depend onT and but not in the sequence

fkgsatisfying

(52).

Step 3:

Observability of eigenfunctions. Using the uniform observability of the eigenfunctions stated in Proposition 6 we easily obtain the existence of constants C1;C2 > 0 such that

C1 Z T

0

X

jkjK()

akeip

kt

2

dt

Z T 0

X

jkjK()

akeip

kt('k)0

(1)

2

dt

C

2 Z T

0

X

jkjK()

akeip

kt

2

dt (54)

for anyT > 0 and (ak)2l

2. Combining Proposition 9 and (54) we obtain the

inequal-ities in (50). In view of the Step 1 this conludes the proof of Proposition 4.

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

0 j'

0

j(x)j 2dx

j' 0

j(1)j 2

Cj: (55)

Consider j the eigenvalue associated to 'j. Then

vj = cos(p

jt)'j(x)

is a solution of (36) which veries:

Z T 0

jv

j

x(1;t)j 2dt =

j' 0

j(1)j 2

Z T 0

cos2( p

jt) =j' 0

j(1)j 2 1

2 + sin(2

p

jT)

4p

j

!

: (56) On the other hand

k(v

j(x;0);vj(x;0))k 2

H1 0

(0;1)L 2

(0;1) = Z

1

0 j'

0

j(x)j 2

dx +Z 1

0

(x)

j'j(x)j 2

dx

=

1 + 1j

Z

1

0 j'

0

j(x)j

2dx: (57)

(14)

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

>

<

>

:

(x)vtt;vxx= 0; 0< x < 1; 0 < t < T;

v(0;t) = v(1;t) = 0; 0< t < T;

v(x;T) = v0(x); vt(x;T) = v1(x); 0 < x < 1:

(58)

Due to the time reversibility of the wave equation, system (58) is well-posed backwards in time. So, given (v0;v1)

2 H 1

0(0;1) L

2(0;1) there exists an unique solution v of

(58) in the class

v(x;t)

2C(0;T;H 1

0(0;1)) \C

1(0;T;L2(0;1)): (59)

Moreover, the energy of the system is conserved: E(t) = Z

1

0

jvx(x;t)j 2dx +

Z

1

0 (x)

jv(x;t)j 2dx

= Z 1

0 jv

0;x(x) j

2dx + Z

1

0 (x) jv

1(x) j

2dx = E(0);

8t2[0;T]:

Due to Proposition 4, there exist positive constants C1;C2 such that when (v0;v1) 2

LK() LK

() we have

C1 k(v

0;v1) k

2

H1 0

(0;1)L 2

(0;1)

Z T 0

jvx(1;t)j 2dt

C 2

k(v 0;v1)

k 2

H1 0

(0;1)L 2

(0;1) (60)

for all 0< < 1.

Multiplying the equations (1) by v and integrating by parts we easily obtain the

following identity:

< (x)ut(x;T);v

0(x) >H ;1;H

1

0 ;

Z

1

0

(x)u(x;T)v

1(x)dx

; <

(x)u

0(x);v

(x;0) >H

;1;H 1

0 + Z

1

0

(x)u

0(x)vt(x;0)dx + Z T

0

f(t)vx(1;t)

= 0: (61)

Here<;>H;1;H1

0 represents the duality product between H 1

0 and its dualH ;1.

As the eigenfunctions are orthogonal inL2(0;1) and v 1

2LK

() which is a subspace

generated by the rst K() eigenfunctions we can simplify the second term in (61) as

follows Z

1

(x)u(x;T)v

1(x)dx = Z

1

(x)K

()(u

(x;T))v

(15)

15 In the same way, we can simplify the rst term in (61). To do this we observe that both ut(x;T) and v0 can be spanned in terms of the eigenfunctions 'n and that

< 'n;'m >H

;1;H 1

0= Z

1

0

(x)'n(x)'m(x)dx (63)

which is nonzero if and only if m = n. So, as v0 2LK

() we easily obtain

< (x)ut(x;T);v

0(x) >H ;1;H1

0=<

(x)K

()(ut(x;T));v0(x) >H ;1;H1

0 (64)

Using (62) and (64) we can simplify (61) into < (x)K

()(ut(x;T));v0(x) >H ;1;H1

0 ;

Z

1

0

(x)K

()(u

(x;T))v

1(x)dx

; <

(x)u

0(x);v

(x;0) >H

;1;H 1

0 + Z

1

0

(x)u

0(x)vt(x;0)dx + Z T

0

f(t)vx(1;t)

