doi: 10.3934/mcrf.2020026

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

* Corresponding author: Hanxiao Wang

Received  June 2019 Revised  January 2020 Published  June 2020

Fund Project: The first author is supported by NSFC Grant 11901280

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.

Citation: Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020026
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.  Google Scholar

[2]

S. ChenX. 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.  Google Scholar

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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.  Google Scholar

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J. HuangX. 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.  