Linear-quadratic fractional Gaussian control

Abstract

In this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean, square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved. An optimal control is given explicitly as the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and the prediction of the response of a system to the future noise process. The optimal cost is also given. The special case of a noise process that is an arbitrary standard fractional Brownian motion is noted explicitly with an explicit expression for the prediction of the future response of a system to the noise process that is used the optimal control.