Optimality of Distributed Control for <i>n × n</i> Hyperbolic Systems with an Infinite Number of Variables

In this paper, we study the existence of solutions for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems 
involving hyperbolic operators with an infinite number of variables and with 
variable coefficients. The necessary and sufficient conditions for optimality 
of the distributed control with constraints are obtained and the set of 
inequalities that defining the optimal control of these systems are also 
obtained.

In addition, optimal control problems for systems involving operators with an infinite number of variables for non-standard functional and time delay have been introduced in [12,13].
Furthermore, time-optimal control of systems with an infinite number of variables has been studied in [14][15][16].
Making use of the theory of Lions [22] and Berezanskiĭ [23], we consider the optimal control problem of distributed type for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients. We first prove the existence and uniqueness of the state for these systems, then we find the set of equations and inequalities that characterize the optimal control of these systems. Finally, we impose some constraints on the control. Necessary and sufficient conditions for optimality with control constraints are derived. This paper is organized as follows. In Section 1, we introduce spaces of functions of an infinite number of variables. In Section 2, we discuss the distributed control problem for these systems with Dirichlet conditions. In Section 3, we consider the problem with Neumann conditions.

Sobolev Spaces with an Infinite Number of Variables
This section covers the basic notations, definitions, and properties, which are necessary to present this work [24]. Let is the space of all square integrable functions on i.e.
We shall set associated to the above norm. We consider a Sobolev space in the case of an unbounded region. For functions which are continuously differentiable l times up to the boundary  of R  and which vanish in a neighborhood of , we introduce the scalar product where D  is defined by and the differentiation is taken in the sense of generalized function on , and after the completion, we obtain the Sobolev space , which is a Hilbert space and dense in . The space forms a positive space. We can construct the negative space with respect to the zero space and then we have the following imbedding Analogous to the above chain we have a chain of the form which is a Hilbert space.
By the Cartesian product, it is easy to construct the following Sobolev spaces with the norm , also we can construct the Cartesian product for the above Hilbert spaces. Finally we have the following chain:

Dirichlet Problem for 2l Order (n × n) Cooperative Hyperbolic System with an Infinite Number of Variables and with Variable Coefficients
In this section, we study the existence and uniqueness of solutions for 2l order   n n  cooperative systems governed by Dirichlet problems involving hyperbolic operators with an infinite number of variables and with variable coefficients, then we find the necessary and sufficient conditions of the optimal control of distributed type.
For 1 i n   , we have the following system: order hyperbolic operator with an infinite number of variables with [23] is given by:

Definition 1:
For each t  (0,T), we define a bilinear form

Lemma 1:
The bilinear form (6) is coercive on , that is, there exists c, c 1  R, such that:

Proof:
We have, which proves the coerciveness condition on .
Under all the a bove consideration, theorems of Lions [22] and using the Lax-Milgram lemma we have proved the following theorem.

Theorem 1:
Under the hypotheses (2), (3) and (7), if by entering the summation on the both sides, we have by comparing the summation with (6), (8) and (9) we obtain: which completes the proof.

Formulation of Dirichlet Problem
Assume that (7) holds and the cost function is given by (11). The necessary and sufficient conditions for to be an optimal control are the following equations and inequalities: The observation equation is give n by for all 1 ,

Control Constraints
The set of admissible controls U ad is a closed convex  , Then the control problem is to ing the genera

Proof:
The optimal control is characterized by [23]: Then us l theory of Lions [22], there ex-, 0     this inequality can be written as 0 .  (14) Now, since that is by using Green's formula, (3) and (10), we have Since the adjoint equation for hyperbolic systems in Lions [22] takes the following form: (15) we obtain the first equation in (12), and from theorem1, system (12) admits a unique solu- Now, we transform (14) by using (12) as follows: using Green's formula, (10) and (12), we obtain which is equivalent to Thus the proof is com

Neumann Problem for 2l Order (n × n) Cooperative Hyperbolic System with an Infinite ables and with Variable Coefficients
In t mal control for 2l order

Number of Vari
  n n  cooperative non-homogenous Neumann systems his section, we discuss the opti involving hyperbolic operators with an infinite number of variables and with variable coefficients.

The Existence and Uniqueness of Solution
w dition on shall in llowing theorem which gives the existence and uniqueness of the state for system (16).   the order of equa problems are po formalism.