Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

TL;DR

This paper develops a finite-parameter feedback control method to stabilize solutions of various dissipative PDEs, leveraging their finite-dimensional long-term behavior for improved control strategies.

Contribution

Introduces a novel finite-parameters feedback control algorithm applicable to multiple dissipative PDEs, emphasizing stabilization via low Fourier modes.

Findings

Successfully stabilizes solutions of Navier-Stokes-Voigt and nonlinear wave equations.

Demonstrates the effectiveness of low Fourier modes feedback control.

Applicable to other spatial coarse mesh interpolants.

Abstract

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes , etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants.