Numerical Stackelberg--Nash Control for the Heat Equation

Abstract

This paper deals with a strategy to solve numerically control problems of theStackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume thatwe can act on the system through several controls, respecting an order and a hierarchy: a first con-trol (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by thechoice of the leader and corresponding to a noncooperative multiple-objective optimization strat-egy, is found (these are the followers). Our method relies on a formulation inspired by the work ofFursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls afunctional that involves weighted integrals of the state and the control, with weights that blow upat the final time. The use of the weights is crucial to ensure the existence of the controls and theassociated state in a reasonable space. We present several mixed formulations of the problems and,then, associated mixed finite element approximations that are easy to handle. In a final step, weexhibit some numerical experiments making use of the Freefem++ package.