The problem of invariant output tracking is considered: given a control system admitting a symmetry group G, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of G. Invariant output errors are defined as a set of scalar invariants of G; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the required “symmetry-preserving” feedback.

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.

In this paper we introduce a numerical approach adapted to the minimizationof the eigenmodes of a membrane with respect to the domain. This method isbased on the combination of the Level Set method of S. Osher and J.A.Sethian with the relaxed approach. This algorithm enables both changing thetopology and working on a fixed regular grid.

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.

Let $L:\Bbb R^N\times\Bbb R^N\rightarrow\Bbb R$ be a Borelian function andconsider the following problems $$\inf\left\{F(y)=\int_a^bL(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\} \qquad\quad\! (P)$$ 
 $$ \inf\left\{F^{**}(y)=\int_a^bL^{**}(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\}\cdot \quad\;\ \! (P^{**})$$ We give a sufficient condition, weaker then superlinearity, under which $\inf F=\inf F^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.

In this paper, we prove that the Lp approximants naturally associated to a supremal functionalΓ-converge to it. This yields a lower semicontinuity result for supremalfunctionals whose supremand satisfy weak coercivity assumptions aswell as a generalized Jensen inequality. The existence of minimizersfor variational problems involving such functionals (together with aDirichlet condition) then easily follows. In the scalarcase we show the existence of at least one absolute minimizer (i.e. localsolution) among these minimizers. We provide two different proofs ofthis fact relying on different assumptions and techniques.

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.

The soil water movement modelgoverned by the initial-boundary value problem for a quasilinear1-D parabolic equation with nonlinear coefficients is considered.The generalized statement of the problem is formulated. Thesolvability of the problem is proved in a certain class offunctional spaces. The data assimilation problem for this model isanalysed. The numerical results are presented.

We consider the pseudo-p-Laplacian, an anisotropicversion of the p-Laplacian operator for $p\not=2$ . We studyrelevant properties of its first eigenfunction for finite p andthe limit problem as p → ∞.

We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

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.

The control of the surface of water in a long canal bymeans of a wavemaker is investigated. The fluid motion is governed by the Korteweg-de Vries equation in Lagrangian coordinates.The null controllability of the elevation of the fluid surface is obtained thanks to a Carleman estimate and some weighted inequalities. The global uncontrollability is also established.

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.

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equationin unbounded domains, typically $\mathbb R_+$ or  $\mathbb R^N$ . Considering an unbounded and disconnected control region of the form $\omega := \cup _n \omega _n$ , we prove two null controllability results:under some technical assumption on the control parts $\omega _n$ , we provethat every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2(Ω) can be controlled to zero usingcontrol functions in a weighted L 2 space.At last we give several examples in which the control region has a finite measure and our null controllability results apply.

The paper is a continuation of a previous work of the same authorsdealing with homogenization processes for some energiesof integral type arising in the modeling of rubber-like elastomers.The previous paper took into account the general case of thehomogenization of energies in presence of pointwise oscillatingconstraints on the admissible deformations.In the present paper homogenization processes are treated in theparticular case of fixed constraints set, in which minimalcoerciveness hypotheses can be assumed, and in which the results canbe obtained in the general framework of BV spaces.The classical homogenization result is established for Dirichlet withaffine boundary data, Neumann, and mixedproblems, by proving that the limit energy is again of integral type,gradient constrained, and with an explicitly computedhomogeneous density.

We establish an approximation theorem for a sequence oflinear elastic strains approaching a compact set in L 1 by thesequence of linear strains of mapping bounded in Sobolev space W 1,p . We apply this result to establish equalities forsemiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

Let H be a real Hilbert space, $\Phi_1: H\to \xR$ a convex function of class ${\mathcal C}^1$ that we wish to minimize under the convexconstraint S.A classical approach consists in following the trajectories of the generalizedsteepest descent system (cf.   Brézis [CITE]) appliedto the non-smooth function  $\Phi_1+\delta_S$ . Following Antipin [1], it is also possible to use a continuous gradient-projection system.We propose here an alternative method as follows:given a smooth convex function  $\Phi_0: H\to \xR$ whose critical points coincidewith Sand a control parameter $\varepsilon:\xR_+\to \xR_+$ tending to zero,we consider the “Steepest Descent and Control” system \[(SDC) \qquad \dot{x}(t)+\nabla \Phi_0(x(t))+\varepsilon(t)\, \nabla \Phi_1(x(t))=0,\] where the control ε satisfies $\int_0^{+\infty} \varepsilon(t)\, {\rm d}t =+\infty$ . This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that $d(x(t), {\rm argmin}\kern 0.12em_S \Phi_1) \to 0\quad (t\to +\infty),$ and we give sufficient conditions under which  $x(t) \to \bar{x}\in \,{\rm argmin}\kern 0.12em_S \Phi_1$ .We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

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 $1\leq p\leq +\infty$ .

Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we givecharacterizationsof the existence of so-called global and local error bounds, for lowersemicontinuous functions defined on complete metric spaces. We thusprovide asystematic and synthetic approach to the subject, emphasizing the specialcaseof convex functions defined on arbitrary Banach spaces (refining theabstract partof Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularityof closed-graph multifunctions between complete metric spaces.