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Some geometrical properties of infinite-dimensional bilinear controlled systems

Naceurdine BensalemFernand Pelletier — 1999

Banach Center Publications

The study of controlled infinite-dimensional systems gives rise to many papers (see for instance [GXL], [GXB], [X]) but it is also motivated by various mathematical problems: partial differential equations ([BP]), sub-Riemannian geometry on infinite-dimensional manifolds ([Gr]), deformations in loop-spaces ([AP], [PS]). The first difference between finite and infinite-dimensional cases is that solutions in general do not exist (even locally) for every given control function. The aim of this paper...

Snakes and articulated arms in an Hilbert space

Fernand PelletierRebhia Saffidine — 2013

Annales de la faculté des sciences de Toulouse Mathématiques

The purpose of this paper is to give an illustration of results on integrability of distributions and orbits of vector fields on Banach manifolds obtained in [5] and [4]. Using arguments and results of these papers, in the context of a separable Hilbert space, we give a generalization of a Theorem of accessibility contained in [3] and [6] for articulated arms and snakes in a finite dimensional Hilbert space.

Sur le théorème de Poincaré-Bendixson

Robert MoussuFernand Pelletier — 1974

Annales de l'institut Fourier

Le but de cet article est de démontrer deux conditions nécessaires de non existence d’ensemble minimal exceptionnel dans un feuilletage F de codimension 1 d’une variété compacte M . La première est métrique ; elle porte sur la croissance des feuilles et elle répond à une conjecture de Plante. La seconde est homotopique, elle porte sur les groupes fondamentaux de M et des feuilles de F . De ces deux conditions, nous déduisons deux conditions nécessaires et suffisantes pour qu’un feuilletage...

Singularities of implicit differential systems and maximum principle

Stanisław JaneczkoFernand Pelletier — 2003

Banach Center Publications

The integrability condition for the Lagrangian implicit differential systems of (TP,ω̇), introduced in [7], is applied for the specialized control theory systems. The Pontryagin maximum principle was reformulated in the framework of implicit differential systems and the corresponding necessary and sufficient conditions were proved. The beginning of the classification list of normal forms for Lagrangian implicit differential systems according to the symplectic equivalence is provided and the corresponding...

Generalized PN manifolds and separation of variables

Fernand PelletierPatrick Cabau — 2008

Banach Center Publications

The notion of generalized PN manifold is a framework which allows one to get properties of first integrals of the associated bihamiltonian system: conditions of existence of a bi-abelian subalgebra obtained from the momentum map and characterization of such an algebra linked with the problem of separation of variables.

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