# American Institute of Mathematical Sciences

October  2018, 11(5): 845-864. doi: 10.3934/dcdss.2018052

## Measure-theoretic Lie brackets for nonsmooth vector fields

 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

Received  January 2017 Revised  June 2017 Published  June 2018

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

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.

Citation: Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052
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