Boundary controllability of the finite-difference space semi-discretizations of the beam equation

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002025

BOUNDARY CONTROLLABILITY OF THE FINITE-DIFFERENCE SPACE SEMI-DISCRETIZATIONS OF THE BEAM EQUATION

Liliana Le´ on

and Enrique Zuazua

Abstract. We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered;b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.

Mathematics Subject Classification. 93C20, 35Q33, 65N06.

Received November 2, 2001.

1. Introduction

The transversal vibrations of a 1-d beam with hinged boundary conditions are modelled by the following equation

















u⁰⁰+∂_x⁴u= 0 in Q= (0,1)×(0, T) u(0, t) =u(1, t) = 0 t∈(0, T)

∂_x²u(0, t) =∂_x²u(1, t) = 0 t∈(0, T) u(x,0) =u⁰(x), u⁰(x,0) =u¹(x) x∈(0,1),

(1)

where ⁰ denotes the derivative ofuwith respect to time.

Keywords and phrases:Beam equation, finite difference semi-discretization, exact boundary controllability.

∗This work was done while the first author was visiting Universidad Complutense de Madrid with a fellowship of the Alpha Program of the European Union “Mod´elisation & Ing´enierie Math´ematique”.

∗∗Partially supported by grant PB96-0663 of the DGES (Spain) and the TMR network of the EU “Homogenization & Multiple Scales”.

1Laborat´orio de Ciˆencias Matem´aticas, Universidade Estadual do Norte Fluminense, 28015 Rio de Janeiro, Brazil;

e-mail: lalm@uenf.br

2Departamento de Matem´aticas, Facultad de Ciencias, Universidad Aut´onoma, Cantoblanco, 28049 Madrid, Spain;

e-mail:enrique.zuazua@uam.es

c EDP Sciences, SMAI 2002

When (u⁰, u¹)∈H₀¹(0,1)×H⁻¹(0,1) there exists a unique solutionusuch that

u∈C([0, T];H₀¹(0,1))∩C([0, T];H⁻¹(0,1)). (2) This solution admits the Fourier expansion

u(x, t) = X∞ k=1

{a_kcosk²π²t+b_ksink²π²t}sinkπx, (3)

with suitable Fourier coefficients depending on the initial data (u⁰, u¹).

The energy associated with (u⁰, u¹)∈H₀¹(0,1)×H⁻¹(0,1) is given by E(t) =1

2

ku(., t)k²₁+ku⁰(., t)k²₋₁

, (4)

where k.k1 andk.k−1 are the canonical norms inH₀¹(0,1) andH⁻¹(0,1), respectively. Namely kuk₁=

Z ₁

0 (∂_xu)²dx _1/2

; kuk₋₁=k(−∂_x²)⁻¹uk₁

where (−∂_x²)⁻¹ denotes the inverse of the operator−∂_x² with homogeneous Dirichlet boundary conditions at x= 0, 1.

It is easy to see that the energyE(t) is conserved along time for the solutions of (1).

Applying multipliers or Fourier series techniques one can prove a boundary observability inequality showing that, for everyT >0, there exists C=C(T)>0 such that

E(0)≤C Z _T

0 |∂_xu(1, t)|²dt, (5)

for every solution of (1) (see Lions [10]).

As a consequence of this observability inequality and Lions’ HUM method [10] the following boundary controllability property may be proved:

For all T > 0 and (y⁰, y¹)∈ F = H₀¹(0,1)×H⁻¹(0,1), there exists a control ν ∈ L²(0, T), such that the solution of

















y⁰⁰+∂_x⁴y= 0 in Q=I×(0, T) y(0, t) =y(1, t) = 0 t∈(0, T)

∂_x²y(0, t) = 0 ∂_x²y(1, t) =ν t∈(0, T) y(x,0) =y⁰(x), y⁰(x,0) =y¹(x) x∈I,

(6)

satisfies

y(x, T, ν) =y⁰(x, T, ν) = 0.

