We study the corrector matrix $P^{\varepsilon}$  to the conductivity equations. We showthat if $P^{\varepsilon}$  converges weakly to the identity, then for any laminate $\det P^{\varepsilon}\geq 0$ at almost every point. This simple property is shown to be false forgeneric microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear].In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal.158 (2001) 155-171]. We use this property of laminates to prove that, in any dimension, the classicalHashin-Shtrikman bounds are not attained by laminates, in certain regimes, when the number ofphases is greater than two. In addition we establish new bounds for the effective conductivity,which are asymptotically optimal for mixtures of three isotropic phases among a certain class ofmicrogeometries, including orthogonal laminates, which we then call quasiorthogonal.

In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized lake.

Il est démontré par Mentagui [ESAIM: COCV9 (2003) 297-315] que, dans le cas des espaces de Banach généraux, la convergenced'Attouch-Wets est stable par une classe d'opérations classiques del'analyse convexe, lorsque les limites des suites d'ensembles et defonctions satisfont certaines conditions de qualification naturelles. Cecitombe en défaut avec la slice convergence. Dans cet article, nousétablissons des conditions de qualification uniformes assurant lastabilité de la slice convergence et de la slice convergence duale par lesmêmes opérations, dont le rôle est fondamental en optimisationconvexe. Nous obtenons comme conséquences certains résultats clés destabilité de l'épi-convergence établis par Mc Linden et Bergstrom [Trans. Amer. Math. Soc.286 (1981) 127-142] en dimension finie. Comme application, nousprésentons un modèle de convergence et de stabilité recouvrant unelarge classe de problèmes en optimisation convexe et en théorie de ladualité. Les éléments clés dans notre démarche sont l'analysed'horizon, les notions de quasi-continuité et d'inf-locale compacité desfonctions convexes, puis la bicontinuité de la transformation deLegendre-Fenchel relativement à la slice convergence et la sliceconvergence duale.

For a Riemannian structure on asemidirect product of Lie groups, the variational problems can bereduced using the group symmetry.Choosing the Levi-Civita connection of a positive definitemetric tensor,instead of any of the canonical connections for the Lie group,simplifies the reduction of the variations but complicates theexpression for the Lie algebra valued covariant derivatives.The origin of the discrepancy is in the semidirect productstructure, which implies that the Riemannianexponential map and the Lie group exponential map do not coincide.The consequence is that the reduced equations look more complicated thanthe original ones.The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups.Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely.The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor.As an example, the case of a rigid body on the Special Euclideangroup is considered in detail.

Using some results proved in De Pascale and Pratelli [Calc. Var. Partial Differ. Equ.14 (2002) 249-274] (and De Pascale et al. [Bull. London Math. Soc.36 (2004) 383-395]) and a suitable interpolation technique, we show that the transport density relative to an Lp source is also an Lp function for any $1\leq p\leq +\infty$ .

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.

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.

We consider an optimal control problem describing a laser-inducedpopulation transfer on a n-level quantum system. For a convex cost depending only on the moduliof controls (i.e. the lasers intensities),we prove that there always exists a minimizer inresonance. This permits to justifysome strategies used in experimental physics. It is also quite importantbecause it permits to reduce remarkablythe complexity of the problem (and extend some of our previous resultsfor n=2 and n=3): instead of looking for minimizers on thesphere $S^{2n-1}\subset\mathbb{C}^n$ one is reduced to look just forminimizers on the sphere $S^{n-1}\subset \mathbb{R}^n$ . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

A hybrid flexible beam equation with harmonicdisturbance at the end where a rigid tip body is attached isconsidered. A simple motor torque feedback control is designedfor which only the measured time-dependent angle of rotation andits velocity are utilized. It is shown that this control can impelthe amplitude of the attached rigid tip body tending to zero astime goes to infinity.

In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments that the stored energy density (surface tension) along a step edge was a smooth symmetric function β of the azimuthal angle θ to the step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$ . The function β provided the crucial thermodynamic parameters needed for the engineering of these materials. Moreover the minimal energy configuration of the step is determined by the values of the stiffness function $\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for these materials. In the 1990's there was a dramatic improvement in electron microscopy which permitted real time observation of the meanderings of a step edge under Brownian heat oscillations. These observations provided much more rapid determination of the relevant thermodynamic parameters for the step edge, even for crystals at temperatures below their roughening temperature. Use of these tools led J. Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive manner as their temperature is varied. The present article is devoted to a model described by a class of variational problems. The main result of the paper describes the solutions of the corresponding problem for a generic integrand.

In this paper, we study the motion planning problem for generic sub-Riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [CITE]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C ∞ case, we study some non-generic generalizations in the analytic case.

Following the Γ-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.

We consider an optimal control problem of Mayer type and prove that,under suitable conditions on the system, the value function isdifferentiable along optimal trajectories, except possibly at theendpoints. We provide counterexamples to show that this property may failto hold if some of our conditions are violated. We then apply our regularityresult to derive optimality conditions for the trajectories of the system.

We consider the eigenvalue problem $$ \begin{array}{l}\displaystyle-{\rm div} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox{in } \Omega u = 0\;\mbox{on } \partial\Omega ,\end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.