= 0: (65)

Now, observe that the condition (21) is equivalent to the following one: < (x)K

()(ut(x;T));v0(x) >H ;1;H

1

0 ;

Z

1

0

(x)K

()(u

(x;T))v

0(x)dx = 0;

8(v 0;v1)

2LK

(): (66)

From (65) and (66) we deduce that the rst statement in Theorem 4 is equivalent to the existence of f 2L

2(0;T) such that

;<

u

1(x);v

(x;0) >H

;1;H 1

0 + Z

1

0

(x)u

0(x)vt(x;0)dx + Z T

0

f(t)vx(1;t)dt = 0 (67)

for all (v0;v1) 2LK

().

Let us introduce the following quadratic functional: J(v0;v1) =

;<

u

1(x);v

(x;0) >H

;1;H 1

0 + Z

1

0

(x)u

0(x)vt(x;0)dx+12 Z T

0

jvx(1;t)j 2dt:

(68) Thanks to (60), J is a continuous, strictly convex and coercive functional in LK(). So,

there exists an unique minimizer (w

0;w

1) 2 LK

() which can be characterized by the

formula

; <

u

1(x);v

(x;0) >H

;1;H1 0 +

Z

1

0

(x)u

0(x)vt(x;0)dx + Z T

0

wx(1;t)vx(1;t)dt = 0;

8(v 0;v1)

2LK

(): (69)

We setf(t) = wx(1;t) where w solves (58) with the minimizer (w

0;w

1) as data. Then

f veries (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

0;w

1) is a minimizer of J, we have

J(w0;w

1)

J(0;0) = 0: (70)

Then,

Z T

jf(t)j 2dt

< u

1(x);w

(x;0) >H;1;H1 0

; Z

1

(x)u

(16)

16

Using the Fourier decomposition of u0 and u1, the fact that w

0;w

1

2 LK

() and the

orthogonality of the eigenfunctions we can write the second term in (71) as < K

()(u

1(x));w

(x;0) >H

;1;H 1

0 ;

Z

1

0

(x)K

()(u

0(x))wt(x;0)dx

= < (K()u0;

K

()u1);(

;wt(x;0);w

(x;0)) >L

2

H ;1;L2

H 1

0

(K

()u0;

K

()u1)

L2 (0;1)H

;1

(0;1) k(w

0;w

1) kH

1

0 (0;1)L

2

(0;1): (72)

Now, in view of (60) we can estimate the last term in (72) by

1

p

C1

(K

()u0;

K

()u1)

L2 (0;1)H

;1

(0;1) Z T

0 jf

(t)

j 2dt

!

1=2

: (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 controlsf are uniformly bounded inL2(0;T) and therefore there exists

a subsequence, still denoted byf, such that

f* g weakly in L2

(0;T): (74)

Let us see thatg = 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;

v(0;t) = v(1;t) = 0; 0< t < T; (75) with nal data (w0;w1), i.e.

v(x;T) = w0(x); vt(x;T) = w1(x); 0< x < 1: (76)

Here, (w0;w1) 2H

1

0 L

2(0;1) is the minimizer of the limit quadratic functional

J(v0;v1) =

;< u

1(x);v(x;0) >H ;1;H

1

0 + Z

1

0

u0(x)vt(x;0)dx

+12Z T 0

jvx(1;t)j

2dt (77)

wherev(x;t) is the solution of (75) with nal data (v0;v1):

Hence, the control f(t) can be characterized by the following two properties: 1. f(t) = wx(1;t) where w is a solution of (75).

2. f(t) veries

; < u

1(x);v(x;0) >H ;1;H1

0 + Z

1

0

u0(x)vt(x;0)dx + Z T

0

f(t)vx(1;t)dt = 0

8(v 0;v1)

2H 1

0 L

2

(0;1): (78)

wherev is the solution of (75) with nal data

(17)

17 We are going to see that g veries 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

0;w

1) is uniformly bounded inH 1

0(0;1) L

2(0;1). So, we can extract a subsequence,

still denoted (w

0;w

1), such that

(w

0;w

1)* (w

0;w1) weakly in H 1

0(0;1) L

2

(0;1): Denew1 the weak limit in L

2(0;1) of w

1(x=).

The theory of Homogenization (see [3], Theorem 3.2) gives us that the solution w

of (58) with nal data (w

0;w

1) converges to the solution w of (75) with nal data

(w0;w1=) in the following sense

w * w weak-* in L1(0;T;H1 0(0;1))

\W

1;1(0;T;L2(0;1)): (80)

Furthermore, for any elementv2L

2(0;1), Z

1

0

(x)wt(x;t)v(x)dx

! Z

1

0

wt(x;t)v(x) strongly in C0([0;T]): (81)

Multiplying the equations (58) by tests functions and integrating we easily obtain the following identity forw:

0 =Z T 0

Z

1

0

(x)wt(x;t)v(x)lt(t)dxdt

; Z T

0 Z

1

0

wx(x;t)vx(x)l(t)dxdt

+Z T 0

wx(1;t)v(1)l(t)dt; 8v(x)2C 1

0((0;1]);l(t) 2C

1

0(0;T): (82)

We can pass to the limit in (82) thanks to (74), (80) and (81), and then w veries 0 =Z T

0 Z

1

0

wt(x;t)v(x)lt(t)dxdt; Z T

0 Z

1

0

wx(x;t)vx(x)l(t)dxdt +

Z T 0

g(t)v(1)l(t)dt

8v(x)2C 1

0((0;1]);l(t) 2C

1

0(0;T): (83)

On the other hand, asw is a solution of the limit system (75), it also veries 0 = Z T

0 Z

1

0

wt(x;t)v(x)lt(t)dxdt; Z T

0 Z

1

0

wx(x;t)vx(x)l(t)dxdt

+Z T 0

wx(1;t)v(1)l(t)dt; 8v(x)2C 1

0((0;1]);l(t) 2C

1

0(0;T): (84)

From (83) and (84) we nally deduce

Z T 0

g(t)l(t)dt =Z T 0

wx(1;t)l(t)dt; 8l(t)2C 1

0(0;T)

and then g = wx(1;t) with w a solution of (75) as we wanted to prove.

Now, we check that g also veries the second property above. Assume that the following Lemma holds:

Lemma 1

Given (v0;v1) 2H

1

0 L

2(0;1); there exists a sequence (v 0;v

1) 2LK

() such

that

(v

0;v

1) !(v

0;v1) in H 1

0 L

2(0;1): (85)

Furthermore, if v is the solution of (58) with initial data (v

0;v

1) and v is the solution

of (75) with initial data (v0;v1) the following holds:

v

!v in L 1

([0;T];H1 0(0;1))

\W 1;1

([0;T];L2

(0;1)); (86)

vx(1;t)!vx(1;t) in L

(18)

18

We prove this Lemma at the end of the section (Step 8). We consider (v0;v1)

2H 1

0 L

2(0;1): By Lemma 1 there exists a sequence (v 0;v

1) 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 1

0(0;1) L

2(0;1) (88)

where v is the solution of (58) with initial data (v

0;v

1) and v is the solution of (75)

with initial data (v0;v1):

Passing to the limit, as ! 0, in formula (69) with (v

0;v1) = (v

0;v

1) and taking

into account (24), (87) and (88) we obtain that g satises

; < u

1(x);v(x;0) >H ;1;H1

0 + Z

1

0

u0(x)vt(x;0)dx + Z T

0

g(t)vx(1;t)dt = 0

8(v 0;v1)

2H 1

0 L

2

(0;1): (89)

So, g veries the second property above.

Step 4:

Strong convergence of the controls. We consider formulas (69) and (78) with (v0;v1) = (w

0;w

1) and (v

0;v1) = (w0;w1) respectively:

;<

u

1(x);w

(x;0) >H

;1;H 1

0 + Z

1

0

(x)u

0(x)wt(x;0)dx + Z T

0

jwx(1;t)j 2

dt = 0; (90)

;< u

1(x);w(x;0) >H ;1;H1

0 + Z

1

0

u0(x)wt(x;0)dx + Z T

0

jwx(1;t)j

2dt = 0: (91)

Due to the hypothesis on the strong convergenceu

1 !u

1 inH

;1 and (81), the rst

two terms in (90) converge to the rst two terms in (91) and therefore

Z T 0

jwx(1;t)j 2dt

! Z T

0

jwx(1;t)j

2dt; as

!0: (92)

The strong convergence of the controls inL2(0;T) is a consequence of the weak

conver-gence stated in step 4 and the converconver-gence 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

1(0;T;L

K()) 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;

(x;0) = t(x;0) = 0; 0< x < 1: (93)

Multiplying the equations of (1) by the solution of (93) and integrating by parts

we easily obtain the following identity:

Z T 0

Z

1

0

udxdt =

; Z T

0

f(t) x(1;t)dt;

8(x;t)2L

1(0;T;L

K()): (94)

Here we have used the fact thatu(x;T) = ut(x;T) = 0 because f is a control.

