New results concerning the hierarchical control of linear and semilinear parabolic equations

¹ IMECC, State University of Campinas, 13083-859, Campinas, SP, Brazil.
² Dpto. EDAN and IMUS, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain.
³ Instituto de Matemáticas, Universidad Nacional Autónoma de México.
⁴ Dpto. de Matemática, Universidade Federal da Paraíba, Brazil.

^*** Corresponding author: deteresal@gmail.com

Received: 9 December 2020
Accepted: 2 February 2022

Abstract

This paper deals with the application of multiple strategies to control some parabolic PDEs. We assume that we can act on the system through a hierarchy of distributed controls: with a first control (a follower), we drive the state exactly to zero; then, with an additional control (the leader), we minimize a prescribed cost functional. That means that we invert the roles played by leaders and followers in the recent literature. We study linear and semilinear problems. More precisely, we prove the existence (and uniqueness in the linear case) of a leader-follower couple. Then, we deduce an appropriate optimality system that must be satisfied by the controls and the corresponding state and adjoint states. We also indicate some generalizations to other controls, PDEs and systems. In particular, we establish similar existence and optimality results for hierarchical-biobjective (Pareto-Stackelberg) control problems, where there are two cost functionals and two independent leader controls whose main task is to find an associated Pareto equilibrium and one common follower in charge of null controllability.