Classical and Relaxed Optimization Methods for Nonlinear Parabolic Optimal Control Problems

Abstract

A distributed optimal control problem is considered, for systems defined by parabolic partial differential equations. The state equations are nonlinear w.r.t. the state and the control, and the state constraints and cost depend also on the state gradient. The problem is first formulated in the classical and in the relaxed form. Various necessary conditions for optimality are given for both problems. Two methods are then proposed for the numerical solution of these problems. The first is a penalized gradient projection method generating classical controls, and the second is a penalized conditional descent method generating relaxed controls. Using relaxation theory, the behavior in the limit of sequences constructed by these methods is examined. Finally, numerical examples are given.