Numerical Solution of Optimal Control Problems for Parabolic Systems

Abstract

We discuss the numerical solution of optimal control problems for instationary convection-diffusion and diffusion-reaction equations. Instead of viewing this problem as a large-scale unconstrained optimization problem after complete discretization of the corresponding optimality system, we formulate the problem as abstract linear-quadratic regulator (LQR) problem. Using recently developed efficient solvers for large-scale algebraic Riccati equations, we show how to numerically solve the optimal control problem at a cost proportional to solving the corresponding forward problem. We discuss two different optimization goals: one can be seen as stabilization of the plant model, the second one is of tracking type, i.e., a given (optimal) solution trajectory is to be attained. The efficiency of our approach is demonstrated for a model problem related to an optimal cooling process. Moreover, we discuss how the LQR approach can be applied to nonlinear problems.