Partial Exact Controllability for Spherical Membranes

PARTIAL EXACT CONTROLLABILITY FOR SPHERICALMEMBRANES∗

PAOLA LORETI† AND VANDA VALENTE†

SIAM J. CONTROL OPTIM. c© 1997 Society for Industrial and Applied MathematicsVol. 35, No. 2, pp. 641–653, March 1997 015

Abstract. In this paper the partial exact controllability of an elastic spherical membrane isproved. The reachability problem for the integrodifferential equation, introduced for the vibrationsof the meridional displacement, is solved. The main result is a generalization of a Ingham’s theoremon nonharmonic Fourier series.

Key words. partial exact controllability, reachability problem, almost periodic functions

AMS subject classifications. 35Q99, 49E15

PII. S0363012994269624

Introduction. Following Love’s and Koiter’s linear shell theory [17], [10], in pre-vious works [3], [4], [5], [6] we studied the problem of exact controllability for a thinelastic shell. The mathematical model is a system of partial differential equationswhere the unknown is the displacement vector v of the middle shell surface. We de-noted by Am and Af the operators associated with the membrane energy and flexionenergy, respectively, and by ε a small parameter depending on h (shell thickness) withlimh→0 ε = 0. The spectrum behavior of A = Am + εAf can be utilized to provesome results of exact controllability for the vibrations of thin shells. In the particularcase of an hemispherical shell, the existence of an asymptotic gap for the eigenvaluesλj of A allowed us to give a controllability time (depending on ε) and to prove exactcontrollability in suitable initial data spaces. In the general case we pointed out thatwhen the thickness of the shell goes to zero, the number of eigenvalues of A less thana fixed λ ≥ λ0 goes to infinity; that is,

Nλ(A) =∑λj<λ

1→∞ as h→ 0.

It suggests therefore that an accumulation point for the eigenvalues of the limit prob-lem ε = 0 (the so-called membrane approximation) may occur; moreover, we provedthat exact controllability of the limit problem generally fails, and an example of nonex-act controllability is constructed in the case of hemispherical membrane approximation(see [6]).

In this paper we give a result of partial controllability for a spherical membrane;i.e., we want to control only one of the displacement components without conditionsfor the other components. The axially symmetric vibrations of a spherical membraneare described in section 1 by a pair of partial differential equations in the meridionaland radial displacements u and w. The partial exact controllability problem is givenin terms of the reachability problem for the integrodifferential equation for the vibra-tions of the meridional displacement. We propose the reverse or reachability Hilbertuniqueness method (RHUM) [11], [14], [15], [16] to construct our control function. Insection 2, the well posedness of the corresponding homogeneous problem is proved.In section 4 we give a result of partial exact controllability (PEC) taking into account

∗Received by the editors October 15, 1995; accepted for publication (in revised form) February22, 1996.

http://www.siam.org/journals/sicon/35-2/26962.html†Istituto per le Applicazioni del Calcolo, Consiglio Nationale delle Ricerche, viale del Policlinico,

137, 00161 Rome, Italy.

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642 PAOLA LORETI AND VANDA VALENTE

a generalization of the Ingham theorem [8], which we prove in section 3. Results onthe reachability problem for plate equation with memory can be found in [9], [11],[12], [13].

1. Statement of the problem. We consider the axially symmetric vibrationsof an elastic spherical membrane with middle ray R and opening angle θ0. Themeridional and radial components of the displacement vector v = (u(θ, t), w(θ, t))satisfy in Q = (0, T )× (0, θ0) the system

(1.1)

dutt − L(u)− (1− ν)u+ (1 + ν)w′ = 0,

dwtt − (1+ν)sin θ (u sin θ)′ + 2(1 + ν)w = 0

with

u(0, t) = 0, u(θ0, t) = g(t)

and

v (θ, 0) = v 0, vt(θ, 0) = v 1,

where

(1.2) L(u) =(

(u sin θ)′

sin θ

)′.

The prime stands for the first derivative with respect to θ, ν ∈ (−1, 1/2), and d =d0 R2(1 − ν2)/E, where E is an elastic positive constant and d0 is the density ofthe material. In what follows we change t =

√d t. We introduce the following

spaces:

L2 = L2(0, θ0; sin θ dθ) =

{f :∫ θ0

0|f |2 sin θ dθ < +∞

},

U ={u :

∂u

∂θ, u cot θ ∈ L2(0, θ0; sin θ dθ) , u(0) = u(θ0) = 0

}.

