Swann Marx

Output feedback control of the linear Korteweg-de Vries equation

This paper presents the design of an output feedback control for a linear Korteweg-de Vries equation. This design is based on the backstepping method which uses a Volterra transformation. An appropriate observer is introduced and the exponential stability of the closed-loop system is proven.

Stabilization of a linear Korteweg-de Vries equation with a saturated internal control

This article deals with the design of saturated controls in the context of partial differential equations. It is focused on a linear Korteweg-de Vries equation, which is a mathematical model of waves on shallow water surfaces. In this article, we close the loop with a saturating input that renders the equation nonlinear. The well-posedness is proven thanks to the nonlinear semigroup theory. The proof of the asymptotic stability of the closed-loop system uses a Lyapunov function.

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

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.

Semi-global stabilization by an output feedback law from a hybrid state controller

This article suggests a design method of a hybrid output feedback for SISO continuous systems. We focus on continuous systems for which there exists a hybrid state feedback law. A local hybrid stabilizability and a (global) complete uniform observability are assumed to achieve the stabilization of an equilibrium with a hybrid output feedback law. This is an existence result. Moreover, assuming the existence of a robust Lyapunov function instead of a stabilizability assumption allows to design explicitly this hybrid output feedback law. This last result is illustrated for linear systems with reset saturated controls.

Output feedback stabilization of the Korteweg-de Vries equation

This paper presents an output feedback control law for the Korteweg-de Vries equation. The control design is based on the backstepping method and the introduction of an appropriate observer. The local exponential stability of the closed-loop system is proven. Some numerical simulations are shown to illustrate this theoretical result.

Using a high-gain observer for a hybrid output feedback: Finite-time and asymptotic cases for SISO affine systems

This article suggests a design method of hybrid output feedbacks for affine systems under observability and stabilizability assumptions. Our aim is to use the separation principle on systems controlled by hybrid feedback laws. We investigate two constructive methods for the high-gain observer: the first one is based on a finite-time convergence of the observation error, the second one is based on an asymptotic convergence of the observation error. We illustrate one of our main results on a well-known example: integrators chain.

Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

This work studies the influence of some constraints on a stabilizing feedback law. An abstract nonlinear control system is considered for which we assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. This controller is then modified via a cone-bounded nonlinearity. Well-posedness and stability theorems are stated. The first theorem is proved thanks to the Schauder fixed-point theorem and the second one with an infinite-dimensional version of LaSalle’s invariance principle. These results are illustrated on a linear Korteweg-de Vries equation by some simulations and on a nonlinear heat equation.