Exact Boundary Observability and Controllability of the Wave Equation in an Interval with two Moving Endpoints

TL;DR

This paper analyzes the wave equation in an interval with moving endpoints, providing exact solutions, energy decay rates, observability, and controllability results, advancing understanding of wave control in dynamic domains.

Contribution

It introduces exact solutions and sharp observability and controllability results for wave equations with moving boundaries, a novel extension in dynamic domain control theory.

Findings

Energy decays at rate 1/t

Established observability in sharp time

Derived exact boundary controllability results

Abstract

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at one or two endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.