The main objective of this work is to study the controllability of the classical semi-discrete space approximation by finite differences of (6). We also study the convergence of controls and controlled solutions as the mesh-size tends to zero. Our work provides two alternative methods for the numerical approximation of the exact control ν of equation (6).

In Section 2 we analyze the problem of the observability of the finite-difference space semi-discretization of the beam equation (1). In Section 3, combining the results of the previous section, we find two results of uniform controllability (as the mesh size tends to zero) of the semi-discrete space approximation by finite differences of (6). The first one concerns the partial control obtained by filtering the high frequencies and the second one the control of the semi-discrete solutions by means of a suitable modification of the boundary control. In Section 4, we study the convergence of the solutions of the previous semi-discrete problems as the mesh-size tends to zero.

2. Analysis of the boundary observability problem

For each N ∈N we consider a partition of I= (0,1), P ={x0= 0, . . . , x_j=jh, . . . , x_N+1 = 1}, where the mesh-size ish= 1/(N+ 1).

To get a discrete definition of the boundary conditions of the problem (1) using centered finite differences, we also introduce two external points x₋₁ = x₀−h and x_N+2 = x_N+1 +h. We denote by u_h,j(t) the approximation of the solution of (1) at the point x_j. We also setu_h,−1=−u_h,1 and u_h,N+2=−u_h,N.

The semi-discretization by finite differences of (1) is then given by the following system of N ordinary differential equations,

























u⁰⁰_h,j=− 1

h⁴[u_h,j+2−4u_h,j+1+ 6u_h,j−4u_h,j−1+u_h,j−2], 0< t < T j= 1,2, . . . N

u_h,0=u_h,N+1= 0, 0< t < T

u_h,−1=−u_h,1 u_h,N+2=−u_h,N, 0< t < T u_h,j(0) =u⁰_h,j u⁰_h,j(0) =u¹_h,j, j= 1,2, . . . N.

(7)

Here the initial conditions (u⁰_j, u¹_j) of (7) are suitable approximations of the initial conditions of (1) at the points x_j of the mesh. It is easy to see that the scheme (7) is convergent as h → 0 in the classical sense, i.e.it is consistent and stable.

The eigenvalue problem associated to (7) is as follows











h1⁴B ~φ^k(h) =β ~φ^k(h) φ_k,0 = φ_k,N+1;

φ_k,−1=−φk,1 φ_k,N+2=−φk,N

(8)

where

B =

















5 −4 1 0 . . . . 0

−4 6 −4 1 0 . . . . 0 1 −4 6 −4 1 0 . . . . 0

0 1 −4 6 −4 1 0 . . . 0

. .. ... ... ... ... ... ...

0 . . . 0 1 −4 6 −4 1 0

0 . . . . 0 1 −4 6 −4 1 0 . . . . 0 1 −4 6 −4 0 . . . . 0 1 −4 5

















N×N

· (9)

Figure 1. Graph of discrete eigenvalues β_k(h) for different values of h and the continuous eigenvalues β_k =k⁴π⁴ of the beam equation.

Note thatB=A² with

A=













−2 1 0 . . . . 0 1 −2 1 0 . . . . 0

0 1 −2 1 0 . . . 0

. .. ... ... ... ...

0 . . . 0 1 −2 1 0

0 . . . . 0 1 −2 1 0 . . . . 0 1 −2













N×N

· (10)

The matrix A arises in the semi-discrete approximation of Laplace’s equation in one space dimension and its eigenvalues and eigenvectors are well known (see [6]);

λ_k(h) = 4 h²sin²

kπh 2

k= 1,2, . . . , N (11)

φ~^k(h) = (φ_k,1, φ_k,2, . . . , φ_k,N) k= 1,2, . . . , N (12)

φ_k,j(h) = sin(jkπh). (13)

Then, the eigenvectors ofB are the same, and their eigenvalues are β_k(h) =λ²_k(h) = 16

h⁴sin⁴ kπh

2

k= 1,2, . . . , N.

The discrete eigenvaluesβ_k(h) approximate the eigenvalues of the continuous modelβ_k=k⁴π⁴forkfixed, when the size of the meshhtends to zero, and its eigenvectorsφ~^k(h) coincide with the eigenfunctionsφ^k(x) = sin(kπx) of the continuous model (1) on the mesh points.

