In this article we are interested in the following problem: tofind a map $u: \Omega \to \mathbb{R}^2$ that satisfies $$\left\{\begin{array}{ll}D u \in E\,\, &\mbox{{\it a.e.} in } \Omega\\ u(x)=\varphi(x) &x \in \partial \Omega\end{array}\right.$$ where Ω is an open set of $\mathbb{R}^2$ and E is acompact isotropic set of $\mathbb{R}^{2\times 2}$ . We will show anexistence theorem under suitable hypotheses on φ.

In this paper, we consider a class of stochasticmathematical programs with equilibrium constraints (SMPECs) thathas been discussed by Lin and Fukushima (2003). Based on areformulation given therein, we propose a regularization methodfor solving the problems. We show that, under a weak condition, anaccumulation point of the generated sequence is a feasible pointof the original problem. We also show that such an accumulationpoint is S-stationary to the problem under additional assumptions.

We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -Young measures and $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -varifolds. The energyfunctional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. Theinterest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classicalformulation. The second formulation moreover accounts for concentration effects.

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 study the homogenization of parabolic or hyperbolic equations like \[\rho_\varepsilon{\partial^n u_\varepsilon\over \partial t^n}- {\rm div}(a_\varepsilon\nabla u_\varepsilon) =f \ \ \hbox{ in } \quad {\O\times(0,T)}+\ \ \hbox{\rm boundary conditions}, \quad n \in \{1,2\},\] when the coefficients $\rho_\varepsilon$ , $a_\varepsilon$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure.Memory effects arise in the limit problem.

This work is devoted to prove the exponential decay for the energyof solutions of the Korteweg-de Vries equation in a bounded intervalwith a localized damping term. Following the method in Menzala (2002)which combines energy estimates, multipliers and compactnessarguments the problem is reduced to prove the unique continuation ofweak solutions. In Menzala (2002) the case where solutions vanish on aneighborhood of both extremes of the bounded interval where equationholds was solved combining the smoothing results by T. Kato (1983)and earlier results on unique continuation of smooth solutions byJ.C. Saut and B. Scheurer (1987). In this article we address thegeneral case and prove the unique continuation property in twosteps. We first prove, using multiplier techniques, that solutionsvanishing on any subinterval are necessarily smooth. We then applythe existing results on unique continuation of smooth solutions.

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.

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 concerns constrained dynamic optimization problemsgoverned by delay control systems whose dynamic constraints are described by bothdelay-differential inclusions and linear algebraic equations. This is a new class ofoptimal control systems that, on one hand, may be treated as a specific type ofvariational problems for neutral functional-differential inclusions while, on the otherhand, is related to a special class of differential-algebraic systems with a generaldelay-differential inclusion and a linear constraint link between “slow” and “fast”variables. We pursue a twofold goal: to study variational stability for this class ofcontrol systems with respect to discrete approximations and to derive necessaryoptimality conditions for both delayed differential-algebraic systems under considerationand their finite-difference counterparts using modern tools of variational analysis andgeneralized differentiation. The authors are not familiar with any results in thesedirections for such systems even in the delay-free case. In the first part of the paperwe establish the value convergence of discrete approximations as well as the strongconvergence of optimal arcs in the classical Sobolev space W -1,2. Then using discreteapproximations as a vehicle, we derive necessary optimality conditions for the initialcontinuous-time systems in both Euler-Lagrange and Hamiltonian forms via basicgeneralized differential constructions of variational analysis.

In this paper we study linear conservative systems of finitedimensioncoupled with an infinite dimensional system of diffusive type. Computing the time-derivative of anappropriate energy functional along the solutions helps us toprove the well-posedness of the systemand a stability property.But in order to prove asymptotic stability we need to applya sufficient spectral condition. We also illustrate the sharpness of thiscondition by exhibiting some systems for which we do not have the asymptoticproperty.

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.