Measure-theoretic Lie brackets for nonsmooth vector fields

Giulia Cavagnari; Antonio Marigonda

doi:10.3934/dcdss.2018052

Article Contents

2018, Volume 11, Issue 5: 845-864. Doi: 10.3934/dcdss.2018052

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Measure-theoretic Lie brackets for nonsmooth vector fields

Giulia Cavagnari^1, , and
Antonio Marigonda^2,

1.
Department of Mathematical Sciences, Rutgers University - Camden, 311 N. 5th Street, Camden, NJ 08102, USA
2.
Department of Computer Science, University of Verona, Strada Le Grazie 15, I-37134 Verona, Italy

* Corresponding author: Giulia Cavagnari
* Corresponding author: Giulia Cavagnari

Received: December 31, 2016

Revised: May 31, 2017

Published: June 2018

The authors have been supported by INdAM-GNAMPA Project 2016: Stochastic Partial Differential Equations and Stochastic Optimal Transport with Applications to Mathematical Finance.

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Abstract

In this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used.

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Mathematics Subject Classification: Primary: 34A60, 49J15; Secondary: 46G10.

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