# American Institute of Mathematical Sciences

November  2010, 14(4): 1361-1373. doi: 10.3934/dcdsb.2010.14.1361

## Some topics in stochastic control

 1 Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, United States

Received  November 2009 Revised  January 2010 Published  August 2010

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.
Citation: Tyrone E. Duncan. Some topics in stochastic control. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1361-1373. doi: 10.3934/dcdsb.2010.14.1361
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