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Simultaneous exact control of piezoelectric systems in multilayered media

Abstract

This paper considers a pair of transmission problems for the system of piezoelectricity having piecewise constant coefficients. Under suitable monotonicity conditions on the coefficients and certain geometric conditions on the domain and the interfaces where the coefficients have a jump discontinuity, results on simultaneous boundary observation and simultaneous exact control are established.

Piezoelectricity; transmission problem; simultaneous exact controllability


Simultaneous exact control of piezoelectric systems in multilayered media

Boris V. KapitonovI, * * Visiting Researcher at the National Laboratory of Scientific Computation (LNCC/MCT). ; Marco Antonio RauppII

ISobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences

IILaboratory of Scientific Computation, LNCC/MC, 25651-070 Quitandinha, Petrópolis, RJ, Brasil

E-mail: borisvk@lncc.br

ABSTRACT

This paper considers a pair of transmission problems for the system of piezoelectricity having piecewise constant coefficients. Under suitable monotonicity conditions on the coefficients and certain geometric conditions on the domain and the interfaces where the coefficients have a jump discontinuity, results on simultaneous boundary observation and simultaneous exact control are established.

Mathematical subject classification: 35L50; 35Q60; 35B40.

Key words: Piezoelectricity; transmission problem; simultaneous exact controllability.

1 Introduction

Throughout this paper W will be a bounded domain in

3 with sufficiently smooth boundary . For k = 1,2,...,n, let k be open, bounded and connected subsets of W with smooth boundary k, and such that k Ì k+1. We set

Assume that W is occupied by a linear multilayered piezoelectric body whose motion is governed by the following system ([4], [6])

where r is the mass density, u is the displacement vector, u is the symmetric part of Ñu, T is the stress tensor, H is the magnetic field, E is the electric field, is the electric displacement vector, m is the magnetic permeability, c, e, b are the elastic, piezoelectric and electric permittivity tensors respectively whose Cartesian components satisfy the following properties:

for any real vector x = (x1, x2, x3) Î 3,

for any real symmetric matrices {eij} of order 3.

We introduce the following matrices of order 3

where

It follows from the symmetry of cijkl that

Using these notations, we write the system (1.1) in the matrix form

We note that

where

It is assumed that

for any real vector hi Î 3. Here (·,·) denotes the inner product in 3. We remark that this assumption holds for an isotropic medium (cijkh = dijdkh + dikdjh + dihdjk) with a constant C0 = :

It is assumed that cijkh(x), bij(x), m(x) are piecewise constant functions which lose the continuity on 1,2,...,n, r and ekhi are constants, r > 0.

We consider the following transmission problems

and

where [·,·] is the vector product, n = n(x) = (n1,n2,n3) (for x Î m, x Î ) is the unit normal vector pointing into the exterior of m or W; , u(m), v(m), E(m), F(m), H(m), Y(m) are the restrictions of the corresponding matrices and vector-functions on Wm. In (1.5) b = b(x) is a continuously differentiable positive function on .

The problem of exact boundary control for the system (1.2)-(1.5) ((1.6)-(1.9)) is formulated as follows:

Given the initial distribution f = {f1,f2,f3,f4} (j = {j1, j2,j3,j4}), time T > 0, and a desired terminal state g = { g1, g2, g3, g4} (y = {y1, y2, y3, y4}) with f, g(j,y) in appropriate function spaces, find a vector-valued functions Q(x,t), G(x,t) ((x,t), (x,t)) in a suitable function spaces such that the solution of (1.2)-(1.5) ((1.6)-(1.9)) satisfies the conditions

Our purpose is to obtain simultaneous exact boundary control of these problems, {(x,t),(x,t)} serving as a control in problem (1.6)-(1.9), while the vector-valued functions

is a control in (1.2)-(1.5).

Spatial energy estimates for a semi-infinite piezoelectric beam have been studied by A. Borrelli and M.C. Patria [2].

Boundary controllability for some partial case of the system (1.1) with another boundary and interface conditions was investigated in [14].

For Ai º 0, the piezoelectric system (1.2) decouples into a pair of hyperbolic systems: the Maxwell system and the hyperbolic system of second order.

The exact controllability problem for the Maxwell system has been studied by D. Russell [27] for a circular cylindrical region, by K. Kime [16] for a spherical region, and by J. Lagnese [20] for a general region. Stabilization for the Maxwell system with the Silver-Müller absorbing boundary conditions and exact controllability for corresponding initial boundary value problem have been studied by V. Komornik [17], P. Martinez [24] and N. Weck [28]. The uniform exponential decay of solutions of Maxwell's equations with boundary dissipation and exact boundary controllability was proved in [7], [8].

