Finite Horizon Linear Quadratic Gaussian Density Regulator with Wasserstein Terminal Cost

Abstract: We formulate and solve an optimal control problem where 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. The resulting controller is termed as linear quadratic Gaussian density regulator. Our derivation for the necessary conditions of optimality finds that unlike the standard linear quadratic Gaussian (LQG) control problem, the Lyapunov matrix differential equation for covariance is coupled with the Riccati matrix differential equation for covariance costate, via nonlinear boundary conditions. We prove that the LQG control problem can be recovered as a special case of our density regulator problem. A numerical example is worked out to elucidate the formulation.