In this paper, we establish Carleman estimates for the twodimensional isotropic non-stationary Lamé system with the zero Dirichlet boundaryconditions. Using this estimate, we prove the uniqueness and thestability in determining spatially varying density and two Lamécoefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomainω satisfies a non-trapping condition.

In this paper, we prove the genericity of the observability for discrete-time systems with more outputs than inputs.

We give the precise behaviour of some solutions of a nonlinearelliptic B.V.P. in a bounded domain when a parameter approaches aneigenvalue of the principal part. If the nonlinearity has someregularity and the domain is for example convex, we also prove anonlinear version of Courant's Nodal theorem.

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let $H=\bigl(\begin{smallmatrix} \sigma& 00 & \sigma^{-1} \end{smallmatrix}\bigr)$ for $\sigma>0$ . Let $0<\zeta_1<1<\zeta_2<\infty$ . Let $K:=SO\left(2\right)\cup SO\left(2\right)H$ .Let $u\in W^{2,1}\left(Q_{1}\left(0\right)\right)$ be a $\xCone$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<\zeta_2$ , $\mathrm{Lip}\left(u^{-1}\right)<\zeta_1^{-1}$ . 
There exists positive constants $\mathfrak{c}_1<1$ and $\mathfrak{c}_2>1$ depending only on σ, $\zeta_1$ , $\zeta_2$ such that if $\epsilon\in\left(0,\mathfrak{c}_1\right)$ and u satisfies the following inequalities \[ \int_{Q_{1}\left(0\right)} {\rm d}\left(Du\left(z\right),K\right) {\rm d}L^2 z\leq \epsilon\] \[ \int_{Q_{1}\left(0\right)} \left|D^2 u\left(z\right)\right| {\rm d}L^2 z\leq \mathfrak{c}_1,\] then there exists $J\in\left\{Id,H\right\}$ and $R\in SO\left(2\right)$ such that \[ \int_{Q_{\mathfrak{c}_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| {\rm d}L^2 z\leq \mathfrak{c}_2\epsilon^{\frac{1}{800}}.\]

A model representing the vibrations of a fluid-solid coupled structure is considered. Following Hilbert Uniqueness Method (HUM) introduced by Lions, we establish exact controllability results for this model with an internal controlin the fluid part and there is no control in the solid part. Novel features which arise because of the coupling are pointed out. It is a source of difficulty in the proof of observability inequalities, definition of weak solutions and the proof of controllability results.

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

Given two measured spaces $(X,\mu)$ and $(Y,\nu)$ , and a third space Z,given two functions u(x,z) and v(x,z), we study the problem of finding twomaps $s:X\rightarrow Z$ and $t:Y\rightarrow Z$ such that the images $s(\mu)$ and $t(\nu)$ coincide, and the integral $\int_{X}u(x,s(x))d\mu-\int_{Y}v(y,t(y))d\nu$ is maximal. We give condition on u and v for whichthere is a unique solution.

We study the asymptotic behavior of $\lambda v_\lambda$ as $\lambda\rightarrow 0^+$ , where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) \[ \lambda v_\lambda + H(x,Dv_\lambda)=0,\] with \[H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}.\] We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions,or in the closure of a bounded connected domain $\Omega\subset{\xR}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat boththe case of the Neumann boundary conditions(reflection on the boundary) andthe case of state constraints boundary conditions.Under the uniform approximate controllability assumption of one player, we extendthe uniform convergence result of the value function to a constant as $\lambda\rightarrow 0^+$ to differential games.As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.

In this work we are interested in the study of controllability andstabilization of the linearized Benjamin-Ono equation withperiodic boundary conditions, which is a generic model forthestudy of weakly nonlinear waves with nonlocal dispersion. It iswell known that theBenjamin-Ono equation has infinite number ofconserved quantities, thus we consider only controls acting in theequation such that the volume of the solution is conserved. Westudy also the stabilization with a feedback law which gives us anexponential decay of the solutions.

We prove the conical differentiability of the solution to a boneremodeling contact rod model, for given data (applied loads andrigid obstacle), with respect to small perturbations of the crosssection of the rod. The proof is based on the special structure ofthe model, composed of a variational inequality coupled with anordinary differential equation with respect to time. Thisstructure enables the verification of the two followingfundamental results: the polyhedricity of a modified displacementconstraint set defined by the obstacle and the differentiabilityof the two forms associated to the variational inequality.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e. $$\left\{\begin{array}{l} \varphi^{i}_{tt} - \varphi^{i}_{xx} = \exp(\varphi^{i+1} -\varphi^{i}) - \exp( \varphi^{i} - \varphi{i-1} ) \quad 0<x<\pi, \quad t \in \rm I\hskip-1.8pt R, i \in Z\!\!\!Z\quad (TC) \varphi^i (0,t) = \varphi^i (\pi,t) = 0 \quad\forall t, i. \end{array}\right.$$ We consider the case of “closed chains" i.e. $ \varphi^{i+N} = \varphi^i \forall i \in Z\!\!\!Z$ and some $ N \in {I\!\!N}$ and look for solutions which are peirodicin time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of thelower Bloch spectrum. For the three dimensional linear elasticity system,the first eigenvalue is degenerate of multiplicity three and henceexistence of such a regular Bloch spectrum is not guaranteed. Theaim here is to develop all necessary spectral tools to overcome thesedifficulties. The existence of a directionally regular Bloch spectrum isproved and isused in the homogenization. As a consequence an interesting relation betweenhomogenization process and wave propagation in the homogenized medium isobtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequenceit is proved that higher modes apart from the first three do not contribute to the homogenization process.

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.

Sliding mode control of ordinary differential equations isconsidered. A key robustness property, called approximability, is studied froman optimization point of view. It is proved that Tikhonov well-posedness of asuitably defined optimization problem is intimately related to approximability.Making use of this link, new approximability criteria are obtained for nonlinear sliding modecontrol systems.

Dans cet article nous proposonsdifférents algorithmes pour résoudre une nouvelle classe deproblèmes variationels non convexes. Cette classe généraliseplusieurs types d'inégalités variationnelles (Cho etal. (2000), Noor (1992), Zeng (1998), Stampacchia(1964)) du cas convexe au cas non convexe. La sensibilitéde cette classe de problèmes variationnels non convexes a été aussi étudiée.

In the framework of transport theory, we are interested in the following optimization problem: given the distributions µ+ of working people and µ- of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of µ+ from µ- with respect to a metric which depends on the transportation network.

We consider an Hamilton-Jacobi equation of the form

 $$ H(x,Du)=0\quad x\in\Omega\subset\mathbb R^N,\qquad\qquad (1) $$ 
where H(x,p) is assumed Borel measurable and quasi-convex inp. The notion of Monge solution, introduced by Newcomb and Su,is adapted to this setting making use of suitable metric devices.We establish the comparison principle for Monge sub andsupersolution, existence and uniqueness for equation ([see full text])coupled with Dirichlet boundary conditions, and a stability result. Therelation among Monge and Lipschitz subsolutions is also discussed.

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

We study the null controllability by one control force of some linear systems of parabolic type.We give sufficient conditions for the null controllabilityproperty to be true and, in an abstract setting, we prove that itis not always possible to control.

We consider an optimal control problem for a class of non-linearelliptic equations. A result of existence and uniqueness of the state equation is proven under weaker hypotheses than in theliterature. We also prove the existence of an optimalcontrol. Applications to some lubrication problems and numericalresults are given.