Therefore, the solution of system (7) may be expressed as

~ u_h(t) =

XN k=1

n

a_kcosp β_k(h)t

+ b_ksinp β_k(h)t

oφ~^k(h) (14)

with ~u_h(t) = (u_h,1(t), u_h,2(t), . . . u_h,N(t)), for suitable Fourier coefficientsa_j,b_j depending on the initial data.

We set A_h = _h¹2A, where A is as in (10) and we denote by A⁻¹_h its inverse. We define the energy E^h(t) associated to problem (7) by

E^h(t) = h 2

XN j=1





u_h,j+1(t)−u_h,j(t) h

²+

A⁻¹_h u~_h⁰(t)

j+1− A⁻¹_h u~_h⁰(t)

j

h

2



 (15) which is an approximation of the continuous energyE(t). Note thatE_h(t) is conserved along time for solutions of (7).

It is then natural to analyze the following semi-discrete version of the observability inequality (5):

E^h(0) ≤ C(T, h) Z _T

0

u_h,N(t) h

²dt (16) where C(T, h) is independent of the solution of (7).

Theobservability inequality (16) is said to beuniform, if the constantsC(T, h) are bounded uniformly in h, as h→0.

However, as we shall see below in Section 2.1, whatever T >0 is, the inequality (16) may not be uniform.

In order to restore the uniformity of the observability inequality with respect tohthere are two possibilities:

(a) to restrict the class of solutions of (7) under consideration; (b) to reinforce the observed quantity on the right hand side of (16).

Once the lack of uniform observability is proved in Section 2.1 the rest of this section will be devoted to prove the two uniform observability properties mentioned above.

2.1. Non-uniform observability

Let us first recall the following observability identity for the eigenvectors ofB (see Lem. 1.1 in [5]) h

XN j=0

φ_k,j+1 − φ_k,j h

² = 2 (4 − p

β_k(h)h²) φ_k,N

h

² k= 1,2, . . . N. (17)

This identity allows to show that the observability inequality may not be uniform as h→0 for anyT. More precisely, we have the following negative result:

Theorem 2.1. For anyT > 0, we have

~uhsol. of (7)sup

E^h(0) R_T

0 u_h,N(t)/h²dt

!

−→ ∞, as h→0. (18)

Proof. Forh >0, consider~u_h(t) = (u_h,1(t), u_h,2(t), . . . , u_h,N(t)) the solution of (7), associated to the eigenfunc- tionφ~^k(h):

~

u_h(t) = eⁱ

√βk(h)tφ~^k(h) = e^iλk(h)tφ~^k(h). (19)

According to (17) Z _T

0

u_h,N(t) h

²dt= φ_k,N

h ²Z _T

0

e^iλk(h)t2dt=T

2 4−λ_k(h)h² h

XN j=0

φ_k,j+1−φ_k,j h

²· (20)

Note that 4−λ_k(h)h²= 4 cos²(^kπh₂ ).

On the other hand,

E^h(0) =h XN j=0

φ_k,j+1−φ_k,j h

²·

Therefore

E^h(0) = 1 2Tcos²(^kπh₂ )

Z _T

0

u_h,N(t) h

²dt. (21)

Thus for any T >0, takingk=N and using that cos²

N πh 2

= cos² π

2 −πh 2

−→0 when h→0,

equation (18) holds immediately.

Remark 2.1. Letδ∈(0,1) be given. The counterexample above may not be found in the class of low frequency solutions with Fourier components corresponding to indexes k≤δN. Indeed in that case, the quotient in (21) may be bounded below by 1/

2Tcos²δ (π−πh)/2

which is bounded ash−→0.

This observation motivates the uniform observability result we state in the following section for filtered solutions in which the high frequency components have been filtered.