‖f‖0 is the norm induced by the scalar product (f, g)0 =∫ θ0

0 f ·g sin θ dθ, and ‖u‖2U =‖u′‖20 + ‖u cot θ‖20. The exact controllability problem requires that we find a controlfunction g(t) that drives the system to the rest in a finite time T . In [6] we provedthat the membrane approximation is not exactly controllable for any {v 1,v 0} ∈(U ′×L2)× (L2×L2). We observed that for the hemispherical membrane there existsa subsequence of eigenfunctions v ∗n(θ) = (u∗n(θ), w∗n(θ)) such that

limn→∞

u∗n(π/2)− (1 + ν)w∗n(π/2) = 0.

Then the sequence {v ∗n , 0} with ‖v ∗n‖U×L2 = 1 (initial data for the homogeneousproblem associated with (1.1)) does not satisfy the necessary (and sufficient) conditionof exact controllability.

Since the exact controllability for membrane approximation generally fails, welook for a partial result; i.e., we look for a PEC result.

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PARTIAL EXACT CONTROLLABILITY FOR SPHERICAL MEMBRANES 643

The problem (1.1) is equivalently written in the form

utt − L(u)− (1− ν)u+ (1 + ν)(w0)′ cos t√

2(1 + ν) +(1 + ν)√2(1 + ν)

(w1)′

· sin t√

2(1 + ν) +(1 + ν)3/2√

2

∫ t

0sin((t− s)

√2(1 + ν))L(u(θ, s)) ds = 0.

We consider the following partial controllability (PC) problem.

(PC) Given T > 0 and two functions u0(θ), u1(θ), find g(t) such that the system

(1.3)

utt − L(u)− (1− ν)u+ (1 + ν)w′ = 0,

wtt − (1+ν)sin θ (u sin θ)′ + 2(1 + ν)w = 0,

with boundary conditions

(1.4) u(0, t) = 0, u(θ0, t) = g(t)

and null initial data, i.e.,

(1.5) v (θ, 0) = 0, v t(θ, 0) = 0,

verifies the final conditions

(1.6) u(θ, T ) = u0, ut(θ, T ) = u1

or, equivalently,

(PC)′ Given T > 0 and two functions u0(θ), u1(θ), find g(t) such that the system

utt − L(u)− (1− ν)u+(1 + ν)3/2√

2

∫ t

0sin((t− s)

√2(1 + ν))L(u(θ, s)) ds = 0,

with

u(0, t) = 0, u(θ0, t) = g(t)

and

v (θ, 0) = 0, v t(θ, 0) = 0,

verifies the conditions

u(θ, T ) = u0, ut(θ, T ) = u1.

To solve the problem (PC)′ we apply the RHUM method [11], [16].We consider the adjoint system

(1.7) ztt − L(z)− (1− ν)z +(1 + ν)3/2√

2

∫ T

t

sin((s− t)√

2(1 + ν))L(z(θ, s)) ds = 0

with homogeneous boundary conditions

(1.8) z(0, t) = 0, z(θ0, t) = 0

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and final data

(1.9) z(θ, T ) = z0, zt(θ, T ) = z1.

Then we consider the problem

φtt − L(φ)− (1− ν)φ+(1 + ν)3/2√

2

∫ t

0sin((t− s)

√2(1 + ν))L(φ(θ, s)) ds = 0

with

φ(0, t) = 0, φ(θ0, t) = − (1 + ν)3/2√

2

∫ T

t

sin((s− t)√

2(1 + ν))z′(θ0, s) ds+ z′(θ0, t)

and

φ(θ, 0) = 0, φt(θ, 0) = 0.

We define µ{z0, z1} = {φ0t (T ),−φ0(T )} and put

Gz(θ, t) =∫ t

0sin((t− s)

√2(1 + ν))z(θ, s) ds

and the adjoint operator of G by G∗, where

(1.10) G∗z(θ, t) =∫ T

t

sin((s− t)√

2(1 + ν))z(θ, s) ds.

With some simple computations we can prove

〈µ{z0, z1}, {z0, z1}〉 =∫ T

0sin θ0φ(θ0, t)z′(θ0, t) dt

− (1 + ν)3/2√

2

∫ T

0G∗z′(θ0, t)φ(θ0, t) sin θ0 dt.

Our aim is to prove that ν{z0, z1} is invertible in a suitable function space.

2. Analysis of the homogeneous problem. In order to prove existence anduniqueness results for (1.7), (1.8), and (1.9), we introduce the energy

E(t) =12

∫ ϑ0

0(z2t + (z′)2 + z2 cot2 ϑ+ νz2) sinϑ dϑ

and

ET =12

∫ ϑ0

0(z12

+ (z0′)2 + (z0)2 cot2 ϑ+ ν(z0)2) sinϑ dϑ.

We have the following proposition.PROPOSITION 2.1. There exist two constants C1(T, ν) and C2(T, ν) such that

E(t) ≤ C1(T, ν)ET · eC2(T,ν)(T−t).

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Proof. To prove Proposition 2.1 we argue in similar way to [16, vol. 2, p. 242].To this end we introduce

r0(t) =∫ ϑ0

0

∫ T

t

sin((s− t)√

2(1 + ν))(z(ϑ, s) sinϑ)′

· 1√sinϑ

ds[z(ϑ, t) sinϑ]′1√

sinϑdϑ

and

r1(t) =∫ ϑ0

0

∫ T

t

cos((s− t)√

2(1 + ν))(z(ϑ, s) sinϑ)′

· 1√sinϑ

ds[z(ϑ, t) sinϑ]′1√

sinϑdϑ.

Applying the Schwarz inequality after simple computations we obtain

|r0(t)| ≤ 4E(t)√

27+√

2716

{T − t

2− 1

4sin 2(T − t)

√2(1 + ν)√

2(1 + ν)

}∫ T

t

E(s) ds,

and similarly we obtain

|r1(t)| ≤ E(t) +14

(T − t

2+

14

sin 2(T − t)√

2(1 + ν)√2(1 + ν)

)∫ T

t

E(s) ds.

On the other hand, it is easy to show that

d

dt

(E(t) +

√2

2

√(1 + ν)3r0(t)

)= −(1 + ν)2r1(t).

Hence,

E(t) ≤ ET +√

22

√(1 + ν)3|r0(t)|+ (1 + ν)2

∫ T

t

|r1(s)| ds.

We compute ∫ T

t

|r1(s)| ds ≤(

1 +14

∫ T

t

M21 (s) ds

)∫ T

t

E(s) ds,

where

M21 =

T − t2

+14

sin 2(T − t)√

2(1 + ν)√2(1 + ν)

,

E(t) ≤ ET + 2√

2(1 + ν)3E(t)√27

+√

27164

(T +

12√

2(1 + ν)

)√2(1 + ν)3

∫ T

t

E(s) ds

+ (1 + ν)2

(1 +

14

∫ T

t

M21 (s) ds

)∫ T

t

E(s) ds.

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It is straightforward to find∫ T

t

M21 (t) dt ≤ T 2

4+

18(1 + ν)

.

From the above estimates we conclude choosing

c1(T, ν) =ET(

1−√

22

√(1 + ν)3 2√

27

) ,

c2(T, ν) =((1 + ν)2T

2

16+√

27164

√2(1+ν)3T +

(√27

1128

+(1+ν)+18

))(1+ν).

From the above result we obtain the following.PROPOSITION 2.2. The problem (1.2) has a unique solution such that

{z, z′} ∈ C([0, T ]; U × L2).

Now we shall find the explicit solution of the equation (1.2) with boundaryhomogeneous conditions and the final state z0, z1.

So we consider the eigenvalues problem

(2.1) −L(zk) = λkzk,

where L is the operator given in (1.2) and zk(0) = zk(ϑ0) = 0.We assume that the solution of our problem can be written as

(2.2) z(ϑ, t) =+∞∑k=1

fk(t)zk(ϑ),

where zk is an orthonormal base in (0, θ0).Substituting z, given by (2.2), using the eigenvalue problem (2.1), and multiplying

the equation by zk we find the following integrodifferential equation in the unknownf (here the dot denotes the derivative with respect to t):

fk(t) + (λk − 1 + ν)fk(t)− λk(1 + ν)2√2(1 + ν)

∫ t

0sin((t− s)

√2(1 + ν))fk(s) ds = 0.

Using [18, p. 149] and the method of Evans [18, p. 67], we find the solution

fk(t) = (fTk + fTk t)(

1 +b1ka+k

+b2ka−k

)− fTk

(b1ka+k

cos(a+k t) +

b2ka−k

cos(a−k t))

− fTk(

b1k(a+k )2

sin(a+k t) +

b2k(a−k )2

sin(a−k t))

and b1k, b2k, a+k , a−k given, respectively, by

b1k = − (λk − 1 + ν)[(a−k )2 − (λk − 1 + ν)]− λk(1 + ν)2

a+k [(a−k )2 − (a+

k )2],(2.3)

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b2k =(λk − 1 + ν)[(a+

k )2 − (λk − 1 + ν)]− λk(1 + ν)2

a−k [(a−k )2 − (a+k )2]

,(2.4)

a±k =

√(λk + 1 + 3ν)±

√(λk + 1 + 3ν)2 − 4(1− ν2)(λk − 2)

2.(2.5)

Doing the substitution t = T − t in (2.2) we find the solution of (1.7)–(1.9): z =∑+∞k=1 fk(t)zk(θ) with

fk(t) = (fTk − fTk (T − t))(

1 +b1ka+k

+b2ka−k

)− fTk

(b1ka+k

cos(a+k (T − t)) +

b2ka−k

cos(a−k (T − t)))

+ fTk

(b1k

(a+k )2

sin(a+k (T − t)) +

b2k(a−k )2

sin(a−k (T − t))).

Taking into account that

b1ka+k

+b2ka−k

= −1 ∀ ν ∀ k,

we have

fk(t) = −fTk{b1ka+k

cos(a+k (T − t)) +

b2ka−k

cos(a−k (T − t))}

+ fTk

{b1ka+2k

sin(a+k (T − t)) +

b2k(a−k )2

sin(a−k (T − t))}.

PROPOSITION 2.3. The following properties of the coefficients a+k and a−k hold:

(a+k )2 = λk − 1 + ν + c+k (1 + ν)2,

(a−k )2 = 2(1 + ν) + c−k (1 + ν)2,

with

limk→∞

c+k = +1, limk→∞

c−k = −1,

limk→∞

a+n = +∞, lim

k→∞a−k =

√1− ν2.

Remark 2.1. The solution of this problem can be easily given at least in thecase ϑ0 = π

2 . More precisely we have zk = akP′k and λk = k(k + 1) with k =

2, 4, 6, . . . , and Pk is the k–Legendre polynomial and ak is the normalization factor;i.e., (akP ′k, ajP

′j) = δkj .

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3. Trigonometrical inequalities for almost periodic functions. The fol-lowing theorem, on almost periodic functions [2], is a generalization of a result due toIngham (see [8, Theorem 1]).

THEOREM 3.1. Let

f(t) =+∞∑

n=−∞A+n e−ia+

n t +A−n e−ia−n t =

∑n

A+n e−ia+

n t +A−n e−ia−n t,(3.1)

where we assume∑nA

+n e−ia+

n t +A−n e−ia−n t is uniformly convergent in [−T, T ] and

∃γ1 > 0 : |a+n − a+

n−1| ≥ γ1 ∀n,(3.2)

∃γ2 > 0 : |a+n − a−j | ≥ γ2 ∀n ∀ j,(3.3)

∃α ≥ 1 , C2 > 0 : ∀ k |A−k | ≤C2

kα|A+k |.(3.4)

Then ∃T0 : ∀T > T0 : ∃C3(T ), C4(T ) > 0 such that

C3(T )∑n

|A+n |2 ≤

∫ T

−T|f |2 dt ≤ C4(T )

∑n

|A+n |2.

