First order necessary condition for stochastic evolution control systems with random generators

Abstract

The main purpose of this paper is to establish the first order necessary optimality condition for optimal control problems of stochastic evolution systems with random generators. For our controlled system, both the drift and the diffusion terms contain the control variable, and the control region is a nonempty closed subset of a separable Hilbert space. Compared with the existing works, the main difficulties here are to prove the well-posedness of the control and the adjoint systems. We obtain the well-posedness of the adjoint system in the sense of mild solutions and that of the control system by means of the stochastic transposition method. In addition, the variational analysis approach is employed to handle the nonconvexity of the control region when deriving our optimality condition.