Finite horizon linear quadratic Gaussian density regulator with Wasserstein terminal cost

Abstract

We formulate and solve an optimal control problem in which a finite dimensional linear time invariant (LTI) control system steers a given Gaussian probability density function (PDF) close to another in fixed time, while minimizing the trajectory-wise expected quadratic cost. We measure the “closeness” between the actual terminal PDF and the desired terminal PDF as the squared Wasserstein distance between the two density functions, and penalize the lack of closeness as terminal cost. We find that unlike the standard linear quadratic Gaussian (LQG) control problem, the necessary conditions for the resulting linear quadratic Gaussian density regulator lead to nonlinear coupling between the boundary conditions of the covariance Lyapunov matrix differential equation and the covariance costate Riccati matrix differential equation. We show that the LQG control problem can be recovered as a special case of our density regulator problem, and illustrate our formulation on a numerical example.