Remarks on hierarchic control for the wave equation in a non cylindrical domain

TL;DR

This paper develops a hierarchical control framework for the wave equation in non-cylindrical domains, establishing existence, uniqueness, and optimality conditions for leader-follower control strategies.

Contribution

It introduces a novel hierarchical control approach for wave equations in complex domains, including existence, uniqueness, and optimality analysis.

Findings

Existence and uniqueness of hierarchical controls proven.

Optimality system for the control problem derived.

Framework applicable to non-cylindrical domains.

Abstract

In this paper we establish hierarchic control for the wave equation in a non cylindrical domain $\widehat{Q}$ of $\mathbb{R}^{n + 1}$. We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. Von Stackelberg \cite{S}, there are local controls, called followers and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as "the leader" and the other one as "the follower". This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.