Stackelberg exact controllability of a class of nonlocal parabolic equations

TL;DR

This paper develops a Stackelberg control framework for nonlocal parabolic equations, combining controllability to trajectories with optimal control, involving a leader and follower in different domain regions.

Contribution

It introduces a novel multi-objective control strategy for nonlocal parabolic systems, addressing both linear and nonlinear cases with interior and boundary controls.

Findings

Successfully controls system to desired states in both linear and nonlinear cases.

Establishes controllability results for systems with interior and boundary controls.

Demonstrates effectiveness of the Stackelberg approach in nonlocal parabolic equations.

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.