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Direct policy search has been widely applied in modern reinforcement learningand continuous control. However, the theoretical properties of direct policysearch on nonsmooth robust control synthesis have not been fully understood.The optimal $\mathcal{H}_\infty$ control framework aims at designing a policyto minimize the closed-loop $\mathcal{H}_\infty$ norm, and is arguably the mostfundamental robust control paradigm. In this work, we show that direct policysearch is guaranteed to find the global solution of the robust$\mathcal{H}_\infty$ state-feedback control design problem. Notice that policysearch for optimal $\mathcal{H}_\infty$ control leads to a constrainednonconvex nonsmooth optimization problem, where the nonconvex feasible setconsists of all the policies stabilizing the closed-loop dynamics. We show thatfor this nonsmooth optimization problem, all Clarke stationary points areglobal minimum. Next, we identify the coerciveness of the closed-loop$\mathcal{H}_\infty$ objective function, and prove that all the sublevel setsof the resultant policy search problem are compact. Based on these properties,we show that Goldstein's subgradient method and its implementable variants canbe guaranteed to stay in the nonconvex feasible set and eventually find theglobal optimal solution of the $\mathcal{H}_\infty$ state-feedback synthesisproblem. Our work builds a new connection between nonconvex nonsmoothoptimization theory and robust control, leading to an interesting globalconvergence result for direct policy search on optimal $\mathcal{H}_\infty$synthesis.

Global Convergence of Direct Policy Search for State-Feedback  $\mathcal{H}_\infty$ Robust Control: A Revisit of Nonsmooth Synthesis with  Goldstein Subdifferential