Sinkhorn Ambiguity Sets for Distributionally Robust Control: Convexity, Weak Compactness, and Tractability

Classical stochastic control assumes perfect knowledge of uncertainty, but real-world data is often incomplete. Cescon, Martin, and Ferrari-Trecate address this with a distributionally robust control (DRC) framework using ambiguity sets defined by the Sinkhorn divergence. Unlike the popular Wasserstein distance, the Sinkhorn divergence does not constrain the worst-case distribution to be discrete, and allows combining observed data with a reference distribution from prior knowledge. This is especially useful when only few noise samples are available for control design. The authors first establish the convexity and weak compactness of Sinkhorn ambiguity sets under standard assumptions, which are critical for theoretical tractability.

Leveraging these properties, they prove that the DRC linear-quadratic control problem over linear policies can be solved through convex programming—even when distributionally robust safety constraints are present. This represents a significant step forward: robust control synthesis becomes a convex optimization problem rather than an intractable non-convex one. The approach is validated on a trajectory planning example, demonstrating its practical effectiveness. The work bridges optimal transport and control theory, offering a principled way to handle uncertainty with limited data, which has implications for autonomous systems, robotics, and safety-critical control applications.