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

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]. 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 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

A single bounded input‐feedback control to the generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation

“…The boundary and distributed control problem of the Burgers equation, the KdV equation, the KdVB equation, the KS equation, the GKS equation, and the GKdVB equation were treated in previous works. 1,2,5,10,11,15,22,26,31,33,35,[38][39][40][41]43,47,[55][56][57][58][59][60][61] The input-feedback control design to nonlinear infinite-dimensional systems is commonly done through its linearization. However, in general, this method does not usually work unless some conditions hold for the system (see, e.g., previous studies 28,42,55,62 ).…”

“…The boundary and distributed control problem of the Burgers equation, the KdV equation, the KdVB equation, the KS equation, the GKS equation, and the GKdVB equation were treated in previous works 1,2,5,10,11,15,22,26,31,33,35,38–41,43,47,55–61 . The input‐feedback control design to nonlinear infinite‐dimensional systems is commonly done through its linearization.…”

“…Recently, the GKdVBKS equation was derived using the long‐wave approximation and perturbation method for the case $$$ \alpha =3 $$$ to model the propagation of waves in a prestressed thick elastic tube filled with a viscous fluid 1 . In addition, the well‐posedness of its solution on a bounded domain with linear boundary conditions has been explored in Smaoui et al 2 The GKdVBKS equation reduces to the Korteweg–de Vries–Burgers (KdVB) equation when $$$ \alpha =1 $$$ and $$$ {\gamma}_2=0 $$$, which is considered to be one of the simplest nonlinear mathematical models displaying the features of both dispersion and dissipation 3–5 .…”

Observer‐based H∞ control of a stochastic Korteweg–de Vries–Burgers equation

“…It usually appears in the study of the weak effects of nonlinearity, dissipation, and dispersion in waves propagating in a liquid-filled elastic tube (see References 2,3). In recent years, many fruitful works on stabilization of KdVB equation have been studied widely (see References 1,[3][4][5][6][7]. The controllability problem of the KdVB equation on bounded domains (see Reference 4) and unbounded domains (see Reference 6).…”

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

“…Recently, the derivation of the MGKdVB equation (1) when α 3 has been presented by means of the long-wave approximation and perturbation method [6]. Furthermore, the existence and uniqueness of solutions of the MGKdVB equation (1) have been investigated in [7] where linear boundary conditions are considered.…”