Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-Lyapunov function

TL;DR

This paper develops a strict $H^2$-Lyapunov function for a nonlinear wave equation with Neumann boundary feedback, proving exponential decay of solutions near a stationary state, thus advancing control methods for hyperbolic PDEs.

Contribution

It introduces a novel strict $H^2$-Lyapunov function for nonlinear wave equations with boundary feedback, ensuring exponential stability.

Findings

Existence of a strict $H^2$-Lyapunov function for the system

Boundary feedback constant can be chosen for exponential decay

Demonstrates local well-posedness and stability of solutions

Abstract

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.