2.2. Uniform observability of filtered solutions

Givenγ∈(0,16) and h >0, we considerC_h(γ) the class of solutions of (7) generated by the eigenvectors of (8) associated with eigenvalues such that

β_k(h)≤γh⁻⁴. More precisely,

C_h(γ) =



~u_h sol. of (7) : ~u_h= X

βk(h)≤γh⁻⁴

n a_keⁱ

√βk(h)t+b_ke⁻ⁱ

√βk(h)to φ~^k(h)



· (22) Observe that, when γ = 16, C_h(γ) = C_h(16) coincides with the space of all solutions of the semi-discrete problem (7). The following observability result holds in this class.

Theorem 2.2. Let 0< γ <16. For allT >0, there existsC=C(T, γ)>0such that E^h(0) ≤ C

Z _T

0

u_h,N(t) h

²dt ∀~u_h∈C_h(γ), ∀h >0. (23) The proof of this result relies on Ingham’s inequality (see [17] for instance) in which the gap between the consecutive eigenvalues of the semi-discrete system (7) plays a crucial role.

Figure 2. The functionk→γ_k(h) for different values of h.

2.2.1. Analysis of the gap between consecutive eigenvalues

Let us first observe that the gap for the continuous model (1) satisfies pβ_k+1 − p

β_k= (2k+ 1)π²→ ∞.

Moreover, we have the following:

Lemma 2.1. The following properties hold:

(i) p

β_k+1(h) − p

β_k(h) ≥ 3π²

(sin^πh₂

πh2

)₂

−π⁴h² (24)

(ii) lim

h→0 inf

1≤k≤_h¹−2

npβ_k+1(h) − p β_k(h)

o

= 3π² (25)

(iii) lim

h→0 sup

1≤k≤_h¹−2

npβ_k+1(h) − p β_k(h)

o

= ∞. (26)

Proof. (i) We set

γ_k(h) =p

β_k+1(h) − p

β_k(h), for k= 1,2, . . . , N−1.

Using classical trigonometrical identities it follows that γ_k(h) = 2

h²

coskπh−cos (k+ 1)πh

= 4

h²sin(2k+ 1)πh 2 sinπh

2 · (27)

We observe that 0< ^3πh₂ ≤^(2k+1)πh₂ ≤π−^3πh₂ fork= 1,2. . . , N −1.

Thus, for every h > 0, the function k → γ_k(h) describes a concave parabola which is symmetric with respect to k= ^1−h_2h , as shown in Figure 2.

Then, it is enough to considerk∈Nsuch that ^3πh₂ ≤ ^(2k+1)πh₂ ≤ ^π₂, or equivalently 1≤k≤^1−h_2h . For those indexesk we have

γ_k(h)≥ 4 h²sin

3πh 2

sin

πh 2

, ∀k∈N∩

1,1−h 2h

·

Using the trigonometrical identity sin 3α= 3 sinα−4 sin³α in the previous inequality, we conclude that γ_k(h)≥3π²

(sin^πh₂

πh2

)₂

−π⁴h²

(sin^πh₂

πh2

)₄

≥3π²

(sin^πh₂

πh2

)₂

−π⁴h²,

for everyk= 1,2, . . . , N−1.

(ii) From (27) we have that

1≤k≤inf_h¹−2{γ_k(h)}= inf

1≤k≤_h¹−2

4

h²sin(2k+ 1)πh 2 sinπh

2

= 4

h²sin3πh 2 sinπh

2 · (28)

Then, (25) holds.

(iii) Again from the equality (27) we have that sup

1≤k≤_h¹−2

γ_k(h) = sup

1≤k≤¹_h−2

4

h²sin(2k+ 1)πh 2 sinπh

2

= 4 h²sinπ

2sinπh

2 , (29)

and (26) holds.

We also have the following additional property:

Lemma 2.2. For allγ_∞>0, there exist δ >0 andk₀∈Nsuch that:

pβ_k+1(h) − p

β_k(h)≥γ_∞ (30)

for k=k₀, k₀+ 1, . . . ,1/h−1−k₀ and ∀ |h|< δ.

