# 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 and Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052
##### References:
 [1] L. Ambrosio, The flow associated to weakly differentiable vector fields: recent results and open problems, Nonlinear Conservation Laws and Applications, IMA Vol. Math. Appl., Springer, New York, 153 (2011), 181-193. doi: 10.1007/978-1-4419-9554-4_7. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [4] G. Cavagnari, Regularity results for a time-optimal control problem in the space of probability measures, Mathematical Control and Related Fields, 7 (2017), 213-233.  doi: 10.3934/mcrf.2017007. [5] G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability measures, Lecture Notes in Comput. Sci., 9374 (2015), 109-116.  doi: 10.1007/978-3-319-26520-9_11. [6] G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli, Generalized control systems in the space of probability measures, Set-Valued and Variational Analysis (2018). Published online. doi: 10.1007/s11228-017-0414-y. [7] G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a timeoptimal control problem in the space of probability measures, in System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology (eds. L. Bociu, JA. Désidéri, A. Habbal), Springer, Cham, 494 (2016), 200-208. doi: 10.1007/978-3-319-55795-3_18. [8] G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, ESAIM: COCV (2018). Published online. doi: 10.1051/cocv/2017060. [9] G. Cavagnari, A. Marigonda and B. Piccoli, Optimal syncronization problem for a multi-agent system, Networks and Heterogeneous Media, 12 (2017), 277-295.  doi: 10.3934/nhm.2017012. [10] E. Feleqi and F. Rampazzo, Integral representations for bracket-generating multi-flows, Discrete Contin. Dyn. Syst., 35 (2015), 4345-4366.  doi: 10.3934/dcds.2015.35.4345. [11] E. Feleqi and F. Rampazzo, Iterated Lie brackets for nonsmooth vector fields, Nonlinear Differ. Equ. Appl., 24 (2017), Art. 61, 43 pp. doi: 10.1007/s00030-017-0484-4. [12] V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997. [13] A. Marigonda and S. Rigo, Controllability of some nonlinear systems with drift via generalized curvature properties, SIAM J. Control Optim., 53 (2015), 434-474.  doi: 10.1137/130920691. [14] A. Marigonda and T. T. Le Thuy, Small-time local attainability for a class of control systems with state constraints, ESAIM: Control, Optimization and Calc. of Var., 23 (2017), 1003-1021.  doi: 10.1051/cocv/2016022. [15] M. Mauhart and P. W. Michor, Commutators of flows and fields, Arch. Math. (Brno), 28 (1992), 229-236. [16] F. Rampazzo, Frobenius-type theorems for Lipschitz distributions, J. Differential Equations, 243 (2007), 270-300.  doi: 10.1016/j.jde.2007.05.040. [17] F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields, J. Differential Equations, 232 (2007), 134-175.  doi: 10.1016/j.jde.2006.04.016. [18] F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow's theorem, in Proc. of the 40th IEEE Conf. on Decision and Control, Orlando, FL, December 2001, 3 (2001), IEEE Publications, New York, 2613-2618. doi: 10.1109/CDC.2001.980661. [19] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

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##### References:
 [1] L. Ambrosio, The flow associated to weakly differentiable vector fields: recent results and open problems, Nonlinear Conservation Laws and Applications, IMA Vol. Math. Appl., Springer, New York, 153 (2011), 181-193. doi: 10.1007/978-1-4419-9554-4_7. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [4] G. Cavagnari, Regularity results for a time-optimal control problem in the space of probability measures, Mathematical Control and Related Fields, 7 (2017), 213-233.  doi: 10.3934/mcrf.2017007. [5] G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability measures, Lecture Notes in Comput. Sci., 9374 (2015), 109-116.  doi: 10.1007/978-3-319-26520-9_11. [6] G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli, Generalized control systems in the space of probability measures, Set-Valued and Variational Analysis (2018). Published online. doi: 10.1007/s11228-017-0414-y. [7] G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a timeoptimal control problem in the space of probability measures, in System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology (eds. L. Bociu, JA. Désidéri, A. Habbal), Springer, Cham, 494 (2016), 200-208. doi: 10.1007/978-3-319-55795-3_18. [8] G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, ESAIM: COCV (2018). Published online. doi: 10.1051/cocv/2017060. [9] G. Cavagnari, A. Marigonda and B. Piccoli, Optimal syncronization problem for a multi-agent system, Networks and Heterogeneous Media, 12 (2017), 277-295.  doi: 10.3934/nhm.2017012. [10] E. Feleqi and F. Rampazzo, Integral representations for bracket-generating multi-flows, Discrete Contin. Dyn. Syst., 35 (2015), 4345-4366.  doi: 10.3934/dcds.2015.35.4345. [11] E. Feleqi and F. Rampazzo, Iterated Lie brackets for nonsmooth vector fields, Nonlinear Differ. Equ. Appl., 24 (2017), Art. 61, 43 pp. doi: 10.1007/s00030-017-0484-4. [12] V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997. [13] A. Marigonda and S. Rigo, Controllability of some nonlinear systems with drift via generalized curvature properties, SIAM J. Control Optim., 53 (2015), 434-474.  doi: 10.1137/130920691. [14] A. Marigonda and T. T. Le Thuy, Small-time local attainability for a class of control systems with state constraints, ESAIM: Control, Optimization and Calc. of Var., 23 (2017), 1003-1021.  doi: 10.1051/cocv/2016022. [15] M. Mauhart and P. W. Michor, Commutators of flows and fields, Arch. Math. (Brno), 28 (1992), 229-236. [16] F. Rampazzo, Frobenius-type theorems for Lipschitz distributions, J. Differential Equations, 243 (2007), 270-300.  doi: 10.1016/j.jde.2007.05.040. [17] F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields, J. Differential Equations, 232 (2007), 134-175.  doi: 10.1016/j.jde.2006.04.016. [18] F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow's theorem, in Proc. of the 40th IEEE Conf. on Decision and Control, Orlando, FL, December 2001, 3 (2001), IEEE Publications, New York, 2613-2618. doi: 10.1109/CDC.2001.980661. [19] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.
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