Invariants géométriques des espaces localement symétriques et théorèmes de multiplicateurs

Résumé  On prouve un analogue du théorème de multiplicateurs de H?rmander– Mikhlin sur les espaces localement symétriques. Les auteurs ont été soutenus par le programme EPEAK Pythagore II (Grèce).     

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.     

Convexes hyperboliques et fonctions quasisymétriques

Hyperbolic convex sets and quasisymmetric functions Every bounded convex open set of R ^m is endowed with its Hilbert metric d . We give a necessary and sufficient condition, called quasisymmetric convexity, for this metric space to be hyperbolic. As a corollary, when the boundary is real analytic, is always hyperbolic.In dimension 2, this condition is: in affine coordinates, the boundary is locally the graph of a C¹ strictly convex function whose derivative is quasisymmetric.      

Contrôlabilité et observabilité unilatérales de systèmes hyperboliques quasi-linéaires

The known theory on the one-side exact boundary controllability and the one-side exact boundary observability for first-order quasilinear hyperbolic systems requires that the unknown variables should be suitably coupled in the boundary conditions at the non-control or non-observation side. In this Note we illustrate, with an inspiring example, that the one-side exact boundary controllability and the one-side exact boundary observability can still be realized by means of a suitable coupling among the unknown variables in the quasilinear hyperbolic system itself. To cite this article: T. Li et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Temps d'instabilité des systèmes hamiltoniens initialement hyperboliques

We compute the time of drift along a transition chain for an initially hyperbolic Hamiltonian system. We prove that this time is polynomial with respect to the opposite of the perturbing parameter.     

Systèmes quasi-orthogonaux et théorème central limite presque sur

We prove, in the case of irrational rotations, the existence of a generalized moment version of the Almost Sure Central Limit Theorem (ASCLT), with a speed of convergence. In our strategy, we use the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context, which should also permit to be able to treat problems of comparable nature.     

Polynômes hyperboliques et combinaisons linéaires de certains polynômes hyperboliques

We give some results in the theory of hyperbolic polynomials and we study the hyperbolicity of some linear combinations of hyperbolic polynomials.Note présentée par Philippe G. Ciarlet.     

Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation

Blake Temple (Trans. Amer. Math. Soc.280 (1983), 781–795) has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide. In this note, we prove the global existence of weak solutions for such systems, with any bounded variation initial condition. The proof is based upon standard numerical schemes, as well as upon parabolic regularization. The key is that the total variation of the Riemann invariants is decreasing in time. At least, in the case of the initial condition with compact support, we prove by using the Glimm scheme that the system is decoupling in two conservation laws in one unknown, in finite time.     

Analyse d'un schéma décalé pour l'approximation de systèmes hyperboliques non conservatifs

We consider a model system made of two nonlinear equations which are non conservative. A conservation law can be obtained from these equations through linear operations only, which don't modify the shock waves. A numerical scheme based on a different mesh adapted to each variable is proposed. By choosing a shifted mesh, we have un explicit Riemann solver and we can derive a finite volume scheme. We prove a priori estimates in L^∞ norm and Total Variation for the system, which lead to a strong convergence in L¹ norm towards a solution satisfying the associated conservation law.     

Systèmes hyperboliques d'équations aux dérivées partielles linéaires : régularité L2loc et matrices diagonalisables

The well-posedness of a Cauchy problem issue from an hyperbolic linear system is linked to the spectral properties of a real matrix pencil. It is known that such a problem is well posed in L² if and only if the imaginary exponential of the pencil is bounded. We give a condition to have a bounded exponential when the eigenvalues don't have the same multiplicities. For pencils spanned by two 3×3 matrices, we prove that the exponential is bounded if and only if the pencil is analytically diagonable.     

Une généralisation des fonctions quasi-symétriques et des fonctions symétriques non commutatives

We construct a self-dual bialgebra admitting as quotients or subalgebras various recent generalizations of symmetric functions, such that most of the usual constructions can be lifted.     

Stabilisation des systèmes bilinéaires homogènes

We deal with the problem of stabilizing the bilinear homogeneous systems. We give sufficient conditions for a bilinear system to be stabilizable with homogeneous feedback of degree zero. The explicit design of the feedback is given.