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Linear-quadratic-Gaussian mean-field-game with partial observation and common noise
Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability
1. | Department of Mathematics, Southern University of Science and Technology, Shenzhen Guangdong 518055, China |
2. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
This paper is concerned with mean-field stochastic linear-quadratic (MF-SLQ, for short) optimal control problems with deterministic coefficients. The notion of weak closed-loop optimal strategy is introduced. It is shown that the open-loop solvability is equivalent to the existence of a weak closed-loop optimal strategy. Moreover, when open-loop optimal controls exist, there is at least one of them admitting a state feedback representation, which is the outcome of a weak closed-loop optimal strategy. Finally, an example is presented to illustrate the procedure for finding weak closed-loop optimal strategies.
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Mean-field stochastic linear quadratic optimal control problems: Open-loop solvabilities, ESAIM Control Optim. Calc. Var., 23 (2017), 1099-1127.
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J. Sun, X. Li and J. Yong,
Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM J. Control Optim., 54 (2016), 2274-2308.
doi: 10.1137/15M103532X. |
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J. Sun and J. Yong,
Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points, SIAM J. Control Optim., 52 (2014), 4082-4121.
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[11] |
H. Wang, J. Sun and J. Yong,
Weak closed-loop solvability of stochastic linear-quadratic optimal control problems, Discrete Contin. Dyn. Syst., 39 (2019), 2785-2805.
doi: 10.3934/dcds.2019117. |
[12] |
J. Wen, X. Li and J. Xiong, Weak closed-loop solvability of stochastic linear quadratic optimal control problems of Markovian regime switching system, Appl. Math. Optim., (2020). https://doi.org/10.1007/s00245-020-09653-8.
doi: 10.1007/s00245-020-09653-8. |
[13] |
W. M. Wonham,
On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697.
doi: 10.1137/0306044. |
[14] |
J. Yong,
Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.
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[15] |
J. Yong,
Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.
doi: 10.1090/tran/6502. |
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J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
show all references
References:
[1] |
J.-M. Bismut,
Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444.
doi: 10.1137/0314028. |
[2] |
S. Chen, X. Li and X. Y. Zhou,
Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim., 36 (1998), 1685-1702.
doi: 10.1137/S0363012996310478. |
[3] |
S. Chen and J. Yong,
Stochastic linear quadratic optimal control problems, Appl. Math. Optim., 43 (2001), 21-45.
doi: 10.1007/s002450010016. |
[4] |
S. Chen and X. Y. Zhou,
Stochastic linear quadratic regulators with indefinite control weight costs. Ⅱ, SIAM J. Control Optim., 39 (2000), 1065-1081.
doi: 10.1137/S0363012998346578. |
[5] |
J. Huang, X. Li and J. Yong,
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Math. Control Relat. Fields, 5 (2015), 97-139.
doi: 10.3934/mcrf.2015.5.97. |
[6] |
X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: Closed-loop solvability, Probab. Uncertain. Quant. Risk, 1 (2016), 24 pp.
doi: 10.1186/s41546-016-0002-3. |
[7] |
M. A. Rami, J. B. Moore and X. Y. Zhou,
Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., 40 (2001/02), 1296-1311.
doi: 10.1137/S0363012900371083. |
[8] |
J. Sun,
Mean-field stochastic linear quadratic optimal control problems: Open-loop solvabilities, ESAIM Control Optim. Calc. Var., 23 (2017), 1099-1127.
doi: 10.1051/cocv/2016023. |
[9] |
J. Sun, X. Li and J. Yong,
Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM J. Control Optim., 54 (2016), 2274-2308.
doi: 10.1137/15M103532X. |
[10] |
J. Sun and J. Yong,
Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points, SIAM J. Control Optim., 52 (2014), 4082-4121.
doi: 10.1137/140953642. |
[11] |
H. Wang, J. Sun and J. Yong,
Weak closed-loop solvability of stochastic linear-quadratic optimal control problems, Discrete Contin. Dyn. Syst., 39 (2019), 2785-2805.
doi: 10.3934/dcds.2019117. |
[12] |
J. Wen, X. Li and J. Xiong, Weak closed-loop solvability of stochastic linear quadratic optimal control problems of Markovian regime switching system, Appl. Math. Optim., (2020). https://doi.org/10.1007/s00245-020-09653-8.
doi: 10.1007/s00245-020-09653-8. |
[13] |
W. M. Wonham,
On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697.
doi: 10.1137/0306044. |
[14] |
J. Yong,
Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.
doi: 10.1137/120892477. |
[15] |
J. Yong,
Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.
doi: 10.1090/tran/6502. |
[16] |
J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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