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.

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 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.

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

scholarly journals Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Nejib Smaoui ◽  
Rasha H. Al-Jamal
Keyword(s):  
Distributed Control ◽  
Feedback Linearization ◽  
Burgers Equation ◽  
Galerkin Projection ◽  
Decomposition Procedure ◽  
Control Schemes ◽  
Feedback Linearization Control ◽  
De Vries ◽  
The Stability ◽  
Model Reduction Technique

The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory.

Boundary control of the generalized Korteweg–de Vries–Burgers equation

2007 ◽  
Vol 51 (3) ◽  
pp. 439-446 ◽  
Author(s):  
Nejib Smaoui ◽  
Rasha H. Al-Jamal
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
De Vries
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