Two Approaches for the Stabilization of Nonlinear KdV Equation With Boundary Time-Delay Feedback

Abstract

This article concerns the nonlinear Korteweg-de Vries equation with boundary timedelay 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.

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

“…There are two main difficulties to study (1): the nonlinear character of this system (due to the presence of yy x ) and the delay in the internal feedback. In particular we have to prove that the delay in the feedback will not destabilize the system, which can be the case for other delayed systems, see for instance [11].…”

“…In particular we have to prove that the delay in the feedback will not destabilize the system, which can be the case for other delayed systems, see for instance [11]. Very recently, the robustness with respect to the delay of the boundary stability of the nonlinear KdV equation has been study in [1], where the boundary condition is y x (L, t) = αy x (0, t) + βy x (0, t − h). The authors obtain, under an appropriate condition on the feedback gains with and without delay (i.e.…”

Event-triggered control of Korteweg–de Vries equation under averaged measurements

“…The study of KdV/ KdVB systems has been an active research topic because of its potential applications, see e.g. [2][3][4]6,14,16]. In the field of automatic control, a backstepping approach has been applied in [4,6,16] for the feedback stabilization of KdV equation, and Lyapunov-based arguments have been employed to ensure the stability of the original system under the proposed control law.…”

“…On the other hand, the survey paper [3] gives a detailed overview of bound-ary controllability and internal stabilization approaches and results for the KdV equation. One can read in [2] two different approaches (from a Lyapunov functional or from an observability inequality) employed to exponentially stabilize the nonlinear KdV equation via delayed boundary damping terms.…”

“…This is achieved thanks to using a V µ term in Lyapunov functional V defined by (4.5). Such a term is borrowed from [2].…”