On the Preconditioning of Optimal Control Problems with State Gradient Constraints

Roland Herzog and Susann Mach^*

Optimal control problems with constraints on the state present a number of challenges, both in terms of analysis and their numerical solution. In this presentation, we consider optimal control problems with constraints acting pointwise on the \emph{gradient} of the state variable. We address a suitable discretization of the problem by mixed finite elements, combined with a path-following penalty approach to make the problem numerically tractable. We shall discuss in detail the saddle-point problems arising during the solution of the penalized problems. Their spectral properties will be analyzed and preconditioners for \emph{Minres} will be presented along with numerical experiments.

Mathematics Subject Classification: 49J20

Keywords: optimal control; state gradient constraints; preconditioning; saddle-point problem

Minisymposion: Preconditioning for PDE-Constrained Optimization