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Quartic Julia sets including any two copies of quadratic Julia sets
Limit value for optimal control with general means
1. | Sorbonne Universités, UPMC Univ Paris 06, IMJ-PRG, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, 4 Place Jussieu, 75005 Paris, France |
2. | Laboratoire de Mathématiques de Bretagne Atlantique, UMR 6205, Université de Brest, 6 Avenue Victor Le Gorgeu, 29200 Brest, France |
3. | TSE (GREMAQ, Université Toulouse 1 Capitole and GDR 2932 Théorie des Jeux), 21 allée de Brienne, 31000 Toulouse |
  For this aim, we introduce an asymptotic regularity condition for a sequence of probability measures on $\mathbb{R}_+$. Our main result is that, for any sequence of probability measures on $\mathbb{R}_+$ satisfying this condition, the associated value functions converge uniformly if and only if this family is totally bounded for the uniform norm.
  As a byproduct, we obtain the existence of a limit value (for general evaluations) for control systems defined on a compact invariant domain and satisfying suitable nonexpansive property.
References:
[1] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular pertubations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp.
doi: 10.1090/S0065-9266-09-00588-2. |
[2] |
M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Belmann equations, Ann. Henri Poincaré, Analyse Nonlinéaire, 14 (1997), 415-438.
doi: 10.1016/S0294-1449(97)80134-5. |
[3] |
M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[4] |
M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493-2498.
doi: 10.1109/CDC.2013.6760255. |
[5] |
A. Bensoussan, Perturbation Methods in Optimal Control, Wiley/Gauthiers-Villas, Chichester, 1988. |
[6] |
R. Buckdahn, D. Goreac and M. Quincampoix, Existence of asymptotic values for nonexpansive stochastic control systems, Applied Mathematics and Optimization, 70 (2014), 1-28.
doi: 10.1007/s00245-013-9230-4. |
[7] |
V. Gaitsgory, On the use of the averaging method in control problems, (Russian) Differentsialnye Uravneniya, 22 (1986), 1876-1886. |
[8] |
D. Goreac, A note on general Tauberian-type results for controlled stochastic dynamics, preprint, hal:01120513. |
[9] |
R. Z. Khasminskii, On the averaging principle for Itô stochastic equations, Kybernetika, 4 (1968), 260-279. |
[10] |
D. Khlopin, On uniform Tauberian theorems for dynamic games, preprint, arXiv:1412.7331. |
[11] |
M. Oliu-Barton and G. Vigeral, A uniform Tauberian theorem in optimal control, in Advances in Dynamic Games, Annals of the International Society of Dynamic Games (eds. P. Cardaliaguet and R. Cressman), Birkhauser, 12 (2012), 199-215. |
[12] |
M. Quincampoix and J. Renault, On the existence of a limit value in some nonexpansive optimal control problems, SIAM Journal on Control and Optimization, 49 (2011), 2118-2132.
doi: 10.1137/090756818. |
[13] |
J. Renault, Uniform value in dynamic programming, J. Eur. Math. Soc. (JEMS), 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
[14] |
J. Renault, General long-term values in dynamic programming, Journal of Dynamics and Games, 1 (2014), 471-484. |
[15] |
J. Renault and X. Venel, A distance for probability spaces, and long-term values in Markov decision processes and repeated games, preprint, arXiv:1202.6259. |
[16] |
S. Sorin, A First Course on Zero-sum Repeated Games, Springer, 2002. |
[17] |
B. Ziliotto, General limit value in stochastic games, preprint, arXiv:1410.5231. |
show all references
References:
[1] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular pertubations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp.
doi: 10.1090/S0065-9266-09-00588-2. |
[2] |
M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Belmann equations, Ann. Henri Poincaré, Analyse Nonlinéaire, 14 (1997), 415-438.
doi: 10.1016/S0294-1449(97)80134-5. |
[3] |
M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[4] |
M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493-2498.
doi: 10.1109/CDC.2013.6760255. |
[5] |
A. Bensoussan, Perturbation Methods in Optimal Control, Wiley/Gauthiers-Villas, Chichester, 1988. |
[6] |
R. Buckdahn, D. Goreac and M. Quincampoix, Existence of asymptotic values for nonexpansive stochastic control systems, Applied Mathematics and Optimization, 70 (2014), 1-28.
doi: 10.1007/s00245-013-9230-4. |
[7] |
V. Gaitsgory, On the use of the averaging method in control problems, (Russian) Differentsialnye Uravneniya, 22 (1986), 1876-1886. |
[8] |
D. Goreac, A note on general Tauberian-type results for controlled stochastic dynamics, preprint, hal:01120513. |
[9] |
R. Z. Khasminskii, On the averaging principle for Itô stochastic equations, Kybernetika, 4 (1968), 260-279. |
[10] |
D. Khlopin, On uniform Tauberian theorems for dynamic games, preprint, arXiv:1412.7331. |
[11] |
M. Oliu-Barton and G. Vigeral, A uniform Tauberian theorem in optimal control, in Advances in Dynamic Games, Annals of the International Society of Dynamic Games (eds. P. Cardaliaguet and R. Cressman), Birkhauser, 12 (2012), 199-215. |
[12] |
M. Quincampoix and J. Renault, On the existence of a limit value in some nonexpansive optimal control problems, SIAM Journal on Control and Optimization, 49 (2011), 2118-2132.
doi: 10.1137/090756818. |
[13] |
J. Renault, Uniform value in dynamic programming, J. Eur. Math. Soc. (JEMS), 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
[14] |
J. Renault, General long-term values in dynamic programming, Journal of Dynamics and Games, 1 (2014), 471-484. |
[15] |
J. Renault and X. Venel, A distance for probability spaces, and long-term values in Markov decision processes and repeated games, preprint, arXiv:1202.6259. |
[16] |
S. Sorin, A First Course on Zero-sum Repeated Games, Springer, 2002. |
[17] |
B. Ziliotto, General limit value in stochastic games, preprint, arXiv:1410.5231. |
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