Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study

TL;DR

This paper investigates a computational feedback control method that stabilizes solutions of infinite-dimensional dissipative systems by leveraging finite determining parameters, with applications demonstrated on reaction-diffusion and fluid dynamics equations.

Contribution

It introduces a simple finite-dimensional feedback control scheme based on finite determining parameters and provides rigorous stability analysis with numerical demonstrations.

Findings

Successfully stabilizes zero solutions in 1D reaction-diffusion and Kuramoto-Sivashinsky equations.

Provides stability conditions linking control parameters to system properties.

Demonstrates effectiveness of coarse-scale observables in control schemes.

Abstract

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters or 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 that are acting on the coarse spatial scales. We demonstrate our numerical results for the stabilization of the unstable zero solution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky equation. We give rigorous stability analysis for the feedback control algorithm and derive sufficient conditions relating the control parameters and model parameter values to attune to the control objective.