Stackelberg-Nash null controllability for a non linear coupled degenerate parabolic equations

TL;DR

This paper investigates hierarchical control for a coupled nonlinear degenerate parabolic system using Stackelberg-Nash strategies, establishing existence, uniqueness, and null controllability results via Carleman inequalities and fixed point methods.

Contribution

It introduces a novel application of Stackelberg-Nash control to nonlinear degenerate parabolic equations, proving key properties and controllability under complex conditions.

Findings

Existence and uniqueness of Nash quasiequilibrium

Equivalence of Nash quasi-equilibrium and Nash equilibrium under certain conditions

Null controllability of the coupled system using Carleman inequalities

Abstract

The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg-Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasiequilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani's fixed point Theorem, we brough the states of our system to the rest at final time T .