Initial Data Identication in Space Dependent Conservation Laws and Hamilton-Jacobi Equations

Consider a Conservation Law and a Hamilton-Jacobi equation with a ux/Hamiltonian depending also on the space variable. We characterize rst the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed prole. An explicit example then shows the deep dierences between the cases of x-independent and x-dependent uxes/Hamiltonians

Finite-time stabilization of a network of strings

We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node

Dissipative boundary conditions for 2 × 2 hyperbolic systems of conservation laws for entropy solutions in BV

International audienceIn this article, we investigate the BV stability of 2×2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in BV. Our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose BV norms are controlled through a Lyapunov functional. This Lyapunov functional is inspired by the one proposed in J. Glimm's seminal work [J. Glimm, Comm. Pure Appl. Math., 18:697--715, 1965], modified with some suitable weights in the spirit of the previous works [J.-M. Coron, G. Bastin, and B. d'Andréa Novel, SIAM J. Control Optim., 47(3):1460--1498, 2008] and [J.-M. Coron, B. d'Andréa Novel, and G. Bastin, IEEE Trans. Automat. Control, 52(1):2--11, 2007]

Exact Controllability of Scalar Conservation Laws with an Additional Control in the Context of Entropy Solutions

International audienceIn this paper, we study the exact controllability problem for nonlinear scalar con-servation laws on a compact interval, with a regular convex flux and in the framework of entropy solutions. With the boundary data and a source term depending only on the time as controls, we provide sufficient conditions for a state to be reachable in arbitrary small time. To do so we introduce a slightly modified wave-front tracking algorithm

Asymptotic stabilization of stationnary shock waves using a boundary feedback law

38 pages, 33 referencesIn this paper we consider scalar conservation laws with a convex flux. Given a stationnary shock, we provide a feedback law acting at one boundary point such that this solution is now asymptotically stable in L 1-norm in the class of entropy solution

Problèmes de contrôle et équations hyperboliques non-linéaires

In this thesis, we study some problems from control theory on several models from fluid mechanics.\par In chapter one, we study the Camassa-Holm equation on a compact interval. After introducing our boundary conditions and a notion of weak solution, we prove an existence result and a weak-strong uniqueness result for the non-homogeneous initial boundary value problem. In a second part, we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law. In chapter two, we study the exact controllability problem for a 1-D scalar conservation law with convex flux, on a compact interval and in the context of entropy solution. We provide several sufficient conditions for a BV function to be reachable in any time and from any initial data in BV. We control the equation by means of the boundary data and also through a source term acting uniformly in space.\par Finally in chapter three, we investigate the asymptotic stabilization problem of the constant stationary solutions of a scalar conservation laws with a convex flux, on a compact interval and in the context of entropy solutions. Once again we control the equation through the boundary data and a source term acting uniformly in space. We provide two stationary feedback laws (depending on whether the state to stabilize has critical speed or not) which allow us to prove the global asymptotic stabilization property.Dans cette thèse nous étudierons plusieurs problèmes de la théorie du contrôle portant sur des modèles non-linéaires issus de la mécanique des fluides. Dans le chapitre un, nous étudions l'équation de Camassa-Holm sur un intervalle compact de R. Après avoir introduit de bonnes conditions aux bords et une notion de solution faible, nous montrons un théorème d'existence et un théorème d'unicité fort-faible pour le problème mixte. Dans une seconde partie nous fournissons une loi de retour pour les données aux bords qui nous permet de stabiliser asymptotiquement l'état stationnaire naturel de l'équation.\par Dans le chapitre deux, nous étudions le problème de la contrôlabilité exacte d'une loi de conservation scalaire à flux convexe, posée sur un intervalle compact et dans le cadre des solutions entropiques. On fournit des conditions suffisantes sur des fonctions de BV pour qu'elles soient atteignables en temps arbitraire depuis n'importe quelle donnée initiale. On contrôle l'équation via les données aux bords et aussi grâce à un terme source agissant uniformément en espace.\par Enfin le chapitre trois est consacré au problème de la stabilisation asymptotique des états stationnaires constants d'une loi de conservation scalaire à flux convexe, posée sur un intervalle compact et dans le cadre des solutions entropiques. On contrôle à nouveau l'équation via les données aux bords et un terme source agissant uniformément en espace. Nous fournissons deux lois de retour stationnaires (suivant que l'état à stabiliser est de vitesse critique ou non) qui nous permettent de montrer la stabilisation asymptotique globale