Descent-Penalty Methods for Relaxed Nonlinear Elliptic Optimal Control Problems

Abstract

An optimal control problem is considered, described by a second order elliptic partial differential equation, jointly nonlinear in the state and control variables, with high monotone nonlinearity in the state, and with control and state constraints. Since no convexity assumptions are made, the problem may have no classical solutions, and so it is reformulated in the relaxed form. The relaxed problem is discretized by a Galerkin finite element method for state approximation, while the controls are approximated by elementwise constant relaxed ones. The first result is that relaxed accumulation points of sequences of admissible and extremal discrete controls are admissible and extremal for the continuous relaxed problem. We then propose a mixed conditional descent-penalty method, applied to a fixed discrete relaxed problem, and also a progressively refining version of this method that reduces computing time and memory. We show that accumulation points of sequences generated by the fixed discretization (resp. progressively refining) method are admissible and extremal for the discrete (resp. continuous) relaxed problem. Numerical examples are given. This paper proposes relaxed discretization and optimization methods instead of the corresponding classical methods presented in [5]. Considered here problems are with not necessarily convex control constraint sets, and with state constraints and cost functionals depending also on the state gradient. Also, the results of Sections 1 and 2 generalize those of [8] w.r.t. the assumptions made.