On the Global Optimality of Linear Policies for Sinkhorn Distributionally Robust Linear Quadratic Control

The Linear Quadratic Gaussian (LQG) regulator is a cornerstone of optimal control theory, yet its performance can degrade significantly when the noise distributions deviate from the assumed Gaussian model. To address this limitation, this work proposes a distributionally robust generalization of the finite-horizon LQG control problem. Specifically, we assume that the noise distributions are unknown and belong to ambiguity sets defined in terms of an entropy-regularized Wasserstein distance centered at a nominal Gaussian distribution. By deriving novel bounds on this Sinkhorn discrepancy and proving structural and topological properties of the resulting ambiguity sets, we establish global optimality of linear policies. Numerical experiments showcase improved distributional robustness of our control policy.