One-Shot Multigrid for the Optimal Neumann Boundary Control of Time-Periodic Partial Differential Equations

We present a one-shot multigrid method for the optimal Neumann boundary control of time-periodic, linear, parabolic partial differential equations. We focus on optimal control problems of the so-called tracking type, i.e., we try to steer the PDE solution such that it matches a prescribed periodic trajectory, or segment thereof, as closely as possible. To that end we derive the optimality condition of such problems. We develop a one-shot multigrid method for solving the resulting coupled system of forward and backward time-periodic differential equations. Numerical examples are presented to illustrate the behavior of the multigrid algorithm.