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LOCAL STABILIZATION OF VISCOUS BURGERS EQUATION WITH MEMORY
Journal
Evolution Equations and Control Theory
ISSN
21632472
Date Issued
2022-01-01
Author(s)
Akram, Wasim
Mitra, Debanjana
Abstract
In this article, we study the local stabilization of the viscous Burgers equation with memory around the steady state zero using localized interior controls. We first consider the linearized equation around zero which corresponds to a system coupled between a parabolic equation and an ODE. We show the feedback stabilization of the system with any exponential decay −ω, where ω ∈ (0, ω0), for some ω0 > 0, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Ric-cati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, ω0 is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay −ω, where ω > ω0, using any L2-control. Finally, we obtain the local stabilization result for the nonlinear system by means of the feedback control stabilizing the linearized system using the Banach fixed point theorem.
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