OPTIMAL DISTRIBUTED CONTROL OF STOCHASTIC ELLIPTIC SYSTEMS WITH CONSTRAINTS

The objective of this paper is to study the optimality for stochastic non cooperative elliptic systems. A distributed control problem for a stochastic elliptic systems with constraints on states and controls is studied. First, the existence and uniqueness of the state process for these systems are proved. The necessary and sufficient conditions of optimality are derived for the Dirichlet and Neumann problems.


INTRODUCTION
Stochastic systems play an important role in mathematical models of phenomena in many fields of science, engineering, finance, biology, epidemiology and economics. In view probability theory, state of a stochastic system is randomly determined. Recently, Many researchers have been directed to the studies of optimal control of stochastic systems due to their last importance. They deal with stochastic differential equations (SDEs) [6,16].
Zhou was one of the first scientists to have used stochastic partial differential equations (SPDEs). He developed the necessary conditions of optimality (maximum principle) for a very strictly elliptic second order partial differential equations on a d -dimensional space [15]. The theory of backward and forward partial differential equations (PDEs) has been developed quickly, and it became a powerful tool in the study of PDEs. Moreover, backward and forward SPDEs appeared and was studied in many literatures (see e.g. [1,3]). The optimality for SPDEs is discussed in [15]. There are difficulties in the discussion of the solution methods of systems of forward and backward SPDEs. So, the unique method for the solution is the maximum principle. Researchers used the same steps; they discussed the maximum principle to face the difficulties of investigating the solutions.
Our first interest in this work is to implement new strategies that give a full study of the stochastic control problems. Firstly, we derive the optimality conditions. Secondly, we apply the easiest methods to simplify the technique of the stochastic control problem. Finally, we develop the study of the stochastic optimal control without constraints on the stochastic systems and without forward and backward SPDEs method. In this paper, we discuss the optimal control problem for 2 2 stochastic non cooperative elliptic systems with Dirichlet and Neumann conditions. We give the unique solution for these systems (the state process of the system). Then after, we formulate the control problem and proved the existence and uniqueness of the control. Also, we present the set of equations and inequalities that characterizes the distributed control for these systems. Moreover, the optimal control problem of the generalized N N  stochastic Neumann elliptic systems with constraints is studied. In all our considered problems the control of distributed type with state constraints is discussed. This paper is organized as follows: In section 1; we introduce the basic notations and some function spaces defined on . 1 R Section 2 is devoted to study the optimal control for 2 2 stochastic elliptic systems with Dirichlet conditions. In section 3, we also study the problem with Neumann conditions and we introduce the main results of this paper.

Notations
This section covers the basic definitions and notations, which are necessary to present our work. Let G be a nonempty open set in , n R the function R  G :  is said to have a compact support if there exists a compact subset The Sobolev space of order m [4], denoted by ), ; , ,

2 2 Stochastic Elliptic Systems with Constraints and Dirichlet Conditions
In this section, we study the optimal control problem for non cooperative 2 2 stochastic elliptic systems with Dirichlet Condition: is a brownian motion (wiener process). In the following subsection, we present the existence and uniqueness of (2.1).

Existence and Uniqueness for Solution
In this subsection, we study the existence and uniqueness of solution for We define the following bilinear form on the sobolev space are the components of the wiener process. By Lax-Milgram lemma, we prove the following theorem.
Theorem 2.1. The bilinear form (2.4) satisfies the stochastic coerciveness condition and then there exists a unique solution then we get the system (2.1).

Proof. The bilinear form (2.4) can be written as
2 )) ; , , ( By comparison of two sides, we get the system (2.1), with constraints (2.2). This completes the proof. W Now, we formulate the control problem with adding the control and we determine the cost functional.

Formulation of the Optimal Control Problem
Here, we formulate the problem and establish the necessary and sufficient conditions for the optimal control of distributed type.
The space is given by the solution of the following system: the cost functional is given by: Then, the control problem is defined by: Since the cost functional (2.8) can be written as: G P L F  then there exists a unique optimal control from the general theory in [8]. Moreover, we have the following theorem which gives the characterization of the optimal control. such that is the adjoint state process.

Stochastic Elliptic Systems
In this section, we generalize the problem (2.1). (2.12) In this case, the necessary and sufficient condition for y to be an optimal control is that the following equations and inequalities be satisfied. For the state process

2 2 Neumann Stochastic Elliptic Systems with Constraints
In this section, we study the optimal control problem for non cooperative 2 2 stochastic elliptic system with Neumann conditions.

Existence and Uniqueness of Solution
In this subsection, we study the existence and uniqueness of solutions for 3) is equivalent to (3.1) and there exists a unique solution of (3.1).

Formulation of the Optimal Control Problem with Neumann Conditions
Here, we formulate the problem and establish necessary and sufficient conditions for the optimal control of distributed type. The

Generalization N N  for Stochastic Neumann Elliptic Systems with Constraints
Here, we generalize the discussion to N N  stochastic elliptic systems with Neumann condition In this case, the bilinear and linear forms are given by: