Hierarchical exact controllability of the fourth order parabolic equations

TL;DR

This paper develops a hierarchical control framework for fourth order parabolic equations using Stackelberg-Nash strategies, proving existence of equilibria and controllability to arbitrary trajectories.

Contribution

It introduces a novel hierarchical control approach for fourth order parabolic equations, establishing Nash equilibria and exact controllability results.

Findings

Existence of Nash equilibrium pairs for hierarchical control

Establishment of observability inequalities via Carleman estimates

Existence of a leader control achieving exact trajectory tracking

Abstract

This paper is concerned with the application of Stackelberg-Nash strategies to control fourth order linear and semi-linear parabolic equations. We assume that the system is acted through a hierarchy of distributed controls: one main control (the leader) that is responsible for an exact controllability property; and a couple of secondary controls (the followers) that minimize two prescribed cost functionals and provides a pair of Nash equilibria for the two prescribed cost functionals. In this paper, we first prove the existence of an associated Nash equilibrium pair corresponding to a hierarchical bi-objective optimal control problem for each leader by Banach fixed points theorem. Then, we establish an observability inequalities of fourth order coupled parabolic equations by global Carleman inequalities and energy methods. Based on such results, we obtain the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory. Furthermore, we also give the second-order sufficient conditions of optimality for the secondary controls.