Stabilization and exact boundary controllability for the system of elasticity have been studied by J. Lagnese [18], [19], F. Alabau and V. Komornik [1] and M. Horn [5] among others. In [7], [9] boundary observation, stabilization and exact controllability were studied for a class of hyperbolic systems which includes the system of elasticity.

Boundary controllability in transmission problems for a class of second order hyperbolic systems has been studied by J. Lagnese [19]. Uniform stabilization and exact control for the Maxwell system in multilayered media were investigated in [8]. The question of boundary controllability in transmission problems for the wave equation has been considered by J.-L. Lions [23], and S. Nicaise [25], [26].

The main novelty of this note is that we study the simultaneous exact control. Simultaneous exact control for the wave equation has been established by D. Russell [27] for a circular cylindrical region and by J.-L. Lions [22], F. Khodja and A. Bader [15].

In [9]-[13] simultaneous controllability were studied for a class of hyperbolic systems of second order, for a pair of Maxwell's equations and for a class of evolution systems which includes the Schrödinger equation.

This article is organized as follows: simultaneous boundary observation for the problems (1.2)-(1.5) and (1.6)-(1.9) with zero boundary conditions (Q º G º º º 0) is established in Section 2. In Section 3 the simultaneous exact controllability is studied by means of the Hilbert Uniqueness Method, introduced by J.-L. Lions [21], [22].

2 Boundary observability

Throughout this paper Hk(W) and Hq() denote the usual Sobolev spaces.

We denote by

0 the Hilbert space of pairs u = {u1(x),u2(x)} of three-component vector-valued functions

with the inner product

From the results of [3] it follows that the expressions [n, u1], [n, u2], (n = n(x), x Î , x Î m,m = 1,2,...,n) are well defined on , m and belong to .

This enables us to introduce in

0 the closed subspaces 1, 1:

the space

1 is defined just as 1 with the only difference that [n,u2] vanishes on .

We denote by the real Hilbert space of quadruples w = { w1,w2, w3, w4} of three-component vector-valued functions wi(x) such that

where is the restriction of wi on Wm. The inner product in is given by

The space is defined just as with the only difference that the first vector-valued function w1 vanishes on .

In and we define unbounded operators and :

() consists of the elements u = { u1, u2, u3, u4} Î such that

for u = { u1, u2, u3, u4} Î ().

The operator is defined just as with the only difference that elements

satisfy another boundary conditions

The skew-selfadjointness of and can be verified in the standard way.

Let (t) and (t) be the strongly continuous groups of unitary operators generated by and .

We set

Denote by M1 and 1 the orthogonal complements of M and in and respectively.

Let us consider the problem (1.2)-(1.5) with homogeneous boundary conditions (Q º G º 0). The kernel M of * is nonempty, since it contains the quadruples w = {w1,0,Ñg1, Ñg2}, where g1Î H2(W), g2Î H2 (W)Ç(W), w1 is a solution of the following problem

It is obvious that (t) takes M1 Ç () into itself. Indeed, if w Î M and v Î M2 Ç (), then

We remark that element v = {v1,v2,v3,v4} Î M1 Ç () possess the following property:

in the sense of distributions.

Indeed, element w = {w1, 0, Ñg1, 0} where g1Î H2(W), supp g1Ì Wm, w1 is a solution of the problem (2.1), belongs to M. Let v Î M1 Ç (). Then

for an arbitrary g1Î H2(W), supp g1Ì Wm, which implies

in the sense of distributions.

It can be shown in a similar way (element {0,0,0,Ñg2} belongs to M for an arbitrary g2Î H2(W), supp g2Ì Wm) that div = 0 in the sense of distributions.

Let us show that elements v = {v1,v2,v3,v4} Î M1 Ç () satisfy the boundary condition

We note that element w = {0,0,0,Ñg2} belongs to the kernel of * for an arbitrary g2Î H2(W), g2 = 0 in 0È 1È ... È n–1.

Thus, for v = {v1,v2,v3,v4} Î M1 Ç () we have

which implies (2.3).

Our next goal is to show that elements v = {v1,v2,v3,v4} Î M1 Ç () satisfy the following interface conditions

Since w = {0,0,0,Ñg2} belongs to the kernel of * for an arbitrary g2Î H2(W) Ç (W), it follows that

Now we choose g2 such that g2 = 0 on 1,...,m–1, m+1,...,n. Then

and we have

Moreover, element w = {w1,0,Ñg1,0} belongs to the kernel of * for an arbitrary g1Î (W) (w1 is a solution of (2.1)). We have

We choose g1 such that g1 = 0 on 1,...,m–1,m+1,..., n. This gives us that

Let us consider now the problem (1.6)-(1.9) with homogeneous boundary conditions (º º 0). We remark that the kernel of * contains the quadruples w = {w1, 0, Ñg1g2}, where g1Î H2(W), g2Î H2(W) Ç (W), w1 is a solution of (2.1) with the only difference that the functions w1 satisfy the boundary condition:

It can be shown in the same way that elements v = { v1, v2, v3, v4} Î 1Ç () satisfy (2.2), (2.4).

