Received 2 January 2014; revised 19 February 2014; accepted 4 March 2014

ABSTRACT

The purpose of this paper is to study the optimal control of the n × n cooperative elliptic system with variable coefficients under conjugation conditions and mixed boundary condition involving Schrödinger operator defined on. By using Green’s formula, properties of the first Eigen values and Lax-Milgram lemma, we prove the existence and uniqueness of solutions for these Systems. We also study the necessary and sufficient conditions for having a maximum principle and existence of solution for the given linear elliptic system.

Keywords:

Elliptic Systems; Existence and Uniqueness of Solutions; Conjugation Conditions; Mixed Boundary Condition; Schrödinger Operator; Optimal Control

1. Introduction

The necessary and sufficient conditions of optimality for systems governed by elliptic operators have been studi- ed by Lions in [1] . Optimal control problems of distributed systems that include partial differential equations have many mechanical sources and a variety of technological and scientific applications. Much optimal control problems of elliptic systems involving Schrödinger operators have been studied in [2] [3] . The linear quadratic optimal control problem described by a distributed parameter system has a variety of mechanical and technical sources and applications. In this paper, we obtain some necessary (sufficient) conditions for the validity of ma- ximum principle cooperative elliptic systems.

2. Distributed Control of an Elliptic System with Conjugation Conditions Involving Schrödinger Operator

2.1. The Case of 2 × 2 Elliptic Systems with Conjugation Condition and Mixed Boundary Condition

In this section, we consider the following 2 × 2 elliptic system (see [3] [4] )

(1)

where on Γ, = K-the direction cosine of v, v being the normal on the Γ exterior to Ω and, , and conjugation conditions (*)

,

where Ω is an open subset of with smooth boundary is Schrödinger operator,

. The giv-

en elliptic equation is specified in bounded, continuous and strictly Lipschitz domains in.

2.1.1. Existence and Uniqueness of Solution

Since, Then we have chain of the form,

we define the following bilinear form on

Lemma 1(see [3] )

Let, in system (1) satisfy, ,

Lemma 2(see [3] )

First Eigen values characterized by

For and and by lax-Milgram lemma, we prove that:

Theorem 1

For, there exists a unique solution of system (1). Proof

The bilinear form can be written as

We choose m large enough such that a + m > 0 and d + m > 0 and the equivalent norm

By Cauchy Schwartz and Lemma 1 and 2 in the paper

(2)

and is continuous on then by Lax-Milgram lemma there exist a unique solu-

tion, such that for all.

2.1.2. Formulation of the Control Problem

Let being the space of controls.

The energy functional

(3)

A unique state, where is a set of the sobolev functions

are specified on domain to every control. For a control the state of the system y is given by the solution of the following system:

(4)

The observation equation is given by.

The cost functional is given by:

(5)

where, is hermitian positive definite operator defined on

such that:

(6)

The control problem then is to find:

,

where is a closed convex subset from, since the cost function (5) can be written as

where

,

Assume that (2), (6) hold. The cost function being given (5), necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied

(7)

is the adjoint state.

Outline of proof

Since is differentiable and is bounded, then the optimal control u are characterized by which is equivalent to

(8)

Since where A is defined by:

then

By Green^’s formula or derivative in the sense of distribution

where then, is transpose of M,

adjoint state, (8) is equivalent to

From (7), we obtain

Remark 1

Generalization to n × n systems

and conjugation conditions

,

where is an open subset of with smooth boundary, A is n × n diagonal matrix of Schrödinger operator, is an matrix, = K-the direction cosine of v, v being the normal on the Γ exterior to Ω,

In this case, the bilinear form is given by:

The linear form is given by:

The cost functional is given by:

where in, , is hermitian positive definite operator defined

on such that:

In this case the necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied.

, is transpose of M.

If constraints are absent i.e. when the equality then the differential

problem of finding the vector-function

(9)

We can find last relations at as the same at the beginning of the paper.

We can choose (as a special case) then get distributed control of cooperative elliptic systems involving Laplace operator [5] .

3. Boundary Control of an Elliptic System with Conjugation Conditions Involving Schrödinger Operator

Formulation of the Control Problem

Let being the space of controls.

For a control the state of the system is given by the solution of the following system:

(10)

The observation equation is given by.

Since the cost function (5) can be written as:

where

since is continuous on then there exist a unique optimal control [1] . The function y is specified, minimizes the energy functional [6] . Moreover, we have the following theorem which gives the characterization of the optimal control u:

The necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied

(11)

is the adjoint state.

Outline of proof

Since J(v) is differentiable and U_ad is bounded then the optimal control u is characterized which is equivalent to

(12)

Since where A is defined by:

then

By Green’s formula or derivative in the sense of distribution

where then,

From (6), we obtain

We can add the control in the part of Dirichlet condition, i.e.

(13)

The difference appears in characterization

Remark 2

Generalization to n × n systems

and conjugation conditions

,

where Ω is an open subset of with smooth boundary Γ, A is an n × n diagonal matrix of Laplace operator

is an matrix

= K-the direction cosine of v, v being the normal at Γ ex-

terior to Ω,

and.

In this case, the bilinear form is given by:

The linear form is given by.

The cost functional is given by:

,

where in, , is hermitian positive definite operator defined

on such that:

.

In this case the necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied.

(11)

, = transpose of M.

If constraints are absent i.e. when the equality then the differential

problem of finding the vector-function, that satisfies the relations

(12)

We can find last relations at n = 2 and q = 0 [7] .

4. Conclusion

The aim is to find the existence and uniqueness of solution (state of the system) and necessary and sufficient condition for optimality of the system in 2 × 2 and generalization to n × n in the case of distributed and bounda- ry control. In this paper, we connect [3] with [4] to get a new work with following the developments.

References