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.

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 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.

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.

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.

Necessary conditions of optimality in the form ofUnmaximized Inclusions (UI) are derived for optimal controlproblems with state constraints. The conditions presented heregeneralize earlier optimality conditions to problems that may benonconvex. The derivation of UI-type conditions in the absence of the convexity assumption is of particularimportance when deriving necessary conditions for constrainedproblems. We illustrate this feature by establishing, as anapplication, optimality conditions for problems that in additionto state constraints incorporate mixed state-control constraints.

We consider a class ofsemilinear elliptic equations of the form 15.7cm - $\varepsilon^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in{\mathbb{R}}^{2}$ where $\varepsilon>0$ , $a:{\mathbb{R}}\to{\mathbb{R}}$ is a periodic, positive function and $W:{\mathbb{R}}\to{\mathbb{R}}$ is modeled on the classical two well Ginzburg-Landaupotential $W(s)=(s^{2}-1)^{2}$ . We look for solutions to ([see full textsee full text])which verify theasymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in{\mathbb{R}}$ .We show via variationalmethods that if ε is sufficiently small and a is not constant, then ([see full textsee full text])admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

This paper deals with a new method to controlflexible structures by designing non-collocated sensors and actuators satisfying a pseudo-collocation criterion in the low-frequency domain. Thistechnique is applied to a simply supported plate with a point forceactuator and a piezoelectric sensor, for which we give some theoretical andnumerical results. We also compute low-order controllers which stabilizepseudo-collocated systems and the closed-loop behavior show that this approachis very promising.

This paper presents coordination algorithms for groups of mobile agents performing deployment and coverage tasks. As an important modeling constraint, we assume that each mobile agent has a limited sensing or communication radius.
Based on the geometry of Voronoi partitions and proximity graphs, we analyze a class of aggregate objective functions and propose coverage algorithms in continuous and discrete time.
These algorithms have convergence guarantees and are spatially distributed with respect to appropriate proximity graphs. Numerical simulations illustrate the results.