Control of some linear stochastic systems in a Hilbert space with fractional Brownian motions

A control problem for a linear system in a Hilbert space with a fractional Brownian motion and a quadratic cost in the state and the control is solved. The feedback form of the optimal control and the optimal cost are given. The optimal control is the sum of the well known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of an optimal system response to the future noise. Some examples of controlled stochastic partial differential equations are given.