On optimal linear regulator with polynomial process of external excitations

A linear control system over an infinite time-horizon is considered, where external excitations are defined as polynomials based on a time-varying Ornstein–Uhlenbeck process. An optimal control law with respect to long-run average type criteria is established. It is shown that the optimal control has the form of a linear feedback law, where the affine term satisfies a backward linear stochastic differential equation. The normalizing functions in the optimality criteria depend on the stability rate of the dynamic equation for the Ornstein–Uhlenbeck process.