An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator

In this paper, we consider cooperative hyperbolic systems involving Schrödinger operator defined on n R . First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.


Introduction
The optimal control problems of distributed systems involving Schrödinger operator have been widely discussed in many papers.One of the first studies was introduced by Serag [1], which discusses 2 × 2 cooperative systems of elliptic operator.Further research in this area developed the problem by studying different operator types (elliptic, parabolic, or hyperbolic) or higher system degree as in [2]- [6].Many boundary control problems have been introduced in [7]- [10].
In [3], we discussed distributed control problem for 2 × 2 cooperative hyperbolic systems involving Schrödinger operator.
Here, using the theory of [11], we consider the following 2 × 2 cooperative hyperbolic systems involving Schrödinger operator: where a , b , c and d are given numbers such that b , 0 c > , i.e. the system (1) is called cooperative (2)

( )
q x is a positive function and tending to ∞ at infinity, and . The model of the system (1) is given by: , , y y B t y x B t y x y x q y ay by q y cy dy t t , A t y x q y ay by q y cy dy We first prove the existence and uniqueness of the state for these systems, then we introduce the optimality conditions of boundary control, we also discuss them when the observation is on the boundary.

Some Concepts and Results
Here we shall consider some results about the following eigenvalue problem which introduced in [1] and [12]: The associated space is ( ) V R , with respect to the norm: Since the imbedding of ( ) is compact, then the operator ( ) L R is positive self-adjoint with compact inverse.Hence its spectrum consists of an infinite se- quence of positive eigenvalues, tending to infinity; moreover the smallest one which is called the principal eigenvalue denoted by ( ) q λ is simple and is associated with an eigenfunction which does not change sign in n R .It is characterized by: ( ) ( ) We have: which is continuous and compact.
Let us introduce the space ( ) ( ) 2 0, ; which is defined on open interval ( ) 0,T and the variable ( ) , T < ∞ denotes the time.
On ( ) 0,T with Lebesgue measure dt we have the norm: and the scalar product with the scalar product and the norm above is a Hilbert space.

The Existence and Uniqueness for the State of the System (1)
We have the bilinear form: , , , . , The coerciveness condition of the bilinear form (7) in has been proved by Serag [1], by using the conditions for having the maximum principle for cooperative system (1) which have been obtained by Fleckinger [13], and take the form: that means: ; , , 0 Theorem (3.1): Under the hypotheses (2) and ( 9), if , then there exists a unique solution: for system (1). Proof: → be a continuous linear form defined on ( ) ( ) then by Lax-Milgram lemma, there exists a unique element , ; , , , Now, let us multiply both sides of first equation of system (1) by ( ) , and the second equation by: x c ψ then integration over Q , we have: By applying Green's formula: ( ) ( ) ( ) By sum the two equations we get: by comparing the previous equation with ( 7), ( 10) and ( 11) we deduce that:

Formulation of the Control Problem
The space ( ) ( ) Σ is the space of controls.For a control of the system is given by the solution of The observation equation is given by ( ) ( , , z u z u z u y u y u y u = = = .
For a given , , the cost function is given by: ( , , , 0 The control problem then is to find , where ad U is a closed con- vex subset of ( ) ( ) Since the cost function ( 14) can be written as (see [11]): where ( ) , a v v is a continuous coercive bilinear form and ( ) Then there exists a unique optimal control ad u U ∈ such that ( ) ( ) for all ad v U ∈ by using the general theory of Lions [11].Moreover, we have the following theorem which gives the necessary and sufficient conditions of optimality: Theorem (4.1): Assume that ( 9) and ( 14) hold.If the cost function is given by ( 13), the optimal control , u u u L = ∈ Σ is then characterized by the following equations and inequalities: y u p By p u q y u ay u by u t t y u p u q y u cy u dy u t t By u B y u y u y u y u q y u ay u by u q y u cy u dy u t t  by using Green formula and (12), we have:  Thus the proof is complete. ,