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

The aim of this work is to study the exponential stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed internal feedback. We first consider the case where the weight of the feedback with delay is smaller than the weight of the feedback without delay and prove the local exponential stability result by two methods: the first one by a Lyapunov method (which holds for restrictive length of the domain but allow to have an estimation on the decay rate) and the second one by an observability inequality for any length (without estimation of the decay rate). We also prove a semiglobal stabilization result for any length. Secondly we study the case where the support of the feedback without delay is not included in the feedback with delay and give a local exponential stability result if the weight of the delayed feedback is small enough. Some numerical simulations are given to illustrate these results

A remark on the stabilization of the 1-d wave equation

International audienceWe consider the wave equation on an interval of length 1 with an interior damping at and with Dirichlet boundary condition at the two ends. It is well-known that, if is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify

Stabilization of the wave equation on 1-D networks

In this paper we study the stabilization of the wa ve equation on general 1-d networks. For that, we transfer known observability results in the context of control problems of conservative systems (see [R. Dáger and E. Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math. Appl. 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems. Then we use an interpolation inequality similar to the one proved in [P. Bégout and F. Soria, J. Differential Equations, 240 (2007), pp. 324-356] to obtain the explicit decay estimates of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We also give some examples of particular networks in which our results apply, yielding different decay rates. © 2009 Society for Industrial and Applied Mathematics

Detectability and state estimation for linear age-structured population diffusion models

International audienceWe investigate a state estimation problem for an infinite dimensional system appearing in population dynamics. More precisely, given a linear model for age-structured populations with spatial diffusion, we assume the initial distribution to be unknown and that we have at our disposal an observation locally distributed in both age and space. Using Luenberger observers, we solve the inverse problem of recovering asymp-totically in time the distribution of population. The observer is designed using a finite dimensional stabilizing output injection operator, yielding an effective reconstruction method. Numerical experiments are provided showing the feasibility of the proposed reconstruction method

Adaptive observer for age-structured population with spatial diffusion

International audienceWe investigate the inverse problem of simultaneously estimating the state and the spatial diffusion coefficient for an age-structured population model. The time evolution of the population is supposed to be known on a subdomain in space and age. We generalize to the infinite dimensional setting an adaptive observer originally proposed for finite dimensional systems

Numerical approximation of some time optimal control problems

International audienceIn this work we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems

Stabilité de quelques problèmes d'évolution

In this PhD thesis we study the stabilization of some evolution equations by feedback laws. First we consider the stabilization of the wave equation on 1-d networks with nodal feedbacks. In chapter 1, assuming that the weight of the feedback without delay is smaller than the one with delay, we give spectral conditions to obtain the strong, exponential or polynomial stability, by studying an observability inequality for the conservative system. In chapter 2 we transfer known observability results for another conservative system into a weighted observability estimate for the dissipative one without delay. Thanks to an interpolation inequality, we obtain explicit decay rates which depend on the geometric and topological properties of the network. Then we develop, in chapter 3, an abstract theory for second order evolution equation with delay, which generalizes the results of chapter 1. We study the case where the delay depends on time for the heat and wave equations in chapter 4. Using some assumptions about the delay and an appropriate Lyapunov functional, we prove that the energy is exponentially decreasing and we give explicitely its decay rate. Finally, we show, in chapter 4, that a filtering technique allows to obtain a quasi-exponential decay of a finite difference space discretization of the wave equation by pointwise interior stabilization.Dans cette thèse nous étudions la stabilisation de quelques équations d'évolution par rétro-action (feedback). Tout d'abord, nous considérons la stabilisation de l'équation des ondes sur des réseaux 1-d par des feedbacks situés aux nœuds. Dans le premier chapitre, en supposant que le poids du feedback avec retard est plus petit que celui sans retard, nous donnons des conditions spectrales pour obtenir la stabilité forte, exponentielle ou polynomiale en nous ramenant à l'étude d'une inégalité d'observabilité pour le problème conservatif. Dans le second chapitre, nous transférons des inégalités d'observabilité à poids déjà existantes pour un autre problème conservatif en inégalités d'observabilité faibles pour le système dissipé sans retard. Grâce à une inégalité d'interpolation, nous obtenons des taux de décroissance explicites qui dépendent des propriétés géométriques et topologiques du réseau. Nous développons ensuite, dans le chapitre 3, une théorie abstraite pour les équations d'évolution du second ordre avec retard généralisant les résultats du chapitre 1. Nous étudions le cas où le retard dépend du temps pour les équations des ondes et de la chaleur dans le chapitre 4. En émettant certaines hypothèses sur ce retard et en utilisant une fonctionnelle de Lyapunov appropriée, nous prouvons que l'énergie est exponentiellement décroissante et nous donnons explicitement son taux de décroissance. Enfin, nous montrons dans le chapitre 5, qu'une technique de filtrage permet d'obtenir une décroissance quasi-exponentielle de l'équation des ondes discrétisée en espace par différences finies avec un amortissement interne

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

International audienceWe consider the wave equation on an interval of length 1 with an interior damping at ξ . It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3-4):215-240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction ξ = p q , where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197-243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337-365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system