Finite-Parameters Feedback Control for Stabilizing Damped Nonlinear Wave Equations

TL;DR

This paper develops a finite-parameters feedback control method to stabilize solutions of various damped nonlinear wave equations, using finitely many Fourier modes, volume elements, and nodal observables, applicable to different boundary conditions.

Contribution

It introduces a novel finite-parameters feedback control algorithm for stabilizing nonlinear wave equations, including controllers acting in subdomains and applicable to multiple solution types.

Findings

Stabilization of zero steady state for various damped nonlinear wave equations.

Effective feedback control using finitely many Fourier modes, volume elements, and nodal points.

Applicability to different boundary conditions and solution types.

Abstract

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we also establish the stabilization of the zero steady state solution to initial boundary value problem for the damped nonlinear wave equation with a controller acting in a proper subdomain. Notably, the feedback controllers proposed here can be equally applied for stabilizing other solutions of the underlying equations.