Convergence of ParaOpt for General Runge–Kutta Time Discretizations

Abstract. ParaOpt is a time parallel method based on Parareal for solving optimality systems arising in optimal control problems. The method was presented in [M. J. Gander, F. Kwok and J. Salomon, SIAM J. Sci. Comput., 42 (2020), A2773–A2802] together with a convergence analysis in the case where implicit Euler is used to discretize the differential equations governing the system dynamics. However, its convergence behavior for higher-order time discretizations has not been considered. In this paper, we use an operator norm analysis to prove that the convergence rate of ParaOpt applied to a linear-quadratic optimal control problem has the same order as the Runge–Kutta time integration method used, provided that a few auxiliary order conditions are satisfied. We illustrate our theoretical results with numerical examples, before showing an additional test case not covered by our analysis, namely, a nonlinear optimal control problem involving a Schrödinger type system.