Optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations

Abstract

a b s t r a c tWe study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.

Existence of the Optimal Control for Stochastic Boundary Control Problems Governed by Semilinear Parabolic Equations

“…for every ∈ 1 ( ) and almost every ∈ (0, ]. [19]). Let the conditions ( 1 ) and ( 2 ) hold; then the state equation (1) has a unique (weak) solution ( , ) ∈ 2 F (Ω × (0, ); 1 ( )) for every control ( , ) ∈ .…”

“…About the control problems governed by semilinear parabolic equations, we can see [17,18]. In this paper, the necessary conditions for our problem have been worked out by a member of our team in the paper [19]. So, in this paper, we only consider the existence of the optimal control as a complement for [19].…”