Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods

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-Lojasiewicz (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.