Proof. In view of (29) there exists k₀ andδ >0 such that

γ_k0(h)≥γ_∞ for all |h|< δ. (31)

Then, taking into account that the parabolak−→γ_k(h) is symmetric with respect tok=^1−h_2h , we deduce that γ_k(h)≥γ_∞ for k=k₀, . . . ,1−2h

h −k₀. (32)

In view of the particular structure of the gap functions described above we need the following variant of Ingham’s inequality, whose proof is very close to that of Theorem 3.4 given in [13].

Lemma 2.3. Let f(t) = P

n∈Zd_ne^iµnt where

µ_{n n∈Z} is a sequence of real numbers, such that there exist N ∈N, γ >0 andγ_∞>0such that

(i) µ_n+1−µ_n ≥γ ∀n∈Z.

(ii) Card{n∈Z:µ_n+1−µ_n≤γ_∞}=N.

Then, for any interval J = [0, T]⊂R with T > _γ^2π

∞ there exist two positive constants C₁, C₂>0 such that C₁X

n∈Z

|dn|²≤ Z _T

0

X

n∈Z

d_ne^iµnt

2

dt≤C₂X

n∈Z

|dn|², (33)

for all sequence {d_n} ∈l².

More precisely C₁ =C₁(N) and C₂ =C₂(N), where C_i(j), i= 1,2, are given by the following recurrent formulas

C₁(j+ 1) =

"

2C₂(j) T + 1

4

C₁(j) (T γ_∞−2π)²γ² + 2 T

#₋₁ , C₂(j) = 2{j T +C₂(0)}, j= 1,2, . . . ,

and C₁(0), C₂(0) are such that (33) holds in the particular case in whichN = 0.

Remark 2.2. The main difference between Lemma 2.3 and that proved in [13] is that, here, the set of badly separatedµ_n-s is not necessarily constituted by the first n-s such that |n| ≤k₀for some finite k₀as in [13].

Proof. We proceed as in [3] and [13]. The proof is divided in two steps.

Step 1. We fix anyT >2π/γ_∞. Let us consider the set

Y ={n∈Z:µ_n+1−µ_n≤γ_∞} · (34)

Denote

g(t) =X

n/n∈Z∈Y

d_ne^iµnt.

Applying Ingham’s inequality [13] tog(t), we have that there exist two constantsC₁(0)>0 and C₂(0)>0 such that:

C₁(0) X

n/n∈Z∈Y

|dn|²≤ Z _T

0 |g(t)|²dt≤C₂(0) X

n/n∈Z∈Y

|dn|². (35)

For

f(t) =g(t) +X

n∈Yn∈Z

d_ne^iµnt,

we have Z _T

0 |f(t)|² dt = Z _T

0

X

n∈Y

d_ne^iµnt+X

n/∈Y

d_ne^iµnt

2

≤2 Z _T

0





X

n∈Y

d_ne^iµnt

2

+

X

n/∈Y

d_ne^iµnt

2

 dt. (36)

According to (35) we obtain Z _T

0

f(t)²dt ≤ 2C₂(0) X

n/∈Y

|d_n|²+ 2T X

n∈Y

|d_n|

!₂

≤ 2C₂(0) X

n/∈Y

|d_n|²+ 2T N X

n∈Y

|d_n|²≤(2C₂(0) + 2T N)X

n∈Z

|d_n|².

This provides the second inequality in (33).

Step 2. We now argue by induction onN.

IfN = 0, equation (33) follows from (35). When N= 1, ifY ={dn1}, then f(t) =X

n/n∈Z∈Y

d_ne^iµnt+d_n1e^iµn1t. (37)

In Theorem 3.4 of [13], it was proved that forη >0 and T⁰=T−η, when f(t) =g(t) +d_n1, then γ²η⁴C₁(0)X

n/n∈Z∈Y

|dn|²≤ Z _T⁰

0

Z _η

0 (f(t+ψ)−f(t)) dψ

² dt≤4η² Z _T

0 |f(t)|²dt. (38) On the other hand,

|d_n1|² ≤ |f(t)−g(t)|²= 1 T

Z _T

0 |f(t)−g(t)|²dt≤ 2 T

(Z _T

0 |f(t)|²dt+ Z _T

0 |g(t)|²dt )

