Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient

Abstract

In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called \textit{leader} and two controls called \textit{followers}. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, L\'imaco and Viana \cite{DemarqueLimacoViana_deg_sys2020} and the local controllability of the non-linear system is obtained using Liusternik's inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem.