On the Parametrized Integral of a Multifunction: The Unbounded Case
Hess, Christian (2007), On the Parametrized Integral of a Multifunction: The Unbounded Case, Set-Valued Analysis, 15, 1, p. 1-20. http://dx.doi.org/10.1007/s11228-006-0032-6
TypeArticle accepté pour publication ou publié
Nom de la revueSet-Valued Analysis
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Résumé (EN)Integration of set-valued maps (alias multifunctions) depending on a parameter is revisited. Results of Artstein, and of Saint-Pierre and Sajid are extended to the case of set-valued maps whose values may be unbounded. In the general case, this is achieved assuming that the transition probabilities involved in the integration procedure are absolutely continuous with respect to some fixed probability measure. However, when the integrating probability measure does not depend on the parameter this hypothesis is shown to be unnecessary. On the other hand, an alternative proof of a result of Saint-Pierre and Sajid is provided for convex compact-valued multifunctions. An application is given to the control of chattering systems. It is an extension of a result of Artstein to the case of set-valued maps with unbounded values. The proof of the main results is simple and essentially relies on measurable selections arguments.
Mots-clésset-valued maps; integration of multifunctions; Aumann integral; transition probability; measurable selections; chattering controls; Radon–Nikodym property
Affichage des éléments liés par titre et auteur.
The Largest Class of Closed Convex Valued Multifunctions for which Effros Measurability and Scalar Measurability Coincide Hess, Christian; Barbati, Alberto (1998) Article accepté pour publication ou publié