Advanced Search
Article Contents
Article Contents

Structure of approximate solutions of dynamic continuous time zero-sum games

Abstract / Introduction Related Papers Cited by
  • In this paper we study a turnpike property of approximate solutions for a class of dynamic continuous-time two-player zero-sum games. These properties describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.
    Mathematics Subject Classification: Primary: 49J99, 91A05; Secondary: 91A25.


    \begin{equation} \\ \end{equation}
  • [1]

    O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131-152.doi: 10.1007/978-0-8176-4553-3_7.


    B. D. O. Anderson and J. B. Moore, "Linear Optimal Control," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971.


    J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.


    S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Physica D, 8 (1983), 381-422.doi: 10.1016/0167-2789(83)90233-6.


    M. Bardi, On differential games with long-time-average cost, in "Advances in Dynamic Games and their Applications," Ann. Internat. Soc. Dynam. Games, 10, Birkhäuser Boston, Inc., Boston, MA, (2009), 3-18.


    J. Baumeister, A. Leitäo and G. N. Silva, On the value function for nonautonomous optimal control problem with infinite horizon, Systems Control Lett., 56 (2007), 188-196.doi: 10.1016/j.sysconle.2006.08.011.


    J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl., 106 (2000), 411-419.doi: 10.1023/A:1004611816252.


    J. Blot and N. Hayek, Sufficient conditions for infinite-horizon calculus of variations problems, ESAIM Control Optim. Calc. Var., 5 (2000), 279-292.doi: 10.1051/cocv:2000111.


    P. Cartigny and P. Michel, On a sufficient transversality condition for infinite horizon optimal control problems, Automatica J. IFAC, 39 (2003), 1007-1010.doi: 10.1016/S0005-1098(03)00060-8.


    L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.


    I. V. Evstigneev and S. D. Flåm, Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators, Set-Valued Anal., 6 (1998), 61-81.doi: 10.1023/A:1008606332037.


    D. Gale, On optimal development in a multisector economy, Rev. of Econ. Studies, 34 (1967), 1-19.


    M. K. Ghosh and K. S. Mallikarjuna Rao, Differential games with ergodic payoff, SIAM J. Control Optim., 43 (2005), 2020-2035.doi: 10.1137/S0363012903404511.


    X. Guo and O. Hernández-Lerma, Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates, Bernoulli, 11 (2005), 1009-1029.doi: 10.3150/bj/1137421638.


    H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369.doi: 10.1007/s00245-007-9025-6.


    O. Hernández-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria, SIAM J. Control Optim., 39 (2000), 1520-1539.doi: 10.1137/S0363012999361962.


    V. Kolokoltsov and W. Yang, The turnpike theorems for Markov games, Dynamic Games and Applications, 2 (2012), 294-312.doi: 10.1007/s13235-012-0047-6.


    A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt., 13 (1985), 19-43.doi: 10.1007/BF01442197.


    A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106 (1989), 161-194.doi: 10.1007/BF00251430.


    V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl., 340 (2008), 498-510.doi: 10.1016/j.jmaa.2007.08.008.


    V. L. Makarov and A. M. Rubinov, "Mathematical Theory of Economic Dynamics and Equilibria," Springer-Verlag, New York-Heidelberg, 1977.


    M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 16 (1999), 593-629.doi: 10.1016/S0294-1449(99)80029-8.


    L. W. McKenzie, Turnpike theory, Econometrica, 44 (1976), 841-865.doi: 10.2307/1911532.


    L. W. McKenzie, "Classical General Equilibrium Theory," MIT press, Cambridge, MA, 2002.


    B. Sh. Mordukhovich, Minimax sythesis of a class of control systems with distributed parameters, Automat. Remote Control, 50 (1989), 1333-1340.


    B. S. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in "Optimal Control, Stabilization and Nonsmooth Analysis," Lecture Notes Control Inform. Sci., 301, Springer, Berlin, (2004), 121-132.doi: 10.1007/978-3-540-39983-4_8.


    S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet, 37 (2008), 451-468.


    T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Methods Oper. Res., 61 (2005), 437-454.doi: 10.1007/s001860400392.


    P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496.


    C. C. von Weizsacker, Existence of optimal programs of accumulation for an infinite horizon, Rev. Econ. Studies, 32 (1965), 85-104.


    A. J. Zaslavski, Ground states in a model of Frenkel-Kontorova type, Math. USSR-Izvestiya, 29 (1987), 323-354.doi: 10.1070/IM1987v029n02ABEH000972.


    A. J. Zaslavski, Optimal programs on infinite horizon. I, II, SIAM Journal on Control and Optimization, 3 (1995), 1643-1660, 1661-1686.doi: 10.1137/S036301299325726X.


    A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis, 27 (1996), 895-931.doi: 10.1016/0362-546X(95)00029-U.


    A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games, Abstract and Applied Analysis, 4 (1999), 21-48.doi: 10.1155/S1085337599000020.


    A. J. Zaslavski, "Turnpike Properties in the Calculus of Variations and Optimal Control," Nonconvex Optimization and its Applications, 80, Springer, New York, 2006 .


    A. J. Zaslavski, A turnpike result for a class of problems of the calculus of variations with extended-valued integrands, J. Convex Analysis, 15 (2008), 869-890.


    A. J. Zaslavski, "Optimization on Metric and Normed Spaces," Springer Optimization and Its Applications, 44, Springer, New York, 2010.doi: 10.1007/978-0-387-88621-3.


    A. J. Zaslavski, The existence and structure of approximate solutions of dynamic discrete time zero-sum games, Journal of Nonlinear and Convex Analysis, 12 (2011), 49-68.


    A. J. Zaslavski and A. Leizarowitz, Optimal solutions of linear control systems with nonperiodic integrands, Mathematics of Operations Research, 22 (1997), 726-746.doi: 10.1287/moor.22.3.726.

  • 加载中

Article Metrics

HTML views() PDF downloads(63) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint