Dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation with periodic boundary conditions

The paper deals with the dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) with periodic boundary conditions. First, the dynamics of the MGKdVB equation is studied using the Fourier Galerkin and the Karhunen–Loève (K–L) Galerkin methods. The Fourier Galerkin approach is used to generate a system of nine ordinary differential equations (ODEs) from a partial differential equation (PDE), and the K–L Galerkin method is used as a model reduction technique for nonlinear systems to derive a reduced-order system of two ODEs that imitates the dynamics of the MGKdVB equation. It is shown that the two-dimensional reduced-order ODE system based on the K–L Galerkin method is superior to the Fourier Galerkin n-dimensional ODE system for any dimension n<2N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n<2N+1$$\end{document}, where N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} is the number of unstable eigenvalues, and it is comparable to the traditional higher nine-dimensional Fourier Galerkin system. Then, in order to enhance the stability of the MGKdVB equation, we propose to use state feedback linearization control schemes for both systems and show that the reduced-order system based on K–L Galerkin method is less expensive to control as compared to the Fourier Galerkin system. Finally, numerical simulations of the controlled systems are shown to illustrate the developed theory.