Hierarchical exact controllability of hyperbolic equations and Dunford-Pettis' theorem

Abstract

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.