Free vibrations of axially moving strings: Energy estimates and boundary observability

TL;DR

This paper analyzes the small vibrations of axially moving strings, deriving explicit energy estimates and boundary observability inequalities for a wave equation with moving endpoints, providing new insights into energy conservation and control.

Contribution

It introduces explicit formulas for solutions and energy expressions, and establishes boundary observability inequalities with explicit constants for moving string vibrations.

Findings

Explicit solution series for vibrating strings with moving endpoints

Energy conservation law for the system

Boundary observability inequalities with explicit constants

Abstract

We study the small vibrations of axially moving strings described by a wave equation in an interval with two endpoints moving in the same direction with a constant speed. The solution is expressed by a series formula where the coefficients are explicitly computed in function of the initial data. We also define an energy expression for the solution that is conserved in time. Then, we establish boundary observability inequalities with explicit constants.