Some topics in stochastic control

Some stochastic optimal control problems in a Hilbert space are formulated and solved. The controlled equations are abstract equations in a HIlbert space that can model stochastic partial differential equations and stochastic delay equations. Both linear and semilinear equations are considered where the cylindrical Brownian motion can occur as distributed, boundary, or at discrete points in the domain. For the linear equations, the cost is an ergodic, quadratic functional of the state and the control. An optimal linear feedback control is given explicitly. For the semilinear equations, the cost is an ergodic functional. Some results for the null controllability of a stochastic parabolic equation are given. A control problem for a finite dimension linear stochastic system with an arbitrary fractional Brownian motion and a quadratic cost functional is formulated and explicitly solved.