Finite-Horizon Covariance Control of Linear Time-Varying Systems

We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of diffusion-less systems. This reformulation leads to a simple condition for the solution, which can be effectively solved using numerical methods. We show that the resulting solution coincides with a LQG problem with particular terminal cost weight matrix. This fact provides an additional justification for using a linear in state controller. In addition, it allows an efficient iterative implementation of the controller.