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October 20, 2025Open Access

Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods

YWYuto WatanabeShinshu University FLFeng-Yi LiaoUniversity of California, San Diego YZYang ZhengUC San Diego Health System

Key Points

Weak convexity in robust control allows establishing a convergence rate for subgradient methods.
The $ ext{H}_ ext{infinity}$ cost showcases weak convexity over convex subsets, highlighting optimization challenges.
A weak Polyak-Łojasiewicz inequality indicates all stationary points in state-feedback are globally optimal.
Numerical examples validate theoretical results, reinforcing the applicability of robust control frameworks.

Abstract

Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with H_ control as a classical formulation. It is known that policy optimization of robust H_ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advances in nonsmooth optimization to analyze discrete-time static output-feedback H_ control. We show that the H_ 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.

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