Maximum Entropy Density Control of Discrete-Time Linear Systems with Quadratic Cost

This paper addresses the problem of steering the distribution of the state of a discrete-time linear system to a given target distribution while minimizing an entropy-regularized cost functional. This problem is called a maximum entropy density control problem. Specifically, the running cost is given by quadratic forms of the state and the control input, and the initial and target distributions are Gaussian. We first reveal that our problem boils down to solving two Riccati difference equations coupled through their boundary values. Based on them, we give the closed-form expression of the unique optimal policy. Next, we show that the optimal density control of a backward system can be obtained simultaneously with the forward-time optimal policy. The backward solution gives another expression of the forward solution. Finally, by considering the limit where the entropy regularization vanishes, we derive the unregularized density control in closed form.