Optimization Letters

, Volume 13, Issue 8, pp 1913–1925 | Cite as

Program strategies for a dynamic game in the space of measures

  • Nikolay PogodaevEmail author
Original Paper


The continuity equation describes the transport of a distributed quantity along a vector field. If two independent players affect the vector field we arrive at a game with dynamics given by the continuity equation, or a game in the space of measures. For this game, we discuss a notion of program strategy, provide an existence theorem for the equilibrium, and prove a necessary equilibrium condition.


Continuity equation Liouville equation Dynamic game Program strategies 



The work was supported by the Russian Science Foundation, Grant No. 17-11-01093.


  1. 1.
    Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws, Lect. Notes Unione Mat. Ital., vol. 5, pp. 3–57. Springer, Berlin (2008)zbMATHGoogle Scholar
  2. 2.
    As Soulaimani, S.: Viability with probabilistic knowledge of initial condition, application to optimal control. Set-Valued Anal. 16(7–8), 1037–1060 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007). CrossRefzbMATHGoogle Scholar
  4. 4.
    Bressan, A.: Noncooperative differential games. Milan J. Math. 79(2), 357–427 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, vol. 2. American Institute of Mathematical Sciences (AIMS), Springfield (2007)zbMATHGoogle Scholar
  6. 6.
    Cardaliaguet, P., Jimenez, C., Quincampoix, M.: Pure and random strategies in differential game with incomplete informations. J. Dyn. Games 1(3), 363–375 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cardaliaguet, P., Souquière, A.: A differential game with a blind player. SIAM J. Control Optim. 50(4), 2090–2116 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kipka, R.J., Ledyaev, Y.S.: Extension of chronological calculus for dynamical systems on manifolds. J. Differ. Equ. 258(5), 1765–1790 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krasovskiĭ, N.N., Subbotin, A.I.: Pozitsionnye Differentsial\(^{\prime }\) nye Igry. Izdat. Nauka, Moscow (1974)Google Scholar
  10. 10.
    Marigonda, A., Quincampoix, M.: Mayer control problem with probabilistic uncertainty on initial positions. J. Differ. Equ. (2017). MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pogodaev, N.: Optimal control of continuity equations. NoDEA Nonlinear Differ. Equ. Appl. 23(2), 24 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Scorza Dragoni, G.: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un’altra variabile. Rend. Sem. Mat. Univ. Padova 17, 102–106 (1948)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Villani, C.: Optimal transport, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin (2009). Old and newCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsYekaterinburgRussia
  2. 2.Matrosov Institute for System Dynamics and Control TheoryIrkutskRussia

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