Optimality of Linear Policies for Distributionally Robust Linear Quadratic Gaussian Regulator with Stationary Distributions

We prove that output-feedback linear policies remain optimal for solving the Linear Quadratic Gaussian regulation problem in the face of worst-case process and measurement noise distributions when these are independent, stationary, and known to be within a radius (in the Wasserstein sense) to some reference zero-mean Gaussian noise distributions. Additionally, we establish the existence of a Nash equilibrium of the zero-sum game between a control engineer, who minimizes control cost, and a fictitious adversary, who chooses the noise distributions that maximize this cost. For general (possibly non-Gaussian) reference noise distributions, we establish a quasi closed-form solution for the worst-case distributions against linear policies. Our work provides a less conservative alternative compared to recent work in distributionally robust control.