# American Institute of Mathematical Sciences

June  2017, 7(2): 213-233. doi: 10.3934/mcrf.2017007

## Regularity results for a time-optimal control problem in the space of probability measures

 Department of Mathematics, University of Trento, Via Sommarive 14, Ⅰ-38123 Povo (Trento), Italy

Received  April 2016 Revised  September 2016 Published  April 2017

Fund Project: The author has been supported by INdAM-GNAMPA Project 2015: Set-valued Analysis and Optimal Transportation Theory Methods in Deterministic and Stochastics Models of Financial Markets with Transaction Costs

This paper investigates some regularity properties of the minimum time function for a time-optimal control problem in the space of probability measures endowed with the topology induced by the Wasserstein metric. The main motivation leading us to the generalization of the classical theory to this framework is to model situations in which we have only a probabilistic knowledge of the initial state, as it happens in real settings where noises and measurement errors may occur. We consider a deterministic evolution for a system ruled by a controlled continuity equation and, pursuing the goal of studying a generalization of the classical results for this setting, we prove an attainability result and a locally Lipschitz continuity property for the generalized minimum time function.

Citation: Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007
##### References:
 [1] L. Ambrosio, The flow associated to weakly differentiable vector fields: recent results and open problems, Contribute in 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. Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, 2nd edition, Birkhäuser Verlag, Basel, 2008. Google Scholar [3] P. Bernard, Young measures, superpositions and transport, Indiana Univ. Math. J., 57 (2008), 247-275. doi: 10.1512/iumj.2008.57.3163. Google Scholar [4] 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 [5] 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 [6] G. Cavagnari and A. Marigonda, Measure-theoretic Lie brackets for nonsmooth vector fields, submitted.Google Scholar [7] G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability measures, in Large-scale scientific computing, Lecture Notes in Computer Science, Springer, Heidelberg, 9374(2015), 109-116. doi: 10.1007/978-3-319-26520-9_11. Google Scholar [8] 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, to appear.Google Scholar [9] G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a time-optimal control problem in the space of probability measures, in Bociu L. , Désidéri JA. , Habbal A. (eds) System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology, Springer, Cham, 494(2016), 200-208. doi: 10.1007/978-3-319-55795-3_18. Google Scholar [10] G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, submitted.Google Scholar [11] 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 [12] S. Lisini and A. Marigonda, On a class of modified Wasserstein distance induced by concave mobility functions defined on bounded intervals, Manuscripta Mathematica, 133 (2010), 197-224. doi: 10.1007/s00229-010-0371-3. Google Scholar [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. Google Scholar [14] A. Marigonda and T. Thi Thien Le, Small-time local attainability for a class of control systems with state constraints ESAIM: Control, Optimization and Calc. of Var. , (2016), to appear. doi: 10.1051/cocv/2016022. Google Scholar [15] C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften, 338 Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

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##### References:
 [1] L. Ambrosio, The flow associated to weakly differentiable vector fields: recent results and open problems, Contribute in 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. Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, 2nd edition, Birkhäuser Verlag, Basel, 2008. Google Scholar [3] P. Bernard, Young measures, superpositions and transport, Indiana Univ. Math. J., 57 (2008), 247-275. doi: 10.1512/iumj.2008.57.3163. Google Scholar [4] 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 [5] 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 [6] G. Cavagnari and A. Marigonda, Measure-theoretic Lie brackets for nonsmooth vector fields, submitted.Google Scholar [7] G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability measures, in Large-scale scientific computing, Lecture Notes in Computer Science, Springer, Heidelberg, 9374(2015), 109-116. doi: 10.1007/978-3-319-26520-9_11. Google Scholar [8] 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, to appear.Google Scholar [9] G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a time-optimal control problem in the space of probability measures, in Bociu L. , Désidéri JA. , Habbal A. (eds) System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology, Springer, Cham, 494(2016), 200-208. doi: 10.1007/978-3-319-55795-3_18. Google Scholar [10] G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, submitted.Google Scholar [11] 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 [12] S. Lisini and A. Marigonda, On a class of modified Wasserstein distance induced by concave mobility functions defined on bounded intervals, Manuscripta Mathematica, 133 (2010), 197-224. doi: 10.1007/s00229-010-0371-3. Google Scholar [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. Google Scholar [14] A. Marigonda and T. Thi Thien Le, Small-time local attainability for a class of control systems with state constraints ESAIM: Control, Optimization and Calc. of Var. , (2016), to appear. doi: 10.1051/cocv/2016022. Google Scholar [15] C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften, 338 Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar
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