Optimal control governed by stochastic elliptic equations with regular states

Abstract

This study deals with an optimal control problem subject to a stochas- tic elliptic equation with Dirichlet boundary condition and in which the state process is regular on a stochastic Hilbert space. We prove the ex- istence and uniqueness of the optimal control and provide furthermore necessary and sufficient optimality conditions. The optimal solution is obtained in the case where there is no constraint. Our method is based on variational theory of elliptic boundary problems in Hilbert spaces.