Taking into account the Fourier decomposition of the space L2(0;1) given in (16)

and the fact that(:;t) 2LK

() we can write (94) as Z T

0 Z

1

0

K

()(u

)dxdt = ; Z T

0

f(t) x(1;t)dt; 8(x;t)2L

1(0;T;L

K()): (95)

(19)

19

Proposition 10

Let 2 L

1(IR) be a periodic function such that 0 <

m (x)

M <1a.e. x2R;and considerT > 0. Then, for all ((x;t);(x;t))2C

1(0;T;L

K())

there exists C2 > 0 such that for any 0< < 1 the solution

of (93) veries

Z T 0

j x(1;t)j 2

dtC 2

kk 2

L1 (0;T;L

2

(0;1)): (96)

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: (97)

satises Z T

0

jx(1;t)j 2dt

C 2

kk 2

L1 (0;T;H

1

0

(0;1)): (98)

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

0

f(t) x(1;t)dt

Ckf

kL 2

(0;T) kkL

1

(0;T;L 2

(0;1)): (99)

Then, the map

!; Z T

0

f(t) x(1;t)dt is linear and continuous from L1(0;T;L

K())

! IR and therefore K ()u

is indeed

characterized by (95).

Furthermore, from (95) and (99) we have

K ()u

L1 (0;T;L

2

(0;1))

CkfkL 2

(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

Z T 0

Z

1

0

utdxdt =; Z T

0

f(t)x(1;t)dt; 82C

1(0;T;H1

0(0;1)): (101)

Hence,

Z T 0

< ut; >H

;1;H 1

0 dt = Z T

0

f(t)x(1;t)dt; 8 2C

1(0;T;H1

0(0;1)): (102)

Taking into account the Fourier decomposition of the spaceH;1(0;1) and the fact that

(:;t)2LK

() we can simplify (102) into Z T

< K

()ut; >H ;1;H1

0 dt = Z T

f(t)x(1;t)dt; 82C

1(0;T;H1

(20)

20

By Proposition 10 the right hand side of (103) can be bounded by

Z T

0

f(t)x(1;t)dt

Ckf

kL 2

(0;T) kkL

1

(0;T;H 1

0

(0;1)): (104)

Then, the map

!

Z T 0

f(t)x(1;t)dt is linear and continuous from L1(0;T;L

K())

! IR and then K

()ut can be indeed

characterized by formula (103). Moreover, from (103) and (104) we deduce that

K ()ut

L1 (0;T;H

;1

(0;1))

CkfkL 2

(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;L2 H

;1(0;1)).

So, there exists a subsequence, still denoted by (K()u

;K

()ut) such that

(K()u

;K

()ut)* (p;q) weakly-* in L

1(0;T;L2 H

;1(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

Z T 0

Z

1

0

K

()(u

)dxdt =

; Z T

0

f(t) x(1;t)dt;

82L 1

(0;T;L2

(0;1)) (106) where is the solutions of (93).

In a similar way u can be characterized by

Z T 0

Z

1

0

udxdt =; Z T

0

f(t) x(1;t)dt; 82L 1

(0;T;L2

(0;1)) (107) 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: (108)

At this point, we need the following Lemma:

Lemma 2

Given(;)2C

1(0;T;L2(0;1) H

1

0(0;1));there exists a sequence (

;)2

C1(0;T;L

K() LK

()) such that

(;)!(;) in C

1(0;T;L2(0;1) H

1

0(0;1)): (109)

Furthermore, if is the solution of (93) with nonhomogeneous term = and is

the solution of (108) the following holds:

x(1;t)! x(1;t) in L

2(0;T): (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

(21)

21 The proof of this lemma is similar to the proof of Lemma 1 and we omit it.

We consider 2 C

1(0;T;L2(0;1)): By Lemma 2 there exists 2 C

1(0;T;L

K())

such that (109) holds. Then, formula (106) gives, in particular

Z T 0

Z

1

0

K

()(u

)dxdt =

; Z T

0

f(t) x(1;t)dt: (112)

