scholarly journals Finite‐parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

2021 ◽  
Author(s):  
Serap Gumus ◽  
Varga Kalantarov
Keyword(s):  
Burgers Equation ◽  
Feedback Stabilization ◽  
Burgers Equations ◽  
Nonlocal Nonlinearities

Generalized Burgers Equations Transformable to the Burgers Equation

2011 ◽  
Vol 127 (3) ◽  
pp. 211-220 ◽  
Author(s):  
B. Mayil Vaganan ◽  
T. Jeyalakshmi
Keyword(s):  
Burgers Equation ◽  
Burgers Equations

Soliton solutions of Burgers equations and perturbed Burgers equation

2010 ◽  
Vol 216 (11) ◽  
pp. 3370-3377 ◽  
Author(s):  
Anwar Ja’afar Mohamad Jawad ◽  
Marko D. Petković ◽  
Anjan Biswas
Keyword(s):  
Burgers Equation ◽  
Soliton Solutions ◽  
Burgers Equations

Classification and Recursion Operators of Dark Burgers’ Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 175-180 ◽  
Author(s):  
Mei-Dan Chen ◽  
Biao Li
Keyword(s):  
Symbolic Computation ◽  
Free Parameter ◽  
Burgers Equation ◽  
Differential Polynomial ◽  
Higher Order ◽  
Recursion Operators ◽  
Burgers Equations ◽  
Dual Symmetry ◽  
Free Parameters

AbstractWith the help of symbolic computation, two types of complete scalar classification for dark Burgers’ equations are derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark Burgers’ systems; so some special equations including symmetry equation and dual symmetry equation are obtained by selecting the free parameter. Furthermore, two kinds of recursion operators for these dark Burgers’ equations are constructed by two direct assumption methods.

scholarly journals Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2326
Author(s):  
Noufe H. Aljahdaly ◽  
Ravi P. Agarwal ◽  
Rasool Shah ◽  
Thongchai Botmart
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Singular Kernel ◽  
Exact Results ◽  
Burgers Equations ◽  
Kernel Operators ◽  
Natural Decomposition

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results.

scholarly journals Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

2019 ◽  
Vol 29 (02) ◽  
pp. 271-316 ◽  
Author(s):  
Georges Bastin ◽  
Jean-Michel Coron ◽  
Amaury Hayat ◽  
Peipei Shang
Keyword(s):  
Steady State ◽  
Lyapunov Function ◽  
Boundary Conditions ◽  
Burgers Equation ◽  
Feedback Stabilization ◽  
Control Lyapunov Function ◽  
Bounded Interval ◽  
Shock Location ◽  
Inviscid Burgers Equation ◽  
Arbitrary Decay

In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in [Formula: see text]-norm, with an arbitrary decay rate.

Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Miroslav Krstic ◽  
Lionel Magnis ◽  
Rafael Vazquez
Keyword(s):  
Output Feedback ◽  
Burgers Equation ◽  
Nonlinear Partial Differential Equation ◽  
Trajectory Generation ◽  
Output Feedback Control ◽  
Companion Paper ◽  
Feedback Stabilization ◽  
Observer Design ◽  
Nonlinear Observer ◽  
Equilibrium Profiles

In a companion paper we have solved the basic problem of full-state stabilization of unstable “shock-like” equilibrium profiles of the viscous Burgers equation with actuation at the boundaries. In this paper we consider several advanced problems for this nonlinear partial differential equation (PDE) system. We start with the problems of trajectory generation and tracking. Our algorithm is applicable to a large class of functions of time as reference trajectories of the boundary output, though we focus in more detail on the special case of sinusoidal references. Since the Burgers equation is not globally controllable, the reference amplitudes cannot be arbitrarily large. We provide a sufficient condition that characterizes the allowable amplitudes and frequencies, under which the state trajectory is bounded and tracking is achieved. We then consider the problem of output feedback stabilization. We design a nonlinear observer for the Burgers equation that employs only boundary sensing. We employ its state estimates in an output feedback control law, which we prove to be locally stabilizing. The output feedback law is illustrated with numerical simulations of the closed-loop system.

scholarly journals Numerical approximation of coupled 1D and 2D non-linear Burgers’ equations by employing Modified Quartic Hyperbolic B-spline Differential Quadrature Method

2021 ◽  
Vol 15 ◽  
pp. 37-55
Author(s):  
Mamta Kapoor ◽  
Varun Joshi
Keyword(s):  
Present Method ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Differential Quadrature Method ◽  
Quadrature Method ◽  
B Spline ◽  
Numerical Examples ◽  
Burgers Equations ◽  
Weighting Coefficients

In this paper, the numerical solution of coupled 1D and coupled 2D Burgers' equation is provided with the appropriate initial and boundary conditions, by implementing "modified quartic Hyperbolic B-spline DQM". In present method, the required weighting coefficients are computed using modified quartic Hyperbolic B-spline as a basis function. These coupled 1D and coupled 2D Burgers' equations got transformed into the set of ordinary differential equations, tackled by SSPRK43 scheme. Efficiency of the scheme and exactness of the obtained numerical solutions is declared with the aid of 8 numerical examples. Numerical results obtained by modified quartic Hyperbolic B-spline are efficient and it is easy to implement

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>

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