Discretization Methods for Nonlinear Elliptic Optimal Control Problems with State Constraints.

Ion Chryssoverghi, Juergen Geiser , Jamil Al-Hawasy

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 21 pp.

MSC 2000

49J20 Optimal control problems involving partial differential equations

76M10 Finite element methods

Abstract
We consider an optimal control problem described by a second order elliptic boundary value problem, jointly nonlinear in the state and control, with control and state constraints, where the state constraints and cost functionals involve also the state gradient. The problem is first discretized by using a finite element method for state approximation, while the controls are approximated by elementwise constant, or linear, or multilinear, controls. Under appropriate assumptions, we prove that strong accumulation points in L2 of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and extremal) for the continuous problem. We then apply a penalized gradient projection method to each discrete problem, and also a progressively refining version of this method to the continuous problem. We prove that accumulation points of sequences generated by the first method are admissible and extremal for the discrete problem, and that strong accumulation points in L2 of sequences of discrete controls generated by the second method are admissible and extremal for the continuous problem. Finally, numerical examples are given.