Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

TL;DR

This paper proves exponential decay of solutions for one-dimensional damped wave equations with dynamic boundary conditions using Lyapunov functions, supported by numerical simulations and related results.

Contribution

It introduces a novel Lyapunov function approach to establish decay rates for wave equations with dynamic boundary conditions.

Findings

Exponential decay of solutions under certain damping conditions

Effective Lyapunov functions for dynamic boundary conditions

Numerical simulations confirming theoretical results

Abstract

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.