Approximate controllability for a one-dimensional wave equation with the fixed endpoint control

TL;DR

This paper investigates the approximate controllability of a one-dimensional wave equation with a moving boundary, establishing controllability when the boundary speed is below the wave speed, and analyzing related optimal control systems.

Contribution

It provides new results on controllability for wave equations with moving boundaries, including existence, uniqueness, and optimal control formulations.

Findings

Controllability is achieved when boundary speed is less than wave speed.

Existence and uniqueness of Nash equilibrium are established.

Optimal control systems for the boundary control are derived.

Abstract

This paper is devoted to the study of the approximate controllability for a one-dimensional wave equation in domains with 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 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.