Éditeur(s) : HAL CCSD
Résumé : 21 pages Publié sous le titre: Spectral asymptotic analysis in algebras of generalized functions dans Asymptot. Anal. 59/1-2, 83-107, 2008.
International audience
We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.
Asymptot. Anal.

Droits : info:eu-repo/semantics/OpenAccess
hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656/document
https://hal.archives-ouvertes.fr/hal-00140656/file/MicrolocalDelcroixMartiOberguggenbergerArkiv.pdf
ARXIV : 0704.1077

Éditeur(s) : HAL CCSD
Résumé : To appear in the Proceeding of GF2004 in ITSF (8 pages).
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential of a subclass of such operators.
https://hal.archives-ouvertes.fr/hal-00004897

hal-00004897
https://hal.archives-ouvertes.fr/hal-00004897
https://hal.archives-ouvertes.fr/hal-00004897/document
https://hal.archives-ouvertes.fr/hal-00004897/file/GF04BernardDelcroixColombeauNov01.pdf
ARXIV : math.GM/0505179

Éditeur(s) : HAL CCSD
Résumé : International audience
We consider a second-order differential system with Hessian-driven damping . An interesting property of this system is that, after introduction of an auxiliary variable y , it can be equivalently written as a first-order system in time and space. This allows us to extend the analysis to the case of a convex lower semicontinuous potential and so to introduce constraints in the model. When considering the indicator function of a closed convex set, the subdifferential operator takes account of the contact forces. In this setting, by playing with the geometrical damping parameter, we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The system is dissipative; in the convex case each trajectory weakly converges to a minimizer of the global potential energy function. Exponential stabilization is obtained under strong convexity assumptions.
Differential Equations & Applications

hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214/document
https://hal.archives-ouvertes.fr/hal-00803214/file/Attouch-Mainge-Redont-30August2011.pdf

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.
ISSN: 1065-2469

hal-01530753
https://hal.univ-antilles.fr/hal-01530753

Éditeur(s) : HAL CCSD
Résumé : International audience
This paper is concerned with a null controllability problem for nonlinear parabolic equations appearing in mathematical biology. We consider a system of two equations with a nonlocal nonlinearity, in a bounded domain Ω of ℝN0, N0 being a positive integer sufficiently small. Such models can be used in the context of dynamics of biological systems to describe the migration of population (of bacteria in a container). Our aim is to find a control common to both equations, which solves the null controllabillity of the studied system under constraints on the state. A similar control problem has recently been addressed for the Laplace operator, so the present paper extends those results.
2017 Proceedings of the Conference on Control and its Applications
Pittsburgh, United States

Droits : http://hal.archives-ouvertes.fr/licences/copyright/
hal-01567339
https://hal.archives-ouvertes.fr/hal-01567339
DOI : 10.1137/1.9781611975024.3

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
ISSN: 0362-546X

hal-00816761
https://hal.univ-antilles.fr/hal-00816761
DOI : 10.1016/0362-546X(93)90065-Z

Éditeur(s) : HAL CCSD
Résumé : We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.
https://hal.archives-ouvertes.fr/hal-00190006

Droits : info:eu-repo/semantics/OpenAccess
hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006/document
https://hal.archives-ouvertes.fr/hal-00190006/file/Regularity-Local-Microlocal-Analysis5.pdf
ARXIV : 0711.3688

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz’ result is contained in our main theorem.
ISSN: 0022-247X

hal-01530748
https://hal.univ-antilles.fr/hal-01530748
https://hal.univ-antilles.fr/hal-01530748/document
https://hal.univ-antilles.fr/hal-01530748/file/delcroixgskttjan21.pdf
DOI : 10.1016/j.jmaa.2005.01.006

Éditeur(s) : HAL CCSD
Résumé : International audience
Generalized functions 2004
Novi Sad, Serbia

hal-01535463
https://hal.univ-antilles.fr/hal-01535463

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems
ISSN: 0163-0563

hal-00776641
https://hal.univ-antilles.fr/hal-00776641
DOI : 10.1080/01630563.2011.619676