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

Abstract

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters and , where is a positive parameter and is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when , Lyapunov stable when , and globally exponentially stable when . Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear ‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear ‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory.