This paper is concerned with the exponential stability of energy solutions to a nonlinear stochastic delay partial differential equations with finite delay in separable Hilbert spaces. Some exponential stability criteria are obtained by constructing the Lyapunov function. As an application, one example is also given to illustrate our results.

This article is concerned with the existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps due to Martelli.

We 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.

In the present paper,we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions,the continuity and differentiability of the value function are proved.We also define a new type of Hamilton–Jacobi–Bellman(HJB) equation and prove that the value function is a viscosity solution of this HJB equation also defineanewtypeofHamilton–Jacobi–Bellman(HJB)equationandprovethatthevalue

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.