192 research outputs found
Output feedback stabilization of the Korteweg-de Vries equation
International audienceThis 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
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.Comment: European Control Conference, Jul 2015, Linz, Austri
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. 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)Comment: arXiv admin note: text overlap with arXiv:1609.0144
Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback
We use the backstepping method to study the stabilization of a 1-D linear
transport equation on the interval (0, L), by controlling the scalar amplitude
of a piecewise regular function of the space variable in the source term. We
prove that if the system is controllable in a periodic Sobolev space of order
greater than 1, then the system can be stabilized exponentially in that space
and, for any given decay rate, we give an explicit feedback law that achieves
that decay rate
Global stabilization of a Korteweg-de Vries equation with a distributed control saturated in L 2 -norm
International audienceThis article deals with the design of saturated controls in the context of partial differential equations. It is focused on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. The aim of this article is to study the influence of a saturating in L 2-norm distributed control on the well-posedness and the stability of this equation. The well-posedness is proven applying a Banach fixed point theorem. The proof of the asymptotic stability of the closed-loop system is tackled with a Lyapunov function together with a sector condition describing the saturating input. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation
Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation
The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory
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