<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \omega\in (0, \omega_0) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M3">\begin{document}$ \omega_0>0 $\end{document}</tex-math></inline-formula>, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, <inline-formula><tex-math id="M4">\begin{document}$ \omega_0 $\end{document}</tex-math></inline-formula> is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay <inline-formula><tex-math id="M5">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \omega>\omega_0 $\end{document}</tex-math></inline-formula>, using any <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-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.</p>
Finite‐parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities
