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

Abstract

We present a computational study of a simple finite-dimensional feedback\ncontrol algorithm for stabilizing solutions of infinite-dimensional dissipative\nevolution equations such as reaction-diffusion systems, the Navier-Stokes\nequations and the Kuramoto-Sivashinsky equation. This feedback control scheme\ntakes advantage of the fact that such systems possess finite number of\ndetermining parameters or degrees of freedom, namely, finite number of\ndetermining Fourier modes, determining nodes, and determining interpolants and\nprojections. In particular, the feedback control scheme uses finitely many of\nsuch observables and controllers that are acting on the coarse spatial scales.\nWe demonstrate our numerical results for the stabilization of the unstable zero\nsolution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky\nequation. We give rigorous stability analysis for the feedback control\nalgorithm and derive sufficient conditions relating the control parameters and\nmodel parameter values to attune to the control objective.