On absolute exponential stability of the Korteweg–de Vries–Burgers equation under nonlinear boundary controls

Abstract

In this paper, we study the absolute exponential stability of the Korteweg–de Vries–Burgers equation under two distinct types of nonlinear boundary position feedback controls. We propose criteria that adhere to the sector-bounded and parabolic-restricted conditions, thereby encompassing a broad spectrum of nonlinear controllers. For each of these two nonlinear control strategies, we establish the well-posedness of the ensuing closed-loop system through the utilization of the Faedo–Galerkin approximation method. Furthermore, by crafting specific Lyapunov functionals, we demonstrate that the closed-loop systems exhibit global exponential stability in the $L^2$-space and semi-global exponential stability in the $H^m$ spaces for $m=1,2,3$ respectively. The explicit exponential decay rates of the closed-loop system solutions are determined, depending upon the dissipative and diffusive parameters. Some numerical simulations using a finite difference scheme are presented to illustrate the effectiveness of the proposed controls.