Some topics in stochastic control

Abstract

Some stochastic optimal control problems in a Hilbert space areformulated and solved. The controlled equations are abstractequations in a HIlbert space that can model stochastic partialdifferential equations and stochastic delay equations. Both linearand semilinear equations are considered where the cylindricalBrownian motion can occur as distributed, boundary, or at discretepoints in the domain. For the linear equations, the cost is anergodic, quadratic functional of the state and the control. Anoptimal linear feedback control is given explicitly. For thesemilinear equations, the cost is an ergodic functional. Someresults for the null controllability of a stochastic parabolicequation are given. A control problem for a finite dimension linearstochastic system with an arbitrary fractional Brownian motion and aquadratic cost functional is formulated and explicitly solved.