Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type
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doi: 10.3934/eect.2021032
Local stabilization of viscous Burgers equation with memory
Wasim Akram and Debanjana Mitra ,
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai - 400076, India
* Corresponding author: Debanjana Mitra
Received August 2020 Revised March 2021 Early access July 2021
Fund Project: The second author acknowledges the support by an Inspire Faculty Fellowship, RD/0118- DSTIN40-001
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 $ -\omega $, where $ \omega\in (0, \omega_0) $, for some $ \omega_0>0 $, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, $ \omega_0 $ is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay $ -\omega $, where $ \omega>\omega_0 $, using any $ L^2 $-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.
Keywords: Burgers type equation, Heat equation with memory, Local stabilization, Feedback finite dimensional control, Distributed control.
Mathematics Subject Classification: Primary: 93B52, 93D15, 34H15, 45K05.
Citation: Wasim Akram, Debanjana Mitra. Local stabilization of viscous Burgers equation with memory. Evolution Equations & Control Theory, doi: 10.3934/eect.2021032
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Figure 1. Spectrum of $ A $ in one-dimension with $ \Omega = (0, 10) $
Figure 2. Spectrum after feedback stabilization, i.e, Spectrum of $ (A-BB^*P) $ in one dimension with $ \Omega = (0, 10) $
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Wasim Akram Debanjana Mitra