Stackelberg exact controllability of a class of nonlocal parabolic equations

¹ Landry Djomegne, University of Dschang, BP 67 Dschang, West region, Cameroon.
² Cyrille Kenne, Laboratoire des Matériaux et Molécules en Milieu Agressif (L3MA UR4_1), UFR STE et IUT, Université des Antilles, 97275 Schoelcher, Martinique, France.

^* Corresponding author: landry.djomegne@gmail.com

Received: 4 February 2023
Accepted: 25 May 2024

Abstract

This paper deals with a multi-objective control problem for a class of nonlocal parabolic equations, where the non-locality is expressed through an integral kernel. We present the Stackelberg strategy that combines the concepts of controllability to trajectories with optimal control. The strategy involves two controls: a main control (the leader) and a secondary control (the follower). The leader solves a controllability to trajectories problem which consists to drive the state of the system to a prescribed target at a final time while the follower solves an optimal control problem which consists to minimize a given cost functional. The paper considers two cases: in the first case, both the leader and the follower act in the interior of the domain, and in the second case, the leader acts in the interior of the domain and the follower acts on a small part of the boundary. These results are applied to both linear and nonlinear nonlocal parabolic systems.