Global stabilization of a Korteweg-de Vries equation with saturating distributed control

TL;DR

This paper develops a control strategy with saturation for the Korteweg-de Vries PDE, proving well-posedness and asymptotic stability, supported by numerical simulations, addressing actuator limitations in wave models.

Contribution

It introduces a novel approach for stabilizing a Korteweg-de Vries equation with saturated distributed controls, including cases with localized control, using Lyapunov functions and sector conditions.

Findings

Proved well-posedness of the controlled PDE

Established asymptotic stability under saturated control

Validated results with numerical simulations

Abstract

This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability, ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation. 1. Introduction. In recent decades, a great effort has been made to take into account input saturations in control designs (see e.g [39], [15] or more recently [17]). In most applications, actuators are limited due to some physical constraints and the control input has to be bounded. Neglecting the amplitude actuator limitation can be source of undesirable and catastrophic behaviors for the closed-loop system. The standard method to analyze the stability with such nonlinear controls follows a two steps design. First the design is carried out without taking into account the saturation. In a second step, a nonlinear analysis of the closed-loop system is made when adding the saturation. In this way, we often get local stabilization results. Tackling this particular nonlinearity in the case of finite dimensional systems is already a difficult problem. However, nowadays, numerous techniques are available (see e.g. [39, 41, 37]) and such systems can be analyzed with an appropriate Lyapunov function and a sector condition of the saturation map, as introduced in [39]. In the literature, there are few papers studying this topic in the infinite dimensional case. Among them, we can cite [18], [29], where a wave equation equipped with a saturated distributed actuator is studied, and [12], where a coupled PDE/ODE system modeling a switched power converter with a transmission line is considered. Due to some restrictions on the system, a saturated feedback has to be designed in the latter paper. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems ([20],[34],[36]). Let us note that in [36], [34] and [20], the study of a priori bounded controller is tackled using abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and unbounded ([34]) control operators, some conditions are derived to deduce, from the asymptotic stability of an infinite-dimensional linear system in abstract form, the asymptotic stability when closing the loop with saturating controller. These articles use the nonlinear semigroup theory (see e.g. [24] or [1]). The Korteweg-de Vries equation (KdV for short)