Compensation spectrale et taux de décroissance optimal de l'énergie de systèmes partiellement amortis

We study the stabilization of systems of two equations, for which only one equation is damped by a feedback control. We show that a well chosen control can compensate the real parts of the eigenvalues of the system, therefore, giving the optimal polynomial energy decay rate of the system for smooth initial data. To cite this article: P.  Loreti, B. Rao, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Jeux à champ moyen. II – Horizon fini et contrôle optimal

We continue in this Note our study of the notion of mean field games that we introduced in a previous Note. We consider here the case of Nash equilibria for stochastic control type problems in finite horizon. We present general existence and uniqueness results for the partial differential equations systems that we introduce. We also give a possible interpretation of these systems in term of optimal control. To cite this article: J.-M. Lasry, P.-L. Lions, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Solvability of monotone systems of fully nonlinear elliptic PDE's

We study quasimonotone weakly coupled systems of uniformly elliptic equations of Isaac type. We prove results on existence of viscosity solutions of such systems and give a necessary and sufficient condition for such a system to satisfy the comparison principle. To cite this article: A. Quaas, B. Sirakov, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Un algorithme de Schur pour les fonctions de transfert des systèmes surdéterminés invariants dans une direction

We define an analogue of the Schur algorithm for transfer functions of lossless 2D systems which are invariant with respect to one of the variables. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Controllability of some coupled parabolic systems by one control force

In this Note we present a new approach which allows one to prove new controllability results for some coupled parabolic systems considered in a bounded domain Ω of ${\mathbb{R}}^N$ when one controls by a unique distributed control. We analyze, as a model example, the null controllability of a linear phase field system. First, one controls the system by two controls. Then, one eliminates the introduced fictitious control. Global Carleman estimates and the parabolic regularity are used. To cite this article: M. González-Burgos, R. Pérez-García, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

New constructions of domain decomposition methods for systems of PDEs

We propose new domain decomposition methods for systems of partial differential equations in two and three dimensions. The algorithms are derived with the help of the Smith factorization. This could also be validated by numerical experiments. To cite this article: V. Dolean et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On a class of local systems associated to plane curvesSur une classe de systèmes locaux associés aux courbes planes

We study a class of local systems on the complement of a germ of irreducible plane curve. We exhibit local systems which by [8] give rise to regular holonomic $\text{D}$-modules with characteristic variety the union of the zero section with the conormal of the curve. To cite this article: P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 421–426.     

A factorization method for elliptic problems in a circular domain

We present a method to factorize a second order elliptic boundary value problem in a circular domain, in a system of uncoupled first order initial value problems. We use a space invariant embedding technique along the radius of the circle, in a decreasing way. This technique is inspired in the temporal invariant embedding used by J.-L. Lions for the control of parabolic systems. The singularity at the origin for the initial value problems is studied. A formal calculation for more general star-shaped domains is presented. To cite this article: J. Henry et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Existence et unicité de la solutions faible-renormalisée pour un système non linéaire de Boussinesq

We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq systems. We establish regularity results for the heat equation which we combine with the usual techniques for Navier–Stokes equations mixed with the tools involved for renormalized solutions. To cite this article: N. Bruyère, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Évaluation ponctuelle et espace de Hardy : le cas multi-échelle

We define a point evaluation for transfer operators of multiscale causal dissipative systems. We associate to such a system a de Branges Rovnyak space, which serves as the state space of a coisometric realization. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Étude d'un système non linéaire de Boussinesq–Stefan

We give a few existence results for solutions for a class of Boussinesq–Stefan systems, with suitable conditions on the forcing terms in the right-hand side of the momentum equation depending on the temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Existence de solutions pour une classe de systèmes non linéaires de Boussinesq

We give a few existence results of solutions for a class of Boussinesq systems, with suitable conditions on the right-hand side of the momentum equation, the forcing term depending on temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Regularity results for electrorheological fluids: the stationary caseRésultats de régularité pour les fluides électrorhéologiques : le cas stationnaire

We report on some regularity results for weak solutions to systems modelling electrorheological fluids in the stationary case, as proposed in [8]. To cite this article: E. Acerbi, G. Mingione, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 817–822.     

Bound and ground states of coupled nonlinear Schrödinger equations

We prove existence of bound and ground states of some systems of coupled nonlinear Schrödinger equations. To cite this article: A. Ambrosetti, E. Colorado, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Fractional monodromy of resonant classical and quantum oscillatorsMonodromie fractionnelle des oscillateurs résonnants classiques et quantiques

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.     

Optimal distributed-control of vortices in Navier–Stokes flows

Optimal control of time-dependent fluid-flow governed by the incompressible Navier–Stokes model is considered. A cost functional based on a local dynamical systems characterization of vortices is investigated. The resulting functional is a non-convex function of the velocity gradient tensor. The optimality systems based on a Lagrangian formulation and adjoint equations describing first order necessary optimality conditions is provided. The first and the second derivative of the cost functional with respect to the control are derived. To cite this article: S. Chaabane et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the controllability of the N-dimensional Navier–Stokes and Boussinesq systems with N−1 scalar controls

In this Note we present several controllability results for nonlinear systems of the Navier–Stokes and Boussinesq kind. We discuss the existence of particular controls with a small number of degrees of freedom. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

New estimates for the Laplacian,the div–curl,and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Remarks on exact controllability for Stokes and Navier–Stokes systems

This Note deals with the controllability of Stokes and Navier–Stokes systems with distributed controls with support in possibly small subdomains. We first present a new global Carleman inequality for the solutions to Stokes-like systems that leads to the null controllability at any time T>0. Then, we present a local result concerning exact controllability to trajectories of the Navier–Stokes system. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).