{"ID":2862403,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.25633","arxiv_id":"2509.25633","title":"Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods","abstract":"Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with $\\mathcal{H}_\\infty$ control as a classical formulation. It is known that policy optimization of robust $\\mathcal{H}_\\infty$ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advances in nonsmooth optimization to analyze discrete-time static output-feedback $\\mathcal{H}_\\infty$ control. We show that the $\\mathcal{H}_\\infty$ cost is weakly convex over any convex subset of a sublevel set. This structural property allows us to establish the first non-asymptotic deterministic convergence rate for the subgradient method under suitable assumptions. In addition, we prove a weak Polyak-Łojasiewicz (PL) inequality in the state-feedback case, implying that all stationary points are globally optimal. We finally present a few numerical examples to validate the theoretical results.","short_abstract":"Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with $\\mathcal{H}_\\infty$ control as a classical formulation. It is known that policy optimization of robust $\\mathcal{H}_\\infty$ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advanc...","url_abs":"https://arxiv.org/abs/2509.25633","url_pdf":"https://arxiv.org/pdf/2509.25633v1","authors":"[\"Yuto Watanabe\",\"Feng-Yi Liao\",\"Yang Zheng\"]","published":"2025-09-30T01:00:02Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\"]","methods":"[]","has_code":false}