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 satises

Z T

0 Z

1

0

pdxdt =; Z T

0

f(t) x(1;t)dt; 82L 1

(0;T;L2

(0;1)): (113) On the other hand, asu 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

Z T

0

< K

()ut; >H ;1;H

1

0 dt = Z T

0

f(t)x(1;t)dt;

82C

1(0;T;H1

0(0;1)) (114)

where is the solutions of (100). On the other hand, ut can be characterized by

Z T 0

< ut; >H;1;H 1

0 dt = Z T

0

f(t)x(1;t)dt; 8 2C 1

(0;T;H1

0(0;1)) (115)

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 2C

1(0;T;H1

0(0;1)): By Lemma 2 there is

2C

1(0;T;L

K()) such that

(109) holds. Then, formula (114) gives, in particular

Z T 0

< K

()ut;

>H

;1;H1 0 dt =

; Z T

0

f(t)x(1;t)dt: (117)

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 satises

Z T 0

< q; >H;1;H1 0 dt =

Z T 0

f(t)x(1;t)dt; 8 2C

1(0;T;H1

0(0;1)): (118)

On the other hand, asut 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 2L

1(0;T;L2(0;1))

with

* weakly in L1(0;T;L2(0;1)) (119)

the following holds

Z T 0

Z

1

0

K

()u

dxdt

! Z T

0 Z

1

0

udxdt: (120)

Here, can be spanned in terms of the eigenfunctions and then we can write the

rst term in (120) as

Z TZ 1

K

()u

dxdt =Z TZ 1

K

()u

K

()

(22)

22

By formula (106) we have in particular that

Z T

0 Z

1

0

K

()u

K

()

dxdt =

; Z T

0

f(t) x(1;t)dt; (122)

where is the solution of (93) with = K

()

.

On the other hand, formula (107) give us

Z T 0

Z

1

0

udxdt =; Z T

0

f(t) x(1;t)dt; (123)

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 2L

1(0;T;L2(0;1)) which veries (119). Then, the

solution of (93) with = K

()

converges to the solution of (108) in the class

* weakly in C(0;T;H1

0(0;1)) \C

1(0;T;L2(0;1)): (124)

Moreover,

x(1;t) * x(1;t) weakly in L2(0;T): (125)

Proof:

Observe that the solution of (93) with = K

()

can be written as

(x;t) =Z t 0

(x;t

;s;s)ds (126)

where is the solution of

8

>

<

>

:

(x) tt; xx = 0; 0< x < 1; 0 < t;s < T;

(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 ofK

()

is an obvious consequence of the boundness of while for

the identication 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 1

0(0;1)) \C

1(0;T;L2(0;1)) (129)

for all s2[0;T] where is the solution of 8

>

<

>

:

tt; xx = 0; 0< x < 1; 0 < t;s < T;

(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: (130)

Furthermore, for anyl 2L

2(0;1) and s

2[0;T]; Z

1

(x) t(x;t;s)l(x)dx! Z

1

(23)

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 inL2(0;T) for all s. Then

x(1;t;s) * p(t;s) weakly in L2(0;T): (132)

Let us see that p(t;s) = x(1;t;s)

Multiplying (127) by the test functionx(t) and integrating by parts we obtain the following expression for :

Z T 0

Z

1

0

(x) tt(t)dxdt

; Z T

0 Z

1

0

x(t)dxdt =Z T 0

x(1;t;s)(t)dt; 82C 1

0(0;T):

(133) Analogously, multiplying the equations in (130) byx(t) we have

Z T 0

Z

1

0

tt(t)dxdt; Z T

0 Z

1

0

x(t)dxdt =

Z T 0

x(1;t;s)(t)dt; 82C 1

0(0;T):

(134) Taking into account (129) and (131) the rst two terms in (133) converge to the rst two terms in (134). Then

Z T 0

x(1;t;s)(t)dt =Z T 0

x(1;t;s)(t)dt; 82C 1

0(0;T) (135)

and we deduce that p(t;s) = x(1;t;s). Finally observe that for any 2L

2(0;T)

Z T 0

x(1;t;s)(t)dt = Z T 0

Z t 0

x(1;t;s;s)ds(t)dt = Z T

0 Z T

s x(1;t;s;s)(t)dtds

! Z T

0 Z T

s x(1;t;s;s)(t)dtds = Z T

0 Z t

0

x(1;t;s;s)ds(t)dt

= Z T 0

x(1;t;s)(t)dt:

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 2C

1(0;T;H1

0(0;1)) with

* weakly in L1(0;T;H1

0(0;1)) (136)

the following holds

Z T 0

< K

()ut;

>H

;1;H 1

0 dt !

