scholarly journals Local stabilization of viscous Burgers equation with memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wasim Akram ◽  
Debanjana Mitra
Keyword(s):  
Exponential Decay ◽  
Burgers Equation ◽  
Accumulation Point ◽  
Feedback Stabilization ◽  
Banach Fixed Point Theorem ◽  
Finite Dimensional ◽  
Explicit Analysis ◽  
Linearized System ◽  
Linearized Operator ◽  
With Memory

<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&gt;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&gt;\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>

scholarly journals Interior and H-∞ feedback stabilization for sabra Shell model of turbulence

2021 ◽  
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti
Keyword(s):  
Control Problem ◽  
Shell Model ◽  
Turbulent Flows ◽  
Robust Stabilization ◽  
Feedback Stabilization ◽  
Algebraic Riccati Equation ◽  
Finite Dimensional ◽  
Shell Models ◽  
Linearized System ◽  
Unknown Disturbance

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations which exhibit at most resemblance with turbulent flows. One of the mathematically well studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model namely feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus only finitely many modes of the shell model would suffice to stabilize the system. Later we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic riccati equation. Using the same riccati operator we establish asymptotic stability of the nonlinear system.

scholarly journals Feedback stabilization of one-dimensional parabolic systems related to formations

2015 ◽  
Vol 63 (1) ◽  
pp. 295-303
Author(s):  
H. Sano
Keyword(s):  
Finite Dimension ◽  
Time Derivative ◽  
Parabolic Systems ◽  
Feedback Stabilization ◽  
Boundary Values ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Neumann Type ◽  
Sturm Liouville ◽  
Output Operator

Abstract This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller

Resonant Decay Near an Accumulation Point

1997 ◽  
Vol 09 (02) ◽  
pp. 227-241
Author(s):  
Christopher King ◽  
Roger Waxler
Keyword(s):  
Quantum Mechanics ◽  
Exponential Decay ◽  
Accumulation Point ◽  
Model System ◽  
Physical Models ◽  
Stable States ◽  
Perturbed System ◽  
Embedded Eigenvalues

We consider the quantum mechanics of a model system in which meta-stable states arise through perturbation of a sequence of embedded simple eigenvalues with an embedded accumulation point. It is shown that the embedded eigenvalues become resonances in the perturbed system. These resonances also accumulate, and the position of the accumulation point is unchanged. The positions of the resonances are estimated uniformly up to the accumulation point. The meta-stable states associated with these resonances have the usual approximately exponential decay with time. Some applications to physical models are discussed.

scholarly journals Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Lifang Niu ◽  
Jianwen Zhang
Keyword(s):  
Rectangular Plate ◽  
Global Attractor ◽  
Exponential Decay ◽  
Global Attractors ◽  
Memory Term ◽  
Two Dimensional ◽  
Plate Equation ◽  
Memory Kernel ◽  
Nonlinear Plate ◽  
With Memory

A two-dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is exponentially decayed if the memory kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).

SOME CONTROL RESULTS FOR SIMPLIFIED ONE-DIMENSIONAL MODELS OF FLUID-SOLID INTERACTION

2005 ◽  
Vol 15 (05) ◽  
pp. 783-824 ◽  
Author(s):  
ANNA DOUBOVA ◽  
ENRIQUE FERNÁNDEZ-CARA
Keyword(s):  
Fixed Point ◽  
Burgers Equation ◽  
Null Controllability ◽  
Original Model ◽  
Control Analysis ◽  
One Dimensional ◽  
Carleman Inequalities ◽  
Linearized System ◽  
Fluid Solid Interaction ◽  
Observability Estimates

We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. This can be viewed as a first step in the control analysis of fluid-solid systems. The fluid is governed by the Burgers equation and the control is exerted at the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities, observability estimates and fixed point arguments.

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