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, 9 (2007), 131. doi: 10.1007/978-0-8176-4553-3_7.

[2]

B. D. O. Anderson and J. B. Moore, "Linear Optimal Control,", Prentice-Hall, (1971).

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics (New York), (1984).

[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. doi: 10.1016/0167-2789(83)90233-6.

[5]

M. Bardi, On differential games with long-time-average cost,, in, 10 (2009), 3.

[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. doi: 10.1016/j.sysconle.2006.08.011.

[7]

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

[8]

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

[9]

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

[10]

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

[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. doi: 10.1023/A:1008606332037.

[12]

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

[13]

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

[14]

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

[15]

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

[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. doi: 10.1137/S0363012999361962.

[17]

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

[18]

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

[19]

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

[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. doi: 10.1016/j.jmaa.2007.08.008.

[21]

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

[22]

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

[23]

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

[24]

L. W. McKenzie, "Classical General Equilibrium Theory,", MIT press, (2002).

[25]

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

[26]

B. S. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations,, in, 301 (2004), 121. doi: 10.1007/978-3-540-39983-4_8.

[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.

[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. doi: 10.1007/s001860400392.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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.

[37]

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

[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.

[39]

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

show all references

References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games,, in, 9 (2007), 131. doi: 10.1007/978-0-8176-4553-3_7.

[2]

B. D. O. Anderson and J. B. Moore, "Linear Optimal Control,", Prentice-Hall, (1971).

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics (New York), (1984).

[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. doi: 10.1016/0167-2789(83)90233-6.

[5]

M. Bardi, On differential games with long-time-average cost,, in, 10 (2009), 3.

[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. doi: 10.1016/j.sysconle.2006.08.011.

[7]

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

[8]

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

[9]

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

[10]

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

[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. doi: 10.1023/A:1008606332037.

[12]

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

[13]

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

[14]

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

[15]

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

[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. doi: 10.1137/S0363012999361962.

[17]

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

[18]

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

[19]

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

[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. doi: 10.1016/j.jmaa.2007.08.008.

[21]

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

[22]

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

[23]

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

[24]

L. W. McKenzie, "Classical General Equilibrium Theory,", MIT press, (2002).

[25]

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

[26]

B. S. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations,, in, 301 (2004), 121. doi: 10.1007/978-3-540-39983-4_8.

[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.

[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. doi: 10.1007/s001860400392.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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.

[37]

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

[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.

[39]

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

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