On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

We study the convergence properties of an overlapping Schwarz decomposition~algorithm for solving nonlinear optimal control problems (OCPs). The approach decomposes the time domain into a set of overlapping subdomains, and solves subproblems defined over such subdomains in parallel. Convergence is attained by updating primal-dual information at the boundaries of the overlapping regions. We show that the algorithm exhibits local linear convergence and that the convergence rate improves exponentially with the overlap size. Our convergence results rely on a sensitivity result for OCPs that we call "exponential decay of sensitivity" (EDS). Intuitively, EDS states that the impact of parametric perturbations at the boundaries of the domain (initial and final time) decays exponentially as one moves into the domain. We show that EDS holds for nonlinear OCPs under a uniform second-order sufficient condition, a controllability condition, and a uniform boundedness condition. We conduct numerical experiments using a quadrotor motion planning problem and a PDE control problem; and show that the approach is significantly more efficient than ADMM and as efficient as the centralized solver Ipopt.