Boundary control of the Korteweg-de Vries-Burgers equation: further results on stabilization and well-posedness, with numerical demonstration

2000 ◽  
Vol 45 (9) ◽  
pp. 1739-1745 ◽  
Author(s):  
A. Balogh ◽  
M. Krstic
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
Well Posedness ◽  
De Vries

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.

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.

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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