In this paper, using direct and inverse images for fractional stochastic tangent sets, weestablish the deterministic necessary and sufficient conditions which control that thesolution of a given stochastic differential equation driven by the fractional Brownianmotion evolves in some particular sets K. As a consequence, a comparisontheorem is obtained.

On expose les difficultés d'ordremathématique que posent des modèles récents de sédimentation-érosion de bassins élaborés par l'InstitutFrançais du Pétrole et fondés sur la prise en compte dediverses contraintes d'unilatéralité. On présente quelquesrésultats partiels théoriques et des directions de recherche pour larésolution d'un problème inverse posé par l'étudestratigraphique d'une colonne monolithologique.

We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition.We obtain precise estimates on the analyticity of reachable functions.As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C ∞ class, a precise description of all reachable functions is given.

Any two-input left-invariant control affine system of full rank, evolving on theEuclidean group SE (2), is (detached) feedback equivalent to one ofthree typical cases. In each case, we consider an optimal control problem which is thenlifted, via the Pontryagin Maximum Principle, to a Hamiltonian system onthe dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the maintopic of this paper. A qualitative analysis of each reduced system is performed. Thisanalysis includes a study of the stability nature of all equilibrium states, as well asqualitative descriptions of all integral curves. Finally, the reduced Hamilton equationsare explicitly integrated by Jacobi elliptic functions. Parametrisations for all integralcurves are exhibited.

In this article we apply the optimal andthe robust control theory to the sine-Gordon equation. In our casethe control is given by the boundary conditions and we work in a finitetime horizon. We present at the beginning the optimal control problemand we derive a necessary condition of optimality and we continue byformulating a robust control problem for which existence and uniquenessof solutions are derived.

In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are ε-periodic functions modulated by a macroscopic variable, where ε is a small parameter. The mean free path of the particles is also of order ε. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point x 0 where its Hessian matrix is positive definite. This assumption yields a concentration phenomenon around x 0, as ε goes to 0, at a new scale of the order of $\sqrt{\varepsilon}$ which is superimposed with the usual ε oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt{\varepsilon}$ , i.e., of the form $\exp \left (- \frac {1} {2 \varepsilon} M (x-x_0)\cdot (x-x_0) \right )$ , where M is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. We provide a detailed analysis of all abnormal and rigid curves of Goursat structures. We show that neither abnormal curves, if n ≥ 6, nor abnormal curves of all elements of the derived flag, if n ≥ 9, determine the local equivalence class of a Goursat structure. The latter observation is deduced from a generalized version of Bäcklund's theorem. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.

This paper deals with the existence of solutions to the following system:

$$\left\{\begin{array}{l} -\Deltau+u=\frac{\alpha}{\alpha+\beta}a(x)|v|^{\beta} |u|^{\alpha-2}u\quad\mbox{ in}\mathbb{R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta}{\alpha+\beta}a(x)|u|^{\alpha}|v|^{\beta-2}v\quad\mbox{ in }\mathbb{R}^N. \end{array}\right.$$$\left\{\begin{array}{c}\\ \mathrm{-}\mathit{\Delta u}\mathrm{+}\mathit{u}\mathrm{=}\frac{\mathit{\alpha }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }}\mathrm{\right|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }\mathrm{-}\mathrm{2}}\mathit{u}\mathrm{in}{R}^{\mathit{N}}\\ \mathrm{-}\mathit{\Delta v}\mathrm{+}\mathit{v}\mathrm{=}\frac{\mathit{\beta }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }}\mathrm{\right|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }\mathrm{-}\mathrm{2}}\mathit{v}\mathrm{in}{R}^{\mathit{N}}\mathit{.}\end{array}\right.$

With the help of the Nehari manifold and the linking theorem, we prove the existence ofat least two nontrivial solutions. One of them is positive. Our main tools are theconcentration-compactness principle and the Ekeland’s variational principle.

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

A multiphase generalization of the Monge–Kantorovich optimaltransportation problem is addressed. Existence of optimal solutions is established.The optimality equations are related to classical Electrodynamics.

In this paper we study the lower semicontinuity problem for a supremal functional of the form $F(u,\Omega )= \underset{x\in\Omega}{\rm ess\,sup} f(x,u(x),Du(x))$ with respect to the strong convergence in L∞(Ω), furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

The aim of this paper is to establish necessary optimality conditions for optimal controlproblems governed by steady, incompressible Navier-Stokes equations with shear-dependentviscosity. The main difficulty derives from the fact that equations of this type mayexhibit non-uniqueness of weak solutions, and is overcome by introducing a family ofapproximate control problems governed by well posed generalized Stokes systems and bypassing to the limit in the corresponding optimality conditions.

Given the probability measure ν over the given region $\Omega\subset \mathbb{R}^n$ , we consider the optimal location of a setΣ composed by n points in Ω in order to minimize theaverage distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (theclassical optimal facility location problem). The paper compares twostrategies to find optimal configurations: the long-term one whichconsists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one addingat each step at most one point and preserving the configurationbuilt at previous steps. We show that the respective optimizationproblems exhibit qualitatively different asymptotic behavior as $n\to\infty$ , although the optimization costs in both cases have the same asymptotic orders of vanishing.

In this paper we find the optimal regularity for viscositysolutions of the pseudo infinity Laplacian. We prove that thesolutions are locally Lipschitz and show an example that provesthat this result is optimal. We also show existence and uniquenessfor the Dirichlet problem.

This paper analyzes the relation of viability kernels and control sets of control affine systems. A viability kernel describes the largest closed viability domain contained in some closed subset Q of the state space. On the other hand, control sets are maximal regions of the state space where approximate controllability holds. It turns out that the viability kernel of Q can be represented by the union of domains of attraction of chain control sets, defined relative to the given set Q. In particular, with this result control sets and their domains of attraction can be computed using techniques for the computation of attractors and viability kernels.

For systems with slowly varying parameters the controllability behavior is studied and the relation to the control sets for the systems with frozen parameters is clarified.

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

This paper is concerned with the stability of the set of trajectoriesof a patchy vector field, in the presence of impulsiveperturbations. Patchy vector fields arediscontinuous, piecewise smooth vector fieldsthat were introduced in Ancona and Bressan (1999) to study feedback stabilization problems.For patchy vector fields in the plane, with polygonalpatches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term.

We study an optimal boundary control problem for the two dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle. The control acts through the Dirichlet boundary condition. We first establish the existence and uniqueness of the solution for the two-dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle with inhomogeneous Dirichlet boundary data, not necessarily smooth. Then, we prove the existence and uniqueness of the optimal solution over the control set. Finally we derive an optimality system from which the optimal solution can be determined.

Uniqueness of the optimal control is obtained by assuming certain conditions on the crowding effect of the species. Moreover, an approximation procedure for the unique optimal control isdeveloped.