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February  2021, 41(2): 777-812. doi: 10.3934/dcds.2020300

Attainability property for a probabilistic target in wasserstein spaces

Giulia Cavagnari 1,, and  Antonio Marigonda 2,

1. 

Politecnico di Milano, Dipartimento di Matematica "F. Brioschi", Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
2. 

University of Verona, Department of Computer Science, Strada Le Grazie 15, I-37134 Verona, Italy

* Corresponding author: Giulia Cavagnari

Received  April 2019 Revised  April 2020 Published  February 2021 Early access  August 2020

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called multiagent system made of several possible interacting agents.

Keywords: optimal transport, differential inclusions, time optimal control, small time local attainability, multi-agent system.
Mathematics Subject Classification: Primary: 34A60; Secondary: 49J15, 49J53, 90C56.
Citation: Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300
References:
[1]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, 256 (2014), 2194-2245.  doi: 10.1016/j.jde.2013.12.018.  Google Scholar

[2]

L. Ambrosio and W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math., 61 (2008), 18-53.  doi: 10.1002/cpa.20188.  Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition [MR1048347], Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

Y. Averboukh, Viability theorem for deterministic mean field type control systems, Set-Valued and Variational Analysis, 26 (2018), 993-1008.  doi: 10.1007/s11228-018-0479-2.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations, 3 (1995), 273-298.  doi: 10.1007/BF01189393.  Google Scholar

[7]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[8]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Mathematical Models and Methods in Applied Sciences, 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.  Google Scholar

[9]

P. Cardaliaguet, Notes on Mean Field Games, 2013. Google Scholar

[10]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I, Probability Theory and Stochastic Modeling, 83, Springer International Publishing, 2018.  Google Scholar

[11]

G. Cavagnari, Regularity results for a time-optimal control problem in the space of probability measures, Math. Control Relat. Fields, 7 (2017), 213-233.  doi: 10.3934/mcrf.2017007.  Google Scholar

[12]

G. Cavagnari, S. Lisini, C. Orrieri and G. Savaré, Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and Gamma-convergence, preprint. Google Scholar

[13]

G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli, Generalized control systems in the space of probability measures, Set-Valued Var. Anal., 26 (2018), 663-691.  doi: 10.1007/s11228-017-0414-y.  Google Scholar

[14]

G. Cavagnari, A. Marigonda and B. Piccoli, Superposition principle for differential inclusions, in Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science (eds. I. Lirkov and S. Margenov), 10665, Springer, Cham, (2018), 201–209.  Google Scholar

[15]

G. Cavagnari, A. Marigonda and M. Quincampoix, Compatibility of state constraints and dynamics for multiagent control systems, preprint. Google Scholar

[16]

J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.  Google Scholar

[17]

M. Duprez, M. Morancey and F. Rossi, Approximate and exact controllability of the continuity equation with a localized vector field, SIAM J. Control Optim., 57 (2019), 1284-1311.  doi: 10.1137/17M1152917.  Google Scholar

[18]

M. Duprez, M. Morancey and F. Rossi, Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field, Journal of Differential Equations, 269 (2020), 82-124.  doi: 10.1016/j.jde.2019.11.098.  Google Scholar

[19]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U. S. A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar

[20]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131.  Google Scholar

[21]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318–390. doi: 10.1016/j.matpur.2011.11.004.  Google Scholar

[22]

M. Fornasier, S. Lisini, C. Orrieri and G. Savaré, Mean-field optimal control as gamma-limit of finite agent controls, European J. Appl. Math., 30 (2019), 1153-1186.  doi: 10.1017/S0956792519000044.  Google Scholar

[23]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130400, 21 pp. doi: 10.1098/rsta.2013.0400.  Google Scholar

[24]

W. Gangbo and A. Tudorascu, On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations, Journal de Mathématiques Pures et Appliquées, 125 (2019), 119-174.  doi: 10.1016/j.matpur.2018.09.003.  Google Scholar

[25]

W. Gangbo and A. Świech, Optimal transport and large number of particles, Discrete Contin. Dyn. Syst., 34 (2014), 1397-1441.  doi: 10.3934/dcds.2014.34.1397.  Google Scholar

[26]

C. Jimenez, A. Marigonda and M. Quincampoix, Optimal control of multiagent systems in the Wasserstein space,, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 58, 45pp. doi: 10.1007/s00526-020-1718-6.  Google Scholar

[27]

M. I. Krastanov and M. Quincampoix, Local small time controllability and attainability of a set for nonlinear control system, ESAIM Control Optim. Calc. Var., 6 (2001), 499-516.  doi: 10.1051/cocv:2001120.  Google Scholar

[28]

T. T. T. Le and A. Marigonda, Small-time local attainability for a class of control systems with state constraints, ESAIM Control Optim. Calc. Var., 23 (2017), 1003-1021.  doi: 10.1051/cocv/2016022.  Google Scholar

[29]

A. Marigonda and S. Rigo, Controllability of some nonlinear systems with drift via generalized curvature properties, SIAM J. Control and Optimization, 53 (2015), 434-474.  doi: 10.1137/130920691.  Google Scholar

[30]

A. Marigonda and M. Quincampoix, Mayer control problem with probabilistic uncertainty on initial positions, J. Differential Equations, 264 (2018), 3212-3252.  doi: 10.1016/j.jde.2017.11.014.  Google Scholar

[31]

C. Orrieri, Large deviations for interacting particle systems: joint mean-field and small-noise limit, arXiv: 1810.12636. Google Scholar

[32]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, 256 (2014), 2194-2245.  doi: 10.1016/j.jde.2013.12.018.  Google Scholar

[2]

L. Ambrosio and W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math., 61 (2008), 18-53.  doi: 10.1002/cpa.20188.  Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition [MR1048347], Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

Y. Averboukh, Viability theorem for deterministic mean field type control systems, Set-Valued and Variational Analysis, 26 (2018), 993-1008.  doi: 10.1007/s11228-018-0479-2.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations, 3 (1995), 273-298.  doi: 10.1007/BF01189393.  Google Scholar

[7]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[8]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Mathematical Models and Methods in Applied Sciences, 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.  Google Scholar

[9]

P. Cardaliaguet, Notes on Mean Field Games, 2013. Google Scholar

[10]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I, Probability Theory and Stochastic Modeling, 83, Springer International Publishing, 2018.  Google Scholar

[11]

G. Cavagnari, Regularity results for a time-optimal control problem in the space of probability measures, Math. Control Relat. Fields, 7 (2017), 213-233.  doi: 10.3934/mcrf.2017007.  Google Scholar

[12]

G. Cavagnari, S. Lisini, C. Orrieri and G. Savaré, Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and Gamma-convergence, preprint. Google Scholar

[13]

G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli, Generalized control systems in the space of probability measures, Set-Valued Var. Anal., 26 (2018), 663-691.  doi: 10.1007/s11228-017-0414-y.  Google Scholar

[14]

G. Cavagnari, A. Marigonda and B. Piccoli, Superposition principle for differential inclusions, in Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science (eds. I. Lirkov and S. Margenov), 10665, Springer, Cham, (2018), 201–209.  Google Scholar

[15]

G. Cavagnari, A. Marigonda and M. Quincampoix, Compatibility of state constraints and dynamics for multiagent control systems, preprint. Google Scholar

[16]

J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.  Google Scholar

[17]

M. Duprez, M. Morancey and F. Rossi, Approximate and exact controllability of the continuity equation with a localized vector field, SIAM J. Control Optim., 57 (2019), 1284-1311.  doi: 10.1137/17M1152917.  Google Scholar

[18]

M. Duprez, M. Morancey and F. Rossi, Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field, Journal of Differential Equations, 269 (2020), 82-124.  doi: 10.1016/j.jde.2019.11.098.  Google Scholar

[19]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U. S. A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar

[20]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131.  Google Scholar

[21]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318–390. doi: 10.1016/j.matpur.2011.11.004.  Google Scholar

[22]

M. Fornasier, S. Lisini, C. Orrieri and G. Savaré, Mean-field optimal control as gamma-limit of finite agent controls, European J. Appl. Math., 30 (2019), 1153-1186.  doi: 10.1017/S0956792519000044.  Google Scholar

[23]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130400, 21 pp. doi: 10.1098/rsta.2013.0400.  Google Scholar

[24]

W. Gangbo and A. Tudorascu, On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations, Journal de Mathématiques Pures et Appliquées, 125 (2019), 119-174.  doi: 10.1016/j.matpur.2018.09.003.  Google Scholar

[25]

W. Gangbo and A. Świech, Optimal transport and large number of particles, Discrete Contin. Dyn. Syst., 34 (2014), 1397-1441.  doi: 10.3934/dcds.2014.34.1397.  Google Scholar

[26]

C. Jimenez, A. Marigonda and M. Quincampoix, Optimal control of multiagent systems in the Wasserstein space,, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 58, 45pp. doi: 10.1007/s00526-020-1718-6.  Google Scholar

[27]

M. I. Krastanov and M. Quincampoix, Local small time controllability and attainability of a set for nonlinear control system, ESAIM Control Optim. Calc. Var., 6 (2001), 499-516.  doi: 10.1051/cocv:2001120.  Google Scholar

[28]

T. T. T. Le and A. Marigonda, Small-time local attainability for a class of control systems with state constraints, ESAIM Control Optim. Calc. Var., 23 (2017), 1003-1021.  doi: 10.1051/cocv/2016022.  Google Scholar

[29]

A. Marigonda and S. Rigo, Controllability of some nonlinear systems with drift via generalized curvature properties, SIAM J. Control and Optimization, 53 (2015), 434-474.  doi: 10.1137/130920691.  Google Scholar

[30]

A. Marigonda and M. Quincampoix, Mayer control problem with probabilistic uncertainty on initial positions, J. Differential Equations, 264 (2018), 3212-3252.  doi: 10.1016/j.jde.2017.11.014.  Google Scholar

[31]

C. Orrieri, Large deviations for interacting particle systems: joint mean-field and small-noise limit, arXiv: 1810.12636. Google Scholar

[32]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

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