Feedback control of nonlinear dissipative systemsby  finite determining   parameters- A reaction-diffusion paradigm

Abderrahim Azouani; Edriss S. Titi

doi:10.3934/eect.2014.3.579

Article Contents

2014, Volume 3, Issue 4: 579-594. Doi: 10.3934/eect.2014.3.579

This issue Previous Article A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction Next Article Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling

Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm

Abderrahim Azouani^1, and
Edriss S. Titi^2,

1.
Mohammed First University, National School of Applied Sciences Al Hoceima, Ajdir, 32003, Al Hoceima
2.
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science,Rehovot 76100

Received: April 30, 2014

Revised: August 31, 2014

Published: November 2014

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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.

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Mathematics Subject Classification: 35K57, 37L25, 37L30, 37N35, 93B52, 93C20, 93D15.

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