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Attainability property for a probabilistic target in wasserstein spaces
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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