Let us show that elements v = { v1, v2, v3, v4} Î 1Ç () satisfy the additional boundary condition

We remark that element w = { w1, 0, Ñg1, 0} belongs to the kernel of * for an arbitrary g1Î H2(W), g1º 0 in imf450È 1È ... È n–1, w1 is a solution of (2.1) with boundary condition (2.5).

Thus, for { v1, v2, v3, v4} Î 1Ç () we have

which implies (2.6).

We arrive at the following assertion.

Theorem 2.1. Suppose that f = { f1, f2, f3, f4} Î M1Ç () (j = {j1, j2, j3, j4} Î 1Ç ()). Then there exists a unique solution {u(x,t), E(x,t), H(x,t)} (v(x,t), F(x,t),Y(x,t)) of (1.2)-(1.5) ((1.6)-(1.9)) with zero boundary conditions such that for all t > 0

Moreover, {u, E, H} ({v,F,Y}) satisfies the additional interface conditions (2.4), where

and

Let f = {f1, f2, f3, f4} Î and fn = Î (), such that ® 0 as n ® ¥. Then, (t)fn satisfies the following identity

where V(t) Î L2(0,T;(*)), Vt(t) Î L2(0,T;), V(T) = 0.

From this we easily obtain that

i.e., (t)f is the weak solution of the abstract Cauchy problem

We note that (t) takes M1 into itself. Indeed, if g Î M and V(t) = (T – t)g, then from (2.7) it follows that

Thus,

In the same way we get the corresponding properties for (t).

Let us now concern ourselves with the simultaneous boundary observability for a pair of piezoelectric systems. The proof is based on the invariance of the piezoelectric system relative to the one-parameter group of dilations in all variables. This property of the system leads to the following identity:

where g(x) is an arbitrary smooth function, Ñ = . For g(x) = 2–1|x – x0|2, (2.8) represents a conservation law.

Let f = {f1,f2,f3,f4} Î M1Ç () and {u(x,t),E(x,t),H(x,t)} is the corresponding solution of (1.2)-(1.5) with zero boundary conditions.

From (2.8) after integration over Wm × (0,T) and summation over m we get

where

m (u,E,H; g) is the restriction of the last three terms on the right-hand side of (2.8) on Wm and

The next assertion is of a technical nature and can be proved by direct computations.

Lemma 2.2. The following representation holds:

Let us now concern ourselves with an estimate of the integral of

m (u, E,H; g) over Wm × (0,T).

We consider the elliptic problem

which admits a solution W(x) Î C2(W) Ç C1 ().

We set

Direct computations give us that (the index m is omitted for simplicity of notations)

We have

where s1 > 0, C0 is such that

and

We note that C(W) > 1/3.

Next, we get the estimate

where s2 is an arbitrary positive number,

We have

Thus, from (2.12)-(2.14) we get the estimate

We now choose s1 and s2. We set

From the inequality (2.15) it follows that

where d is an arbitrary positive number and

From here on we will assume that W and

m satisfy the following conditions: there exists d1> 0 such that

We note that the above conditions are valid when d1 = 0 for star-shaped surfaces 1, 2, ..., n, and strongly star-shaped surface , i.e.,

Moreover, if

1,2,...,n are strongly star-shaped with respect to a point x0Î W, then, the above conditions hold with d1 > 0 for a class of domains which includes star-shaped domains.

Henceforth we set

where d1 is defined in (2.17).

Our next goal is to estimate the second integral on the left-hand side of (2.9).

The following inequality is proved by standard arguments

where

c2 > 0 is such that

Now, we are concern with an estimate of the surface integral (over × (0,T)) in (2.9).

Using the boundary conditions (1.5) (Q º G º 0) and additional boundary condition

we get

Let c > 0 be such that

We have

Assume that b = b(x) satisfies the following condition

Thus, from (2.20) we obtain the inequality

from which it follows that

Suppose that the coefficients of the systems (1.2) satisfy the following monotonicity conditions

Using these conditions and Lemma 2.2, we obtain

Thus, from the identity (2.9) and the inequalities (2.16), (2.18), (2.21), (2.23) we get

where

We now consider the problem (1.6)-(1.9) with zero boundary conditions. Let j = {j1, j2, j3, j4} Î 1 Ç () and { v,, F, Y} = (t)j.

