January  2014, 1(1): 153-179. doi: 10.3934/jdg.2014.1.153

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

1. 

Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel

Received  April 2012 Revised  June 2012 Published  June 2013

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.
Citation: Alexander J. Zaslavski. Structure of approximate solutions of dynamic continuous time zero-sum games. Journal of Dynamics & Games, 2014, 1 (1) : 153-179. doi: 10.3934/jdg.2014.1.153
References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

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

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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.  Google Scholar

[33]

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.  Google Scholar

[34]

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

[35]

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

[36]

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.  Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

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

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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.  Google Scholar

[33]

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.  Google Scholar

[34]

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

[35]

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

[36]

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.  Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

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