Distributionally Robust GMM Steering Under Wasserstein Ambiguity Sets

This letter addresses the problem of steering a discrete-time linear system from an initial Gaussian Mixture Model (GMM) distribution, where the component weights are uncertain, to a prescribed terminal Gaussian distribution. We formulate a distributionally robust optimal control (DRO) problem by modeling the weight uncertainty via a Wasserstein-type ambiguity set. The resulting non-convex min-max problem is shown to be equivalently transformed into a tractable, deterministic Semidefinite Program (SDP) by leveraging duality theory and convex lifting techniques. The standard nominal (non-robust) controller can fail to satisfy terminal constraints under plausible worst-case weight distributions within the ambiguity set. In contrast, the proposed DRO controller successfully guarantees constraint satisfaction in these scenarios, demonstrating its improved robustness.