Z T 0

< ut; > dt: (137)

Observe that

Z T 0

< K

()ut;

>H

;1;H 1

0 dt = Z T

0

< K

()ut;K()

>H

;1;H 1

0 dt:

By formula (114) we have in particular that

Z T

< K

()ut;K()

>H;1;H1 0 dt =

Z T

(24)

24

where is the solution of (100) with = K

()

.

On the other hand, formula (115) give us

Z T 0

< ut; >H;1;H 1

0 dt = Z T

0

f(t)x(1;t)dt; (139)

where is the solution of (116).

Now we need the following lemma:

Lemma 4

Consider a sequence 2C

1(0;T;H1

0(0;1)) which veries (136). Then, the

solution of (100) with t= K

()t converges to the solution of (116) in the class

! weakly in C(0;T;H

1

0(0;1)) \C

1(0;T;L2(0;1))::

Moreover,

x(1;t)!x(1;t) weakly in L

2(0;T):

Proof:

The solution of (100) with t = K

()t can be written as

(x;t) =Z t 0

Z t 0

@ @x3

(x;r

;s;s)drds (140)

where @

@x3 is the derivative with respect to the third component and

is the solution

of

8

>

<

>

:

(x)tt;xx = 0; 0< x < 1; 0 < t;s < T;

(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 (v0;v1) 2H

1

0(0;1) L

2(0;1):

We recall that the eigenfunctions 'n(x) = sin(nx=p

) of the limit system (75) constitute an orthonormal basis of H1

0(0;1) and !n(x) = ('n; p

n'n) constitute an

orthonormal basis of H1 0(0;1)

L

2(0;1) with the norm

k(v 0;v1)

k 2

H1 0

L 2

(0;1)= Z

1

0 jv

0

0(x) j

2dx + Z

1

0

jv 1(x)

j 2dx:

So, we can expand (v0;v1) in Fourier series as follows:

(v0;v1) = X

n2IN

an!n(x):

Recall also that the eigenfunctions 'n(x) of system (58) constitute an orthonormal basis of H1

0(0;1) and then !n(x) = ('n; q

n'n) constitute an orthonormal basis of H1

0(0;1) L

2(0;1) with the norm

k(v 0;v1)

k 2

H1 0

L 2

(0;1)= Z

1

0 jv

0

0(x) j

2dx + Z

1

0

jv 1(x)

(25)

25 Dene

(v

0;v

1) =

r() X

n=1

an!n(x)

wherer() = ;1=2. It is clear that

(v

0;v

1) 2LK

() LK

():

We now prove the convergence stated in (85). To simplify the notation we only prove the convergence of the rst component. Observe that

kv 0 ;v 0 kH 1 0 (0;1)=

X

n2IN

an'n(x);

r() X

n=1

an'n(x)

H1 0

(0;1)

r() X

n=1

an['n(x);'n(x)] H1 0

(0;1)

+ 1 X

n=r()

an'n(x)

H1 0

(0;1)

0

@

r() X

n=1 janj

2 1

A 1=2

0

@

r() X

n=1

k'n(x);'n(x)k 2

H1 0

(0;1) 1

A 1=2

+

0

@ 1

X

n=r() janj

2 1

A 1=2

: (142) As the last series is convergent, we deduce that

0

@ 1

X

n=r() janj

2 1

A 1=2

!0 as !0: (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 2L

1(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 nC

;1 verify q n; q n cn

32; (144)

k'n;'nkW 1;1

(0;1) cn

3

2

; (145)

where n and 'n are the eigenvalues and eigenfunctions of the limit system (75).

The rst term in (142) can be bounded by

0

@

r() X

n=1 janj

2 1

A 1=2

0

@c 24

r() X

n=1

n6 1

A 1=2

kv 0 kH 1 0 (0;1)+

kv 1

kL 2

(0;1)

c2 1 + Z r

()+1 1

s6ds !

1=2

(146) kv 0 kH 1 0 (0;1)+

kv 1

kL 2

(0;1)

c2 1 + (r() + 1) 7

7

!

1=2

kv 0 kH 1 0 (0;1)+

kv 1

kL 2

(0;1)

c2 1 + (

;1=2+ 1)7

7

!

1=2

Figure

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