In this case we have

In the same way we get the estimate

Our next goal is to obtain the simultaneous boundary observation for a pair of systems (1.2), (1.6).

Let f Î M1 Ç (), j Î 1 Ç () and

We can immediately verify the identity

The following formula can be proved by direct computations:

We have

We set

Then (2.27) implies the inequality

Henceforth we assume that matrices Aij satisfy the following condition

Taking (2.29) into account, from (2.28) we find that

where

From this and the identity (2.26) we obtain

We multiply (2.24) and (2.25) by

respectively, and add the inequalities thus obtained to (2.30); using the inequality

where

we arrive at the estimate

From (2.31) we deduce the following uniqueness property.

Theorem 2.3. Assume that

m and W satisfy conditions (2.17). Suppose that the matrices
, B(m) and the coefficients m(m) satisfy the monotonicity conditions (2.22),
satisfy the condition (2.29),

Let f(x) Î M1Ç (), j(x) Î 1Ç (). Suppose that {u(x,t), E(x,t), H(x,t)} and {v(x,t),F(x,t),Y(x,t)} are solutions of problems (1.2)-(1.5) and (1.6)-(1.9) with zero boundary conditions, respectively, and that

In this case, if T > T* = T1 + max, then

From Theorem 2.3 it follows that for T > T* the expression

defines a norm on the set of initial data f = {f1,f2,f3,f4} and j = {j1, j2,j3,j4} of problems (1.2)-(1.5) and (1.6)-(1.9) with zero boundary conditions. In (2.32)

We denote by the Hilbert space obtained by completing M1 Ç () × 1 Ç () with respect to the norm (2.32). We have

Our next purpose is to prove simultaneous exact controllability for the problems (1.2)-(1.5), (1.6)-(1.9).

3 Exact controllability

We denote by ' the dual space of with respect to . Let us consider the pair of problems: (1.2)-(1.4) with boundary conditions

and (1.6)-(1.8) with boundary conditions

where Q(x,t), (x,t) Î L2( × (0,T)), {f(x),j(x)} Î '.

We rewrite systems (1.2) and (1.6) in the form

By definition,

is a solution of (1.2)-(1.4), (3.1) and (1.6)-(1.8), (3.2) if the identity

holds for all {,} Î , 0 < t < T. In (3.3)

In a similar way we define a solution of (1.2), (1.4), (3.1) and (1.6), (1.8), (3.2) with zero data for t = T:

is a solution of (1.2), (1.4), (3.1), (1.6), (1.8), (3.2) with zero data for t = T if

for all {,} Î , 0 < t < T.

Let {g,y} be an arbitrary element of , and let {u,u',E,H,v,v',F,Y} be solution of (1.2), (1.4), (3.1), (1.6), (1.8), (3.2) with zero data for t = T, T > T*, and boundary functions

where

We set

From (3.4) it follows that

for any {,} Î . This implies that M is an isomorphism of onto the whole of '.

We return to problems (1.2)-(1.4), (3.1) and (1.6)-(1.8), (3.2). Suppose that the initial data {f,j} belong to '. We set

where

From (3.3) with t = T > T* we find that

for any {,} Î . By (3.5), the right-hand side of the last identity is equal to zero; that is, {u(T), u'(T), E(T), H(T), v(T), v'(T), F(T), Y(T)} generates the zero functional on .

We arrive at the following assertion.

Theorem 3.1. Assume that

, B(m), m(m),
m and W satisfy the conditions of Theorem 2.3. If T > T*, then for any initial data {f,j} Î ' of problems (1.2)-(1.5) and (1.6)-(1.9) there exists a control {(x,t),(x,t)} Î C1(0,T;L2(S)) × C0(0,T;L2(S)) such that the corresponding solution of problem (1.6)-(1.9) satisfies

while the vector-valued functions

drive the system (1.2)-(1.5) to a state of rest at the same time T:

To prove this assertion, it suffices to construct functions Q(x,t), (x,t) as before, by setting

We remark that, in view of the linearity of the systems, it suffices to consider controls that reduce the systems to a state of rest.

Received: 17/IX/02.

Supported by FAPERJ (Brazil), project E-26/151.523/01.

#555/02.

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  • *
    Visiting Researcher at the National Laboratory of Scientific Computation (LNCC/MCT).
  • Publication Dates

    • Publication in this collection
      19 July 2004
    • Date of issue
      2003

    History

    • Received
      17 Sept 2002
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