This article aims at studying the controllability of a simplified fluid structure interaction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan, RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995); J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. This interaction is modeled by a wave equation surrounding a harmonic oscillator. Our main result states that, in the radially symmetric case, this system can be controlled from the outer boundary. This improves previous results [J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. Our proof is based on a spherical harmonic decomposition of the solution and the so-called lateral propagation of the energy for 1d waves.

The aim of this paper is to provide a rigorous variational formulation forthe detection of points in 2-d biological images. To this purposewe introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals forwhich we prove the Γ-convergence to the initial one.

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.

Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.

The propagation of fractures in a solid undergoing cyclic loadingsis known as the fatigue phenomenon. In this paper, we present a time continuousmodel for fatigue, in the special situation of the debonding of thin layers,coming from a time discretized version recently proposed byJaubert and Marigo [C. R. Mecanique333 (2005) 550–556].Under very general assumptions on the surface energy density andon the applied displacement, we discuss the well-posedness of ourproblem and we give the main properties of the evolution process.

A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional conditions, it is shown that the value function is actually twice differentiable and the so-called quasi-Riccati equation is derived, whose solution can be used to construct the state feedback representation for the optimal control.

For a hybrid system composed of a cable with masses at both ends, we prove the existence of solutions for a class of nonlinear and nonmonotone feedback laws by means of a priori estimates. Assuming some local monotonicity, strong stabilization is obtained thanks to some Riemann's invariants technique and La Salle's principle.

This article is divided into two chapters. Theclassical problem of homogenization of elliptic operators withperiodically oscillating coefficients is revisited in thefirst chapter. Following a Fourier approach, we discuss someof the basic issues of the subject: main convergence theorem,Bloch approximation, estimates on second order derivatives,correctors for the medium, and so on. The second chapter isdevoted to the discussion of some non-classical behaviour ofvibration problems of periodic structures.

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during themodeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls -neighborhood of a reference function whereby the underlying analysis allows to use weaker norms than $L^\infty$ .

Let ${\cal L}$ (x,u,∇ u) be a Lagrangian periodic of period 1 in x 1,...,x n ,u. We shall study the non self intersectingfunctions u: R n ${\to}$ R minimizing ${\cal L}$ ; non self intersecting means that, if u(x 0 + k) + j = u(x 0) for some x 0∈R n and (k , j) ∈Z n × Z, then u(x) = u(x + k) + j $\;\forall$ x. Moser has shown that each of these functions is at finite distance from a plane u = ρ $\cdot$ x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called $\beta(\rho)$ since it only depends on the slope of u.Aubry and Senn have noticed a connection between $\beta(\rho)$ and the theory of crystals in ${\bf R}^{n+1}$ , interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$ . The polar of β is usually called -α; Senn has shown that α is C 1 and that the dimension of the flat of α which contains c depends only on the “rational space” of $\alpha^\prime$ (c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of α: they are C 1 and their dimension depends only on the rational space of their normals.

The Monge-Kantorovich problem is revisited by means of a variantof the saddle-point method without appealing to c-conjugates. Anew abstract characterization of the optimal plans is obtained inthe case where the cost function takes infinite values. It leadsus to new explicit sufficient and necessary optimality conditions.As by-products, we obtain a new proof of the well-knownKantorovich dual equality and an improvement of the convergence ofthe minimizing sequences.

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.

In this paper we study the compact and convex setsK ⊆ Ω ⊆ ℝ2 that minimize

\begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*}${\mathrm{\int }}_{\mathit{\Omega }}\mathrm{dist}\mathrm{\left(}{x}\mathit{,K}\mathrm{\right)}\mathrm{d}{x}\mathrm{+}{\mathit{\lambda }}_{\mathrm{1}}\mathrm{Vol}\mathrm{\left(}\mathit{K}\mathrm{\right)}\mathrm{+}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{Per}\mathrm{\left(}\mathit{K}\mathrm{\right)}$

In terms of the normal cone and the coderivative,we provide some necessary and/or sufficient conditions of metric subregularity for(not necessarily closed) convex multifunctions in normed spaces. As applications, we present someerror bound results for (not necessarily lower semicontinuous) convex functions on normedspaces. These results improve and extend some existing error bound results.

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$}$\mathrm{\{}{\mathrm{ℰ}}_{\mathit{\epsilon }}{\mathrm{\}}}_{\mathit{\epsilon }}\mathit{,}\hspace{0.17em}\mathrm{\{}\begin{array}{}{}_{\mathrm{􏽥}}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}{\mathrm{\}}}_{\mathit{\epsilon }}$ of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the Γ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰε and \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ strictly contains the domain of the Γ-limit of ℰε.

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.

A distributed optimal control problem for evolutionary Stokes flows isstudied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the correspondingoptimality system are derived based on estimates for the penalizedStokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, theconvergence of the solutions of the penalized optimality systems as ε → 0 is examined.

Let $\Lambda $ be a Lagrangian submanifold of $T^{*}X$ for some closedmanifold X. Let $S(x,\xi )$ be a generating function for $\Lambda $ whichis quadratic at infinity, and let W(x) be the corresponding graph selectorfor $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that thereexists a subset $X_{0}\subset X$ of measure zero such that W is Lipschitzcontinuous on X, smooth on $X\backslash X_{0}$ and $(x,\partial W/\partialx(x))\in \Lambda $ for $X\backslash X_{0}.$ Let H(x,p)=0 for $(x,p)\in\Lambda$ . Then W is a classical solution to $H(x,\partial W/\partialx(x))=0$ on $X\backslash X_{0}$ and extends to a Lipschitz function on thewhole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions.We also prove that these conditions are satisfied in many cases, includingcertain commonly occuring cases where H(x,p) is not convex in p.