Proof. We prove the first inequality. Let h(t) be a nonnegative integrable functionover (−∞,+∞). We consider∫ +∞

−∞h(t)|f(t)−

∑k

A+k e−ia+

kt|2 dt

=∫ +∞

−∞h(t)

(f(t)−

∑k

A+k e−ia+

kt

)·

f(t)−∑j

A+j e

ia+jt

dt

=∫ +∞

−∞h(t)|f(t)|2 dt+

∫ +∞

−∞

∑k

∑j

A+k A

+j e

i(a+j−a+

k)th(t) dt

−∫ +∞

−∞h(t)f(t) ·

∑k

A+k e−ia+

kt dt−

∫ +∞

−∞h(t)f(t)

∑j

A+j e

+ia+jt dt

=∫ +∞

−∞h(t)|f(t)|2 dt+

∑k

∑j

A+k A

+j K(a+

k − a+j )

−∫ +∞

−∞h(t)

∑j

A+j e

ia+jt +A

−j e

ia−jt

·∑k

A+k e−ia+

kt dt

−∫ +∞

−∞h(t)

(∑k

A+k e−ia+

kt +A−k e

−ia−kt

)·∑j

A+j e

ia+jt dt,

where

K(u) =∫ +∞

−∞e−itu · h(t) dt.

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Hence, using∑k

∑′j to denote the sum for k 6= j,

0 ≤ −∑k

|A+k |2K(0)−

∑k

′∑j

A+k A

+j K(a+

k − a+j )

−∑j

∑k

A−j A

+kK(a+

k − a−j )

−∑k

∑j

A−k A+j K(a−k − a

+j ) +

∫ +∞

−∞h(t)|f |2 dt,

we take

h(t) =

{cos πt

2T , |t| ≤ T,

0, |t| > T,

with T > π2γ and γ = min{γ1, γ2}

∫ +∞

−∞h(t)ei(a

+k−a+

j)t dt =

4Tπ cos(a+

k−a+

J)T

π2−4T 2(a+k−a+

j)2 , k 6= j,

4Tπ , k = j.

By assumption (3.2)

∑k

∣∣∣∣∣ 4Tπ cos(a+k − a

+j )

π2 − 4T 2(a+k − a

+j )2

∣∣∣∣∣ ≤ 4πTγ2

1≤ 4πTγ2

and by assumption (3.3)

∑k

∣∣∣∣∣ 4Tπ cos(a+k − a

−j )

π2 − 4T 2(a+k − a

−j )2

∣∣∣∣∣ ≤ 4πTγ2

2≤ 4πTγ2

so that∫ T

−T|f(t)|2 dt ≥ 4T

π

∑k

|A+k |2 +

∑k

′∑j

A+k A

+j K(a+

k − a+j )

+∑k

∑j


+j ) +

∑j

∑k

A−j A

+kK(a+

k − a−j )

≥ 4Tπ

∑k

|A+k |2 −

4πTγ2

1

∑k

|A+k |2

− C2

∑k

∑j

|A+k ||A

+j ||K(a−k − a

+j )|

kα− C2

∑k

∑j

|A+j |jα|A+k ||K(a+

k − a−j )|

≥∑k

|A+k |2(

4Tπ− 4πTγ2

1

)−∑k

∑j

|A+k |2|K(a+

j − a−k )|

− C22

∑k

∑j

|A+j |2|K(a+

j − a−k )|k2α .

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If we put S = C22∑J

1J2α , α ≥ 1, we have∫ T

−T|f(t)|2 dt ≥

(4Tπ− 4πTγ2

1− 4πTγ2

2− S4πTγ2

2

)·∑k

|A+k |2.

This inequality is verified for any T > T0 with

(3.5) T0 ={

inf T :4Tπ− 4πTγ2

1− 4πTγ2

2− S4πTγ2

2> 0}

= π

√1 + S

γ22

+1γ2

1.

Next, we prove the second inequality.We compute

∫ T

−T|f(t)|2h(t) dt

=∫ T

−T

(∑k

A+k e−ia+

kt +A−k e

−ia−kt

)·

∑j

A+J e

+ia+jt +A

−j e

ia−jt

h(t) dt

=∫ T

−T

∑k

∑j

A+k A

+j e

i(a+j−a+

k)th(t) dt+

∫ T

−T

∑k

∑j

A−k A−j e

i(a−j−a−

k)t · h(t) dt

+∫ T

−T

∑k

∑j

A+k A−j e

i(a−j−a+

k)t · h(t) dt+

∫ T

−T

∑k

∑j

A−k A+j e

i(a+j−a−

k)t · h(t) dt,

∫ T

−T|f |2h(t) dt =

∑k

∑j

A+k A

+j K(a+

k − a+j )

