Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

TL;DR

This paper investigates two methods to stabilize the nonlinear KdV equation with boundary time-delay feedback, proving exponential stability under certain conditions and illustrating results through numerical simulations.

Contribution

It introduces two distinct approaches for stabilizing the nonlinear KdV equation with boundary delays, including a Lyapunov method and an observability inequality approach.

Findings

Exponential stability is achieved under specific conditions.

Lyapunov approach provides decay rate estimates but with domain restrictions.

Observability inequality approach guarantees stability without decay rate estimation.

Abstract

This article concerns the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non critical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results.