Abstract
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.
Highlights
The generalized Korteweg-de Vries-Burgers GKdVB equation ut − νuxx μuxxx uαux 0, x ∈ 0, 2π, t ≥ 0, 1.1 u x, 0 u0 x, where ν, μ ≥ 0, and α is a positive integer, is one of the simplest partial differential equations that displays nonlinearity, with fixed level of dissipation and dispersion
If α 1, and ν 0 in 1.1, the GKdVB equation becomes the classical KdV equation which was derived in 1872 by Boussinesq and Korteweg and de Vries to model the unidirectional propagation of waves in many physical systems 2, 3
We present a distributed control scheme for GKdVB equation with periodic boundary conditions and the following initial condition: u0 x fx e−10
Summary
The generalized Korteweg-de Vries-Burgers GKdVB equation ut − νuxx μuxxx uαux 0, x ∈ 0, 2π , t ≥ 0, 1.1 u x, 0 u0 x , where ν, μ ≥ 0, and α is a positive integer, is one of the simplest partial differential equations that displays nonlinearity, with fixed level of dissipation and dispersion. It has depicted many phenomena, for example strain wave and longitudinal deformation in a nonlinear elastic rod 1.
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