We consider the linearized elasticity system in a multidomain of ${\bf R}^3$ . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius $r^{\varepsilon}$ . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\varepsilon}$ tend to zero simultaneously, with $r^{\varepsilon}\gg\varepsilon^2$ , we identify the limit problem. This limit problem involves six junction conditions.

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection ofeither (a) a hysteretic input nonlinearity, an L 2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2-stable, time-invariantlinear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certainconstant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

In this paper we prove a comparison result between semicontinuousviscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in ${\rm I}\!{\rmR}^n\times(0,T)$ where the Hamiltonian H may be noncoercive inthe gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

We consider the approximation of a class ofexponentially stable infinite dimensional linear systems modellingthe damped vibrations of one dimensional vibrating systems or ofsquare plates. It is by now well known that the approximatingsystems obtained by usual finite element or finite difference arenot, in general, uniformly stable with respect to the discretizationparameter. Our main result shows that, by adding a suitablenumerical viscosity term in the numerical scheme, our approximationsare uniformly exponentially stable. This result is then applied toobtain strongly convergent approximations of the solutions of thealgebraic Riccati equations associated to an LQR optimal controlproblem. We next give an application to a non-homogeneous stringequation. Finally we apply similar techniques for approximating theequations of a damped square plate.

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zerowhose support is spread out overthe real line.

We study a parameter (σ)dependent relaxation of the Travelling Salesman Problem on  $\mathbb{R}^2$ .The relaxed problem is reduced to the Travelling Salesman Problemas $\sigma\rightarrow$ 0. For increasing σ it is also anordered clustering algorithm for a set of points in $\mathbb{R}^2$ .A dual formulation is introduced, which reduces the problem to aconvex optimization, provided the minimizer is in the domain ofconvexity of the relaxed functional. It is shown that this lastcondition is generically satisfied, provided σ is largeenough.

In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf{R}^3$ . The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$ . We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf{S}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.

We prove uniform local energy estimates of solutions to the damped Schrödinger equation in exterior domains under the hypothesis of the ExteriorGeometric Control. These estimates are derived from the resolvent properties.

This paper provides KKT and saddle point optimality conditions, dualitytheorems and stability theorems for consistent convex optimization problemsposed in locally convex topological vector spaces. The feasible sets ofthese optimization problems are formed by those elements of a given closedconvex set which satisfy a (possibly infinite) convex system. Moreover, allthe involved functions are assumed to be convex, lower semicontinuous andproper (but not necessarily real-valued). The key result in the paper is thecharacterization of those reverse-convex inequalities which are consequenceof the constraints system. As a byproduct of this new versions of Farkas'lemma we also characterize the containment of convex sets in reverse-convexsets. The main results in the paper are obtained under a suitableFarkas-type constraint qualifications and/or a certain closedness assumption.

Given a one-parameter family $\{g_\lambda\colon\lambda\in [a,b]\}$ of semi Riemannian metrics on ann-dimensional manifold M, a family of time-dependent potentials $\{ V_\lambda\colon \lambda\in [a,b]\}$ and a family $\{\sigma_\lambda\colon \lambda\in [a,b]\} $ of trajectories connecting two points of the mechanical system defined by $(g_\lambda, V_\lambda)$ , we show that there are trajectories bifurcating from the trivial branch $\sigma_\lambda$ if the generalized Morse indices $\mu(\sigma_a)$ and $\mu(\sigma_b)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index inthe case of locally symmetric spaces and a comparison principle of Morse Schöenberg type.