Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting

Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to revealhidden convexityin classical optimal and robust control problems from a modern optimization perspective. Our optimization perspective offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-ordernon-degeneratestationary points to be globally optimal. This ECL framework can cover many benchmark control problems, including LQR, LQG, and$\mathcal {H}_\infty$robust control. We also believe that the new ECL framework will be of independent interest for analyzing nonconvex problems beyond control.