≤ 2 T







 Z _T

0 |f(t)|²dt+C₂(0)X

n/n∈Z∈Y

|dn|²







≤ 2

T + 8C₂(0) T γ²η²C₁(0)

Z _T

0 |f(t)|²dt. (39) Then, from (38) and (39) we have

X

n∈Z

|dn|² ≤

4

γ η²C₁(0)+ 2

T + 8C₂(0) T γ²η²C₁(0)

Z _T

0 |f(t)|²dt

≤ 2

T + 4

γ²η²C₁(0)

2C₂(0)

T + 1

Z _T

0 |f(t)|²dt. (40)

IfN = 2, letY ={n₁, n₂}. In particular, if n₁= 0, we writef(t) as f(t) = X

n/n∈Z∈Y

d_ne^iµnt+d₀e^iµn1t+d_n2e^iµn2t.

Setting

g(t) = X

n6=nn∈Z2

d_ne^iµnt+d₀,

and applying the result in step 1 to g(t) we have X

n6=nn∈Z2

|dn|²≤ 2

T + 4

γ²η²C₁(0)

2C₂(0)

T + 1

Z _T

0 |g(t)|²dt and

(2C₂(0) + 2T)X

n6=nn∈Z2

|dn|²≥ Z _T

0 |g(t)|²dt.

Thus, from the previous estimate it follows that

|dn2|² ≤ |f(t)−g(t)|²= 1 T

Z _T

0 |f(t)−g(t)|²dt≤ 2 T

(Z _T

0 |f(t)|²dt+ Z _T

0 |g(t)|²dt )

≤ 2 T



 Z _T

0 |f(t)|²dt+ (2C₂(0) + 2T) X

k∈Z−{n2}

|d_n|²





≤ 2

T + 4

γ²η²C₁(0)

4C₂(0)

T + 4

Z _T

0 |f(t)|²dt. (41)

Iterating this argument the first inequality in (33) follows.

2.2.2. Proof of Theorem 2.2

In order to represent the solution in C_h(γ) in a simpler way we introduce I_h(γ) =

k∈N: 1≤k < 1

h−2 and β_k(h)≤γh⁻⁴

and the functions

m_k,j =









c⁺_k φ_k,j k∈Z∩I_h c⁻_−k φ_−k,j −k∈Z∩I_h 0 k /∈Z∩I_h

(42)

µ_k(h) =

( pβ_k(h) k∈N

−p

β_−k(h) −k∈N. (43)

The components of~u_h, are then given by

u_h,j=X

k∈Z

m_k,je^iµk(h)t. (44)

Thus,

u⁰_h,j = X

k∈Z

iµ_k,j(h)m_k,j e^iµk(h)t (A⁻¹_h u~_h⁰)_h,j = −X

k∈Z

im_k,je^iµk(h)t. (45)

Substituting identities (44) and (45) in (15) and having into account that the matrixA_h= _h¹₂A is orthogonal, we deduce that

E^h(t) =h XN j=0

X

k∈N

(

|c⁺_k|²

φ_k,j+1−φ_k,j h

²+|c⁻_k|²

φ_k,j+1−φ_k,j h

² )

· (46)

Moreover, from (46) and the identity (17) we have that E^h(t) ≤ 2

4− √γ X

k∈N

|c⁺_k|²+|c⁻_k|² φ_k,N h

²= 2 4− √γ

X

k∈Z

m_k,N h

²· (47)

From Lemma 2.2, for allT >0 existsδ₀>0 andk₀∈N, such that µ_k+1(h)−µ_k(h)≥2π

T ∀k∈K₀(h) =

k₀+ 1, k₀+ 2, . . . ,1−3h h −k₀

with|h| ≤δ₀. (48) Hence Y ={1,2, . . . , k₀,^1−2h_h −k₀, . . . ,^1−2h_h }.

Let us observe that, independently of the size ofh, the set of indicesY is constituted by 2k₀ elements.

