# Observability and controllability of the 1--d wave equation in domains with moving boundary

## TL;DR

This paper establishes sharp energy estimates, observability, and exact boundary controllability for the 1D wave equation in domains with moving boundaries, using Fourier series and Hilbert Uniqueness Method.

## Contribution

It provides explicit observability constants and extends controllability results to domains with moving endpoints using advanced Fourier analysis techniques.

## Key findings

- Derived sharp energy estimates for wave equations with moving boundaries.
- Proved observability at endpoints in minimal time with explicit constants.
- Established exact boundary controllability for the wave equation in dynamic domains.

## Abstract

By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation.