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.