+∑k

∑j

A−k A−j K(a−k − a

−j ) +

∑k

∑j

A+k A−j K(a+

k − a−j )

+∑k

∑j


+j ) ≤

∑k

∑j

|A+k ||A

+j |K(a+

k − a+j )|

+∑k

∑j

|A+k |kα|A+j |jα|K(a−k − a

−j )|+ 2

∑k

∑j

|A+k ||A+j |jα|K(a+

k − a−j )|

≤(

4Tπ

+4πTγ2

)∑k

|A+k |2 +

4TπC2

2

{∑k

|A+k |2∑j

1j2α +

∑j

|A+j |2∑k

1k2α

}

+ C22

∑k

|A+k |2∑j

|K(a−j − a+k )|

j2α +∑j

|A+j |2∑k

|K(a−j − a+k )|

≤(

4Tπ

+4πTγ2 +

8T S

π

)∑k

|A+k |2+

(4πTγ2 S +

4πTγ2

)∑k

|A+k |2 = C4(T )

∑k

|A+k |2.

We conclude the proof choosing C4(T ) = 2√2C4(2T ).D

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.120

.242

.61.

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; see

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://w

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.sia

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jsa.

php


Remark 3.1. In an analogous way to [1] (see also [7]), the assumptions (3.2) and(3.3) can be relaxed with

(3.2′) ∃γ1 > 0 : |a+k+1 − a

+k | ≥ γ1 ∀|k| > K,

(3.3′) ∃γ2 > 0 : |a+k − a

−j | ≥ γ2 ∀|k| > K and ∀j.

If γ2 → +∞ as K → +∞ (as in the case of spherical membranes, see Proposition2.3), we find from (3.5) that the proof of Theorem 3.1 holds with

(3.6) T0 =π

γ.

4. Controllability. In what follows we put α = (1+ν)3/2√

2.

LEMMA 4.1. For every fixed T > 0, there exist two positive constants c0 = c0(T, α)and c1 = c1(T, α) such that

c0

∫ T

0(z′ − αG∗z′(θ0, t))2 dt ≤

∫ T

0(z′(θ0, t))2 dt ≤ c1

∫ T

0(z′ − αG∗z′(θ0, t))2 dt.

Proof. The left inequality follows by simple computations. To prove the rightinequality we assume that there exists a sequence z′n such that ‖z′n‖L2 = 1 and

(4.1) z′n(t)− αG∗z′n(t) = fn(t),

with

limn→∞

‖fn‖L2 = 0.

By simple computation [18, p. 45] we have by (4.1),

(4.2) z′n(t) = fn(t)−∫ T

t

Q(s, t)fn(s) ds,

where Q is a reciprocal kernel of (4.1), (1.10) given by

Q(s, t) =∞∑h=1

P (h)(s, t),

where

|P (h)(s, t)| ≤ |α|h|s− t|h−1

(h− 1)!.

It is easy to prove that

|Q(s, t)| ≤ |α|e|α| |s−t|.

On the other hand,∫ T

t

Q2(s, t) ds ≤ α2∫ T

t

e2|α| |s−t| ds =α2

2|α| (e(T−t)·2|α| − 1)D

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129

.120

.242

.61.

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istr

ibut

ion

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ect t

o SI

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ight

; see

http

://w

ww

.sia

m.o

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urna

ls/o

jsa.

php


and ∫ T

0

∫ T

t

Q2(s, t) ds ≤ |α|2

∫ T

0(e(T−t)2|α| − 1) dt =

14e2T |α| − 1

4− |α|

2T,

so that ∫ T

0(z′n(θ0, t)2 dt ≤ 2

∫ T

0(fn(s))2 ds

{34

+14e2T |α| − |α|

2T

}.

Hence we find a contradiction.THEOREM 4.1. If the generalized assumptions of Theorem 3.1 are verified, then

we can solve the partial controllability problem in a suitable space.Proof. We take

A+k =

b1k(a+k )2

z′k(θ0)[a+k f

Tk +

1ifTk

],

A−k =b2k

(a−k )2z′k(θ0)

[a−k f

Tk +

1ifTk

],

where a+k , a−k , b1k, and b2k are given by (2.3), (2.4), and (2.5). By Lemma 4.1 and

Theorem 3.1 we have that there exist two positive constants C1(T ) and C2(T ) suchthat

C1(T )∑k

|A+k |2 ≤ 〈µ(z0, z1) , (z0, z1)〉 ≤ C2(T )

∑k

|A+k |2.