On the other hand, in (ii) of Lemma 2.1 it was proved that for all∈(0,1), there existsδ₁>0, such that µ_k+1(h)−µ_k(h)≥3π²− ∀k∈Z and |h|< δ₁. (49) Thus, applying Lemma 2.3 in (47) it is immediate to check that, for all T >0, withδ= min{δ₀, δ₁}>0, there exists a constantC=C(T, γ)>0, such that

E^h(0)≤C Z _T

0

u_h,N(t) h

² dt, ∀~u_h∈C_h(γ) with|h|< δ. (50)

This completes the proof of Theorem 2.2.

2.3. Uniform observability by reinforced boundary measurements

The goal of this section is to show that, in view of the gap properties obtained in the previous section, the observability inequality is uniform if the boundary measurement is reinforced in a suitable way.

Theorem 2.3. For allT >0, there existsC=C(T)>0 such that

E^h(0)≤C (Z _T

0

u_h,N(t) h

²dt+h² Z _T

0

u⁰_h,N(t) h

²dt )

(51) for all ~u_hsolution of (7) and 0< h <1.

Proof. Consider~u_h with components as in (44). According to the identities (46) and (17),

E^h(0) = h XN j=1

X

k∈N

|c⁺_k|²+|c⁻_k|² φ_k,j+1−φ_k,j h

²=X

k∈N

2

(4− |µ_k(h)|h²)(|c⁺_k|²+|c⁻_k|²) φ_k,N

h ²

= X

k∈Z^∗

2 (4− |µ_k(h)|h²)

m_k,N h

²· (52)

On the other hand, from Theorems 2.2 and 2.1 we have that, for allT >0 there exists δ >0, such that c₁ X

k∈Z^∗

m_k,N h

²≥ Z _T

0

u_h,N(t) h

² dt≥C₁ X

k∈Z^∗

m_k,N h

², (53)

for all |h|< δ.

Analogously,

c₂ X

k∈Z^∗

|µk(h)|²m_k,N h

² ≥ Z _T

0

u⁰_h,N(t) h

² dt= Z _T

0

X

k∈Z^∗

iµ_k(h) e^iµk(h)t m_k,N h

2

dt

≥ C₂ X

k∈Z^∗

|µ_k(h)|²m_k,N h

²· (54)

Thus, for ˆC= min{C₂, C₁}: Z _T

0

u_h,N(t) h

² dt+h² Z _T

0

u⁰_h,N(t) h

² dt≥Cˆ X

k∈Z^∗

1 +h²|µ_k(h)|² m_k,N h

²

≥Cˆ X

k∈Z^∗

(1 +h²|µ_k(h)|²)

4− |µk(h)|h²

4− |µ_k(h)|h² m_k,N h

²· (55)

On the other hand, for hsufficiently small, 1 +h²|µk(h)|²

4−h²|µk(h)|

= 4

1 + 16 h²sin⁴

|k|πh 2

cos²

|k|πh 2

≥4

cos²

|k|πh 2

+k²π²sin²

|k|πh 2

≥4, for|k|= 1,2, . . . ,1/h−1.

(56) Therefore,

Z _T

0

u_h,N(t) h

²dt+h² Z _T

0

u⁰_h,N(t) h

²dt≥c X

k∈Z^∗

1 (4− |µ_k(h)|h²)

m_k,N h

²

≥c hX

k∈N

XN j=1

(|c⁺_k|²+|c⁻_k|²)

φ_k,j+1−φ_k,j h

²· (57)

This concludes the proof of Theorem 2.3.

2.4. The reverse inequalities

Proposition 2.1. For anyT >0, there exists c=c(T)>0 such that E^h(0) ≥ c

Z _T

0

u_h,N(t) h

²dt, ∀~u_h, 0< h <1. (58)

Proof. According to the identities (17) and (46), we have

E^h(t) = X

k∈Z^∗

2

(4−λ_k(h)h²)|m_k,h(t)|² φ_k,N

h ²·

As 4−λ_k(h)h²= 4 cos²(^kπh₂ )≤4,

E^h(t)≥ 1 2

X

k∈Z^∗

|mk,h(t)|² φ_k,N

h

²· (59)

Then, applying Lemma 2.3, taking (59) into account and properties (48) and (49) on the eigenvalues under

consideration we deduce (58).

