Abstract
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.
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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)
As Soulaimani, S.: Viability with probabilistic knowledge of initial condition, application to optimal control. Set-Valued Anal. 16(7–8), 1037–1060 (2008). https://doi.org/10.1007/s11228-008-0097-5
Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-34514-5
Bressan, A.: Noncooperative differential games. Milan J. Math. 79(2), 357–427 (2011). https://doi.org/10.1007/s00032-011-0163-6
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)
Cardaliaguet, P., Jimenez, C., Quincampoix, M.: Pure and random strategies in differential game with incomplete informations. J. Dyn. Games 1(3), 363–375 (2014). https://doi.org/10.3934/jdg.2014.1.363
Cardaliaguet, P., Souquière, A.: A differential game with a blind player. SIAM J. Control Optim. 50(4), 2090–2116 (2012). https://doi.org/10.1137/100808903
Kipka, R.J., Ledyaev, Y.S.: Extension of chronological calculus for dynamical systems on manifolds. J. Differ. Equ. 258(5), 1765–1790 (2015). https://doi.org/10.1016/j.jde.2014.11.014
Krasovskiĭ, N.N., Subbotin, A.I.: Pozitsionnye Differentsial\(^{\prime }\) nye Igry. Izdat. Nauka, Moscow (1974)
Marigonda, A., Quincampoix, M.: Mayer control problem with probabilistic uncertainty on initial positions. J. Differ. Equ. (2017). https://doi.org/10.1016/j.jde.2017.11.014. https://www.sciencedirect.com/science/article/pii/S0022039617306046
Pogodaev, N.: Optimal control of continuity equations. NoDEA Nonlinear Differ. Equ. Appl. 23(2), 24 (2016). https://doi.org/10.1007/s00030-016-0357-2
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)
Villani, C.: Optimal transport, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin (2009). https://doi.org/10.1007/978-3-540-71050-9. Old and new
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The work was supported by the Russian Science Foundation, Grant No. 17-11-01093.
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Pogodaev, N. Program strategies for a dynamic game in the space of measures. Optim Lett 13, 1913–1925 (2019). https://doi.org/10.1007/s11590-018-1318-y
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DOI: https://doi.org/10.1007/s11590-018-1318-y