On infinite time linear-quadratic Gaussian control of inhomogeneous systems

In this paper we consider infinite-time linear quadratic Gaussian control problems for non-autonomous systems with possibly unbounded coefficients of their inhomogeneous components. It is shown that under standard stabilizability and detectability conditions, along with assumptions on inhomogeneous term growth, there exists an optimal control law. This control law of well-known feedback form appears to be optimal with respect to proposed extended criteria. The applied criteria expand the notions of long-run average and pathwise long-run average, commonly used for ergodic control. We also investigate the discounted tracking problems, where the cost incorporates a discount function penalizing deviations from reference trajectories.