Feedback Control of Nonlinear Dissipative Systems by Finite Determining Parameters - A Reaction-diffusion Paradigm

TL;DR

This paper presents a finite-dimensional feedback control method for stabilizing solutions of infinite-dimensional dissipative systems, leveraging the finite number of determining parameters inherent in such systems, with applications in data assimilation.

Contribution

It introduces a unified feedback control scheme applicable to various dissipative systems using finite determining parameters, demonstrated on a reaction-diffusion model.

Findings

Effective stabilization of reaction-diffusion systems demonstrated

Unified approach applicable to multiple dissipative equations

Potential applications in data assimilation and control

Abstract

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. The designed feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers. This observation is of a particular interest since it implies that our approach has far more reaching applications, in particular, in data assimilation. Moreover, we emphasize that our scheme treats all kinds of the determining projections, as well as, the various dissipative equations with one unified approach. However, for the sake of simplicity we demonstrate our approach in this paper to a one-dimensional reaction-diffusion equation paradigm.