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

2013/01/29 by Abderrahim Azouani, Azouani, Abderrahim, Edriss S. Titi +1 · 8 citations
Engineering · Computer Science · #Stability and Controllability of Differential Equations #Nonlinear Dynamics and Pattern Formation #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1301.6992

Abstract

We introduce here a simple finite-dimensional feedback control scheme for\nstabilizing solutions of infinite-dimensional dissipative evolution equations,\nsuch as reaction-diffusion systems, the Navier-Stokes equations and the\nKuramoto-Sivashinsky equation. The designed feedback control scheme takes\nadvantage of the fact that such systems possess finite number of determining\nparameters (degrees of freedom), namely, finite number of determining Fourier\nmodes, determining nodes, and determining interpolants and projections. In\nparticular, the feedback control scheme uses finitely many of such observables\nand controllers. This observation is of a particular interest since it implies\nthat our approach has far more reaching applications, in particular, in data\nassimilation. Moreover, we emphasize that our scheme treats all kinds of the\ndetermining projections, as well as, the various dissipative equations with one\nunified approach. However, for the sake of simplicity we demonstrate our\napproach in this paper to a one-dimensional reaction-diffusion equation\nparadigm.\n

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