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Featured researches published by Lionel Rosier.


Systems & Control Letters | 1992

Homogeneous Lyapunov function for homogeneous continuous vector field

Lionel Rosier

The goal of this article is to provide a construction of a homogeneous Lyapunov function P associated with a system of differential equations J = f(x), x ~ R ~ (n > 1), under the hypotheses: (1) f ~ C(R n, ~) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function V(x) tends to infinity with 1( x (I, and belongs to C=(R~\{0}, R)n CP(~ ~, ~), with p E (~* as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.


Siam Journal on Control and Optimization | 2006

Global Stabilization of the Generalized Korteweg--de Vries Equation Posed on a Finite Domain

Lionel Rosier; Bing-Yu Zhang

This paper is concerned with the internal stabilization of the generalized Korteweg--de Vries equation on a bounded domain. The global well-posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval [1,4). The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua [Quart. Appl. Math., 60 (2002), pp. 111-129].


Journal of Systems Science & Complexity | 2009

Control and stabilization of the Korteweg-de Vries equation: recent progresses

Lionel Rosier; Bing-Yu Zhang

The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s. Both exact control and stabilization problems have been intensively studied since then and significant progresses have been made due to many peoples hard work and contributions. In this article, the authors intend to give an overall review of the results obtained so far in the study but with an emphasis on its recent progresses. A list of open problems is also provided for further investigation.


Communications in Partial Differential Equations | 2010

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

Camille Laurent; Lionel Rosier; Bing-Yu Zhang

In [38], Russell and Zhang showed that the Korteweg-de Vries equation posed on a periodic domain 𝕋 with an internal control is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of 𝕋. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability is established with the aid of certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system. With Slemrods feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrarily large decay rate.


Inverse Problems | 2008

Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights

Alberto Mercado; Axel Osses; Lionel Rosier

Baudouin and Puel (2002 Inverse Problems 18 1537-54), investigated some inverse problems for the evolution Schr¨odinger equation bymeans of Carleman inequalities proved under a strict pseudoconvexity condition. We showhere that new Carleman inequalities for the Schr¨odinger equationmay be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x * e, where e is some fixed direction in RN. As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schr¨odinger equation from a single boundary or internal measurement.


Siam Journal on Control and Optimization | 2000

Exact Boundary Controllability for the Linear Korteweg--de Vries Equation on the Half-Line

Lionel Rosier

This paper is concerned with the controllability of the linear Korteweg--de Vries equation on the domain


Siam Journal on Control and Optimization | 2009

Local Exact Controllability and Stabilizability of the Nonlinear Schrödinger Equation on a Bounded Interval

Lionel Rosier; Bing-Yu Zhang

\O =(0,+\infty )


Siam Journal on Control and Optimization | 2013

NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE EQUATION WITH MOVING CONTROL

Philippe Martin; Lionel Rosier; Pierre Rouchon

, the control being applied at the left endpoint x=0. It is shown that the {\em exact} boundary controllability holds true in


Mathematics of Control, Signals, and Systems | 1998

Liapunov and Lagrange stability : Inverse theorems for discontinuous systems

Andrea Bacciotti; Lionel Rosier

L^2(0,+\infty )


Siam Journal on Control and Optimization | 2010

Null Controllability of a Parabolic System with a Cubic Coupling Term

Jean-Michel Coron; Sergio Guerrero; Lionel Rosier

provided that the solutions are not required to be in

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Bing-Yu Zhang

University of Cincinnati

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Ademir F. Pazoto

Federal University of Rio de Janeiro

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Vincent Perrollaz

François Rabelais University

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