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On the system of partial differential equations arising in mean field type control
Ergodicity conditions for zero-sum games
1. | INRIA and CMAP, Ecole polytechnique, CNRS, CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, France, France |
References:
[1] |
M. Akian and S. Gaubert, Spectral theorem for convex monotone homogeneous maps and ergodic control, Nonlinear Analysis, T.M.A., 52 (2003), 637-679.
doi: 10.1016/S0362-546X(02)00170-0. |
[2] |
M. Akian, S. Gaubert and R. Nussbaum, Uniqueness of the fixed point of nonexpansive semidifferentiable maps,, Trans. Amer. Math. Soc., ().
doi: 10.1090/S0002-9947-2015-06413-7. |
[3] |
M. Akian, S. Gaubert and C. Walsh, The max-plus Martin boundary, Documenta Mathematica, 14 (2009), 195-240, URL http://www.math.uni-bielefeld.de/documenta/vol-14/09.html. |
[4] |
M. Akian, S. Gaubert and A. Hochart, Fixed point sets of payment-free shapley operators and structural properties of mean payoff games, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, Groningen, The Netherlands, (2014), 1438-1441. |
[5] |
X. Allamigeon, On the complexity of strongly connected components in directed hypergraphs, Algorithmica, 69 (2014), 335-369.
doi: 10.1007/s00453-012-9729-0. |
[6] |
F. Baccelli, G. Cohen, G. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992. |
[7] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original.
doi: 10.1137/1.9781611971262. |
[8] |
G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). |
[9] |
J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operations Research, 40 (2014), 171-191, URL http://arxiv.org/abs/1301.1967, Published online.
doi: 10.1287/moor.2014.0666. |
[10] |
R. Cavazos-Cadena and D. Hernández-Hernández, Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case, Nonlinear Anal., 72 (2010), 3303-3313.
doi: 10.1016/j.na.2009.12.010. |
[11] |
H. Everett, Recursive games, in Contributions to the theory of games Vol. III, vol. 39 of Ann. Math. Studies, Princeton University Press, (1957), 47-78. |
[12] |
A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[13] |
A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. |
[14] |
G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications, Discrete Appl. Math., 42 (1993), 177-201.
doi: 10.1016/0166-218X(93)90045-P. |
[15] |
S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950.
doi: 10.1090/S0002-9947-04-03470-1. |
[16] |
O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, vol. 42 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999, URL http://dx.doi.org/10.1007/978-1-4612-0561-6.
doi: 10.1007/978-1-4612-0561-6. |
[17] |
J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York-Heidelberg, 1976, Reprinting of the 1960 original, Undergraduate Texts in Mathematics. |
[18] |
E. Kohlberg and A. Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math., 38 (1981), 269-275.
doi: 10.1007/BF02762772. |
[19] |
B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012.
doi: 10.1017/CBO9781139026079. |
[20] |
J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66.
doi: 10.1007/BF01769259. |
[21] |
A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Kluwer Academic Publishers, 2003.
doi: 10.1007/978-94-010-0189-2. |
[22] |
R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988), iv+137pp.
doi: 10.1090/memo/0391. |
[23] |
R. D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric. II, Mem. Amer. Math. Soc., 79 (1989), iv+118pp.
doi: 10.1090/memo/0401. |
[24] |
O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513.
doi: 10.1090/S0002-9947-1944-0010555-7. |
[25] |
J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
[26] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel J. Math., 121 (2001), 221-246.
doi: 10.1007/BF02802505. |
[28] |
S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dynamic Systems, 14 (2004), 109-122.
doi: 10.1023/B:DISC.0000005011.93152.d8. |
[29] |
G. Vigeral, A zero-zum stochastic game with compact action sets and no asymptotic value, Dyn. Games Appl., 3 (2013), 172-186.
doi: 10.1007/s13235-013-0073-z. |
[30] |
P. Whittle, Optimization Over Time. Vol. II, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1983, Dynamic programming and stochastic control. |
[31] |
K. Yang and Q. Zhao, The balance problem of min-max systems is co-NP hard, Systems & Control Letters, 53 (2004), 303-310.
doi: 10.1016/j.sysconle.2004.05.009. |
show all references
References:
[1] |
M. Akian and S. Gaubert, Spectral theorem for convex monotone homogeneous maps and ergodic control, Nonlinear Analysis, T.M.A., 52 (2003), 637-679.
doi: 10.1016/S0362-546X(02)00170-0. |
[2] |
M. Akian, S. Gaubert and R. Nussbaum, Uniqueness of the fixed point of nonexpansive semidifferentiable maps,, Trans. Amer. Math. Soc., ().
doi: 10.1090/S0002-9947-2015-06413-7. |
[3] |
M. Akian, S. Gaubert and C. Walsh, The max-plus Martin boundary, Documenta Mathematica, 14 (2009), 195-240, URL http://www.math.uni-bielefeld.de/documenta/vol-14/09.html. |
[4] |
M. Akian, S. Gaubert and A. Hochart, Fixed point sets of payment-free shapley operators and structural properties of mean payoff games, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, Groningen, The Netherlands, (2014), 1438-1441. |
[5] |
X. Allamigeon, On the complexity of strongly connected components in directed hypergraphs, Algorithmica, 69 (2014), 335-369.
doi: 10.1007/s00453-012-9729-0. |
[6] |
F. Baccelli, G. Cohen, G. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992. |
[7] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original.
doi: 10.1137/1.9781611971262. |
[8] |
G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). |
[9] |
J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operations Research, 40 (2014), 171-191, URL http://arxiv.org/abs/1301.1967, Published online.
doi: 10.1287/moor.2014.0666. |
[10] |
R. Cavazos-Cadena and D. Hernández-Hernández, Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case, Nonlinear Anal., 72 (2010), 3303-3313.
doi: 10.1016/j.na.2009.12.010. |
[11] |
H. Everett, Recursive games, in Contributions to the theory of games Vol. III, vol. 39 of Ann. Math. Studies, Princeton University Press, (1957), 47-78. |
[12] |
A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[13] |
A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. |
[14] |
G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications, Discrete Appl. Math., 42 (1993), 177-201.
doi: 10.1016/0166-218X(93)90045-P. |
[15] |
S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950.
doi: 10.1090/S0002-9947-04-03470-1. |
[16] |
O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, vol. 42 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999, URL http://dx.doi.org/10.1007/978-1-4612-0561-6.
doi: 10.1007/978-1-4612-0561-6. |
[17] |
J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York-Heidelberg, 1976, Reprinting of the 1960 original, Undergraduate Texts in Mathematics. |
[18] |
E. Kohlberg and A. Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math., 38 (1981), 269-275.
doi: 10.1007/BF02762772. |
[19] |
B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012.
doi: 10.1017/CBO9781139026079. |
[20] |
J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66.
doi: 10.1007/BF01769259. |
[21] |
A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Kluwer Academic Publishers, 2003.
doi: 10.1007/978-94-010-0189-2. |
[22] |
R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988), iv+137pp.
doi: 10.1090/memo/0391. |
[23] |
R. D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric. II, Mem. Amer. Math. Soc., 79 (1989), iv+118pp.
doi: 10.1090/memo/0401. |
[24] |
O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513.
doi: 10.1090/S0002-9947-1944-0010555-7. |
[25] |
J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
[26] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel J. Math., 121 (2001), 221-246.
doi: 10.1007/BF02802505. |
[28] |
S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dynamic Systems, 14 (2004), 109-122.
doi: 10.1023/B:DISC.0000005011.93152.d8. |
[29] |
G. Vigeral, A zero-zum stochastic game with compact action sets and no asymptotic value, Dyn. Games Appl., 3 (2013), 172-186.
doi: 10.1007/s13235-013-0073-z. |
[30] |
P. Whittle, Optimization Over Time. Vol. II, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1983, Dynamic programming and stochastic control. |
[31] |
K. Yang and Q. Zhao, The balance problem of min-max systems is co-NP hard, Systems & Control Letters, 53 (2004), 303-310.
doi: 10.1016/j.sysconle.2004.05.009. |
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