This ends the proof.Example 4.1 (PEC for the hemispherical shell).For θ0 = π/2, we have

λk = 2k(2k + 1), k = 1, 2, . . . ,

a+k ∼ 2k +

12

as k →∞,

|a+k+1 − a

+k | = 2 , lim

k→∞a−k =

√1− ν2 < a+

1 .

Moreover,

b1k(a+k )2

∼ constk

as k →∞

and

b2k(a−k )2

∼ constk2 as k →∞.

Moreover, we assume the data {fT , fT } are given in a suitable space in order for∑k |A

+k | < +∞. Hence the hypotheses of Theorem 3.1 are verified, and we can apply

Theorem 4.1.

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REFERENCES

[1] J. M. BALL AND M. SLEMROD, Nonharmonic Fourier series and the stabilization of distributedsemi–linear control systems, Comm. Pure Appl. Math., 37 (1979), pp. 555–587.

[2] A. S. BESICOVITCH, Almost Periodic Functions, Cambridge University Press, Cambridge, UK,1932.

[3] G. GEYMONAT, P. LORETI, AND V. VALENTE, Controllabilite exacte d’une modele de coquemince, C. R. Acad. Sci. Paris Ser. I Math., 313 (1991), pp. 81–86.

[4] G. GEYMONAT, P. LORETI, AND V. VALENTE, Exact controllability of a shallow shell model,in Optimization, Optimal Control and Partial Differential Equations (Iasi, 1992), Interna-tional Series of Numerical Mathematics 107, Birkhauser Verlag, Basel, 1992, pp. 85–97.

[5] G. GEYMONAT, P. LORETI, AND V. VALENTE, Exact controllability of a thin elastic hemi-spherical shell via harmonic analysis, in Boundary Value Problems for Partial DifferentialEquations and Applications, Masson, Paris, 1993.

[6] G. GEYMONAT, P. LORETI, AND V. VALENTE, Spectral problem for thin shells and exactcontrollability, in Spectral Analysis of Complex Structures, 49, Hermann, Paris, 1995,pp. 35–59.

[7] A. HARAUX, Series lacunaires et controle semi-interne des vibrations d’une plaque rectangu-laire, J. Math. Pures Appl., 68 (1989), pp. 457–465.

[8] A. E. INGHAM, Some trigonometrical inequalities with applications to the theory of series,Math. Z., 41 (1936), pp. 367–379.

[9] J. U. KIM, Control of a plate equation with large memory, Differential Integral Equations,5 (1992), pp. 261–279.

[10] W. T. KOITER, On the foundations of the linear theory of thin elastic shells, Proc. Kon. Nederl.Akad. Wetensch., B73 (1970), pp. 169–195.

[11] J. E. LAGNESE AND J. L. LIONS, Modeling Analysis and Control of Thin Plates, Masson, Paris,1988.

[12] I. LASIECKA, Controllability of a viscoelastic Kirchhoff plate, in Control and Estimation of Dis-tributed Parameter Systems (Vorau, 1988), International Series of Numerical Mathematics91, Birkhauser Verlag, Basel, 1989, pp. 237–247.

[13] G. LEUGERING, Exact boundary controllability of an integro–differential equation, Appl. Math.Optim., 15 (1987), pp. 223–250.

[14] J. L. LIONS, Controllabilite exacte des systemes distribues, C. R. Acad. Sci. Paris Ser. I Math.,302 (1986), pp. 471–475.

[15] J. L. LIONS, Exact controllability, stabilization and perturbations for distributed systems, SIAMRev., 30 (1988), pp. 1–68.

[16] J. L. LIONS, Controllabilite exacte perturbation et stabilization des systemes distribues, 1 and2, Masson, Paris, 1988.

[17] E. H. A. LOVE, A Treatise on the Mathematical Theory of Elasticity, Cambridge UniversityPress, Cambridge, UK, 1927.

[18] V. VOLTERRA, Theory of Functionals and of Integral and Integro–differential Equations, DoverPublications, New York, 1959.

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Documents

Partial Exact Controllability for Spherical Membranes