Hierarchical Control for the Wave Equation with a Moving Boundary

TL;DR

This paper investigates hierarchical control strategies for a one-dimensional wave equation with a moving boundary, establishing controllability and Nash equilibrium solutions when the boundary moves below the wave's characteristic speed.

Contribution

It introduces a hierarchical control framework for the wave equation with a moving boundary, including existence, uniqueness, and optimality of Nash equilibria.

Findings

Existence and uniqueness of Nash equilibrium.

Approximate controllability with respect to leader control.

Derivation of the optimality system for leader control.

Abstract

This paper addresses the study of the hierarchical control for the one-dimensional wave equation in intervals with a moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We assume that we can act on the dynamic of the system by a hierarchy of controls. According to the formulation given by H. von Stackelberg (Marktform und Gleichgewicht, Springer, Berlin, 1934), 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. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.