BOUNDARY AND POINT CONTROLS FOR SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

In this paper we consider a class of semilinear stochastic partial differential equations with nonhomogeneous boundary conditions including noise and (boundary) control. The system is formulated as an abstract evolution equation in a suitable Hilbert space. We prove existence and regularity of mild solutions. We consider Bolza problem and prove existence of optimal controls for two classes of admissible controls, one being the class of G_t-adapted measurable stochastic processes with values in a weakly compact subset of a suitable Hilbert space, and the other being the class weak star measurable G_t-adapted signed Borel measures containing point controls (or Dirac measures) as a special case.