scholarly journals Boundary linear stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Nejib Smaoui ◽  
Boumediène Chentouf ◽  
Ala’ Alalabi
Keyword(s):  
Exponential Stability ◽  
Numerical Simulations ◽  
Boundary Control ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Global Solutions ◽  
Spatial Variable ◽  
Stabilization Problem ◽  
De Vries ◽  
Adaptive Boundary Control

Abstract The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in $[0,1]$ [ 0 , 1 ] . First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in $L^{2} (0,1)$ L 2 ( 0 , 1 ) . Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.

scholarly journals Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
B. Chentouf ◽  
N. Smaoui ◽  
A. Alalabi
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
Lyapunov Theory ◽  
Physical Parameters ◽  
Control Laws ◽  
Nonlinear Adaptive Control ◽  
Control Schemes ◽  
De Vries ◽  
Adaptive Boundary Control ◽  
Stability Of The Solution

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is 0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.

Exponential Stability Estimate of Fully Nonlinear Aceive Equation by Boundary Control

2011 ◽  
Vol 467-469 ◽  
pp. 1078-1083
Author(s):  
Dian Chen Lu ◽  
Ruo Yu Zhu
Keyword(s):  
Fixed Point ◽  
Exponential Stability ◽  
Dispersion Equation ◽  
Boundary Control ◽  
Existence And Uniqueness ◽  
Stability Estimate ◽  
Banach Fixed Point Theorem ◽  
Operator Semigroups ◽  
Fully Nonlinear ◽  
Well Posed

The well-posed problem for the fully nonlinear Aceive diffusion and dispersion equation on the domain [0, 1] is investigated by using boundary control. The existence and uniqueness of the solutions with the help of the Banach fixed point theorem and the theory of operator semigroups are verified. By using some inequalities and integration by parts, the exponential stability of the fully nonlinear Aceive diffusion and dispersion equation with the designed boundary feedback is also proved.

Adaptive Boundary Control of the Forced Generalized Kortewegde Vries-Burgers Equation

2010 ◽  
Vol 16 (1) ◽  
pp. 72-84 ◽  
Author(s):  
Nejib Smaoui ◽  
Alaa El-Kadri ◽  
Mohamed Zribi
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
Adaptive Boundary Control

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

2018 ◽  
Vol 149 (1) ◽  
pp. 219-260 ◽  
Author(s):  
F. A. Gallego ◽  
A. F. Pazoto
Keyword(s):  
Burgers Equation ◽  
Unique Continuation ◽  
Global Solutions ◽  
Exponential Stabilization ◽  
Well Posedness ◽  
Interpolation Theory ◽  
Damping Term ◽  
Unique Continuation Property ◽  
De Vries ◽  
Compactness Arguments

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

scholarly journals Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
N. Smaoui ◽  
B. Chentouf ◽  
A. Alalabi
Keyword(s):  
Nonlinear Boundary ◽  
Burgers Equation ◽  
Stabilization Problem ◽  
Boundary Stabilization ◽  
Wave Approximation ◽  
Long Wave ◽  
Control Schemes ◽  
De Vries ◽  
Long Wave Approximation ◽  
Stability Of The Solution

Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is $[0,1]$ [ 0 , 1 ] . First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $L^{2} $ L 2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

scholarly journals APPLICATION OF THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL TO KORTEWEG-DE VRIES-BURGERS EQUATION∗

2016 ◽  
Vol 21 (2) ◽  
pp. 188-198 ◽  
Author(s):  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Continuous Solution ◽  
Singular Kernel ◽  
Fractional Differentiation ◽  
De Vries ◽  
Perturbation Parameters ◽  
Derivative Order

In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels to the concepts of fractional differentiation with no singularity. We make use of the newly developed Caputo-Fabrizio fractional derivative with no singular kernel to establish the model. For existence and uniqueness of the continuous solution to the model, conditions on the perturbation parameters ν, µ and the derivative order α are provided. Numerical approximations are performed for some values of the perturbation parameters. This shows similar behaviors of the solution for close values of the fractional order α.

scholarly journals On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Julie Valein
Keyword(s):  
Asymptotic Stability ◽  
Exponential Stability ◽  
Numerical Simulations ◽  
Vries Equation ◽  
Stability Result ◽  
Internal Feedback ◽  
De Vries

<p style='text-indent:20px;'>The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.</p>

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