Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 663–691 | Cite as

Generalized Control Systems in the Space of Probability Measures

  • Giulia Cavagnari
  • Antonio MarigondaEmail author
  • Khai T. Nguyen
  • Fabio S. Priuli


In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on \(\mathbb {R}^{d}\). We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1–16, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system.


Optimal transport Differential inclusions Time-optimal control Set-valued analysis 

Mathematics Subject Classification (2010)

34A60 49J15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio, L.: The flow associated to weakly differentiable vector fields: recent results and open problems. IMA V. Math. 153, 181–193 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Gangbo, W.: Hamiltonian ODEs in the Wasserstein space of probability measures. Commun. Pur. Appl. Math. 61, 18–53 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel, Birkhäuser Verlag (2008)zbMATHGoogle Scholar
  4. 4.
    Ancona, F., Bressan, A.: Patchy vector fields and asymptotic stabilization. ESAIM Contr. Optim. Ca. 4, 445–471 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ancona, F., Bressan, A.: Nearly time optimal stabilizing patchy feedbacks. Ann. I. H. Poincaré,-A 24, 279–310 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. The Clarendon Press Oxford University Press, New York (2000)zbMATHGoogle Scholar
  7. 7.
    Aubin, J.-P., Cellina, A.: Differential Inclusions - Set-Valued Maps and Viability Theory. Springer-Verlag Berlin, Heidelberg, Germany (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Aubin, J.-P., Frankowska, H.: Set-valued analysis. Birkhäuser Boston Inc., Boston, MA (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bernard, P.: Young measures, superpositions and transport. Indiana U. Math. J 57, 247–276 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bressan, A., Priuli, F.S.: Nearly optimal patchy feedbacks. Discrete Cont. Dyn.-A 21, 687–701 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations, p. 207. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1989)zbMATHGoogle Scholar
  12. 12.
    Cannarsa, P., Wolenski, P.R.: Semiconcavity of the value function for a class of differential inclusions. Discrete Cont. Dyn. S 29, 453–466 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cannarsa, P., Marigonda, A., Nguyen, K.T.: OptiMality conditions and regularity results for time optimal control problems with differential inclusions. J. Math. Anal. Appl. 427, 202–228 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cardaliaguet, P., Quincampoix, M.: Deterministic differential games under probability knowledge of initial condition. Int. Game Theor. Rev. 10, 1–16 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cavagnari, G.: Regularity results for a time-optimal control problem in the space of probability measures. Math. Contr. Relat. Field 7(2), 213–233 (2017). doi: 10.3934/mcrf.2017007 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cavagnari, G., Marigonda, A.: Time-optimal control problem in the space of probability measures. Lect. Notes Comput. Sci 9374, 109–116 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998)zbMATHGoogle Scholar
  19. 19.
    Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Partial Dif. 34, 193–231 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gangbo, W., Nguyen, T., Tudorascu, A.: Hamilton-jacobi equations in the Wasserstein space. Methods Appl. Anal. 15, 155–183 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gangbo, W., Święch, A.: Optimal transport and large number of particles. Discrete Cont. Dyn. S 34, 1397–1441 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lions, P.L., Lasry, J-M.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Acad. Sci. Paris 343, 619–625 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lions, P.L., Lasry, J.-M.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Acad. Sci. Paris 343, 679–684 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rampazzo, F., Sussmann, H.: Commutators of flow maps of nonsmooth vector fields. J. Differ. Equ. 232, 134–175 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sussmann, H.: Subanalytic sets and feedback control. J. Differ. Equ. 31, 31–52 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Villani, C.: Topics in optimal transportation, American Mathematical Society, Providence, RI 58 (2003)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoTrentoItaly
  2. 2.Department of Computer ScienceUniversity of VeronaVeronaItaly
  3. 3.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  4. 4.Istituto per le Applicazioni del Calcolo “M.Picone”Consiglio Nazionale delle RicercheRomaItaly

Personalised recommendations