As an immediate consequence of Proposition 2.1 the following holds:

Proposition 2.2. For anyT >0, there existsc=c(T)>0such that

E^h(0) +h²E_∗^h(0)≥c (Z _T

0

u_h,N(t) h

²dt+h² Z _T

0

u⁰_h,N(t) h

²dt )

, ∀~u_h, (60)

where E_∗^h is the energy associated to~u_h⁰ instead of ~u_h, i.e.

E_∗^h(t) =h 2

XN j=0





u⁰_h,j+1−u⁰_h,j h

²+

(A⁻¹_h ~u_h⁰⁰)_h,j+1−(A⁻¹_h ~u_h⁰⁰)_h,j h

2



· (61)

Remark 2.3. Due to the fact that (u~⁰_h)⁰ =~u_h⁰⁰=−A²_h~u_h, the energy E_∗^h satisfies

E_∗^h(t) =h 2

XN j=0

(u⁰_h,j+1−u⁰_h,j h

²+

(A_h~u_h)_h,j+1−(A_h~u_h)_h,j h

² )

· (62)

3. Control of the semi-discrete equation

In this section, we apply the observability results obtained above to analyze the controllability properties of the semi-discrete system.

3.1. The semi-discrete control problem

LetP ={0 = x₀ < x₁ <· · · < x_N+1 = 1}. It is natural to introduce the following approximations of the boundary conditions in (6):

y_h,0=y_h,N+1= 0 y_h,1−2y_h,0+y_h,−1= 0

y_h,N+2−2y_h,N+1+y_h,N =h²ν_h(t). (63)

Let us now consider the following controlled semi-discrete systems



















y⁰⁰_h,j=− 1

h⁴{y_h,j+2−4y_h,j+1+ 6y_h,j−4y_h,j−1+y_h,j−2}, 0< t < T, 1≤j≤N

y_h,0=y_h,N+1= 0, 0< t < T

y_h,−1=−yh,1 y_h,N+2=−yh,N+h²ν_h, 0< t < T y_h,j(0) =y⁰_h,j y⁰_h,j(0) =y¹_h,j, j= 1, . . . , N.

(64)

This system may be viewed as a semi-discretization of (6) but also as the semi-discrete system (7) under the action of a controlν_h.

We denote by~y_h(t) = (y_h,1(t), y_h,2(t), . . . , y_h,N(t)) the solution of (64), with controlν_h. Note that,y_h,j⁰ and y¹_h,j are, as usual, approximations of the initial conditions (y⁰, y¹) of the control problem (6) at the pointsx_j.

3.2. The discrete spaces H

and H

The eigenvectorsφ~^k(h) of the spectral problem (8) satisfy

h XN j=1

|φ_k,j(h)|²=h XN j=1

sin²(jkπh) = 1/2. (65)

However, to simplify the notation, in what follows, we shall normalize them so that

h|φ~^k(h)|²_RN =h XN j=1

|φ_k,j(h)|²= 1. (66)

For every s∈R, introduce the finite dimensional Hilbert spaces H^s_h = span{φ~¹(h), . . . , ~φ^N(h)} endowed with the norm

k~v_hk²_s,h= XN k=1

λ^s_k(h)|ck|², whenever ~v_h= XN k=1

c_k φ~^k(h). (67)

In particular,H⁰_h will be denoted byL²_h.

Remark 3.1. The norm inH⁻¹_h is the dual to that inH¹_h in the sense that k~v_hk_−1,h= sup

~ wh∈H¹_h

|(~v_h, ~w_h)_0,h|

kw~_hk1,h , (68)

where (~v_h, ~w_h)_0,h=h P^N

j=1v_h,jw_h,j denotes the scalar product inL²_h.

We also introduce the discrete energy spaceFh=H¹_h× H⁻¹_h , with norm

k(~u_h, ~v_h)k²_Fh =k~u_hk²_1,h+k~v_hk²_−1,h. (69) The dual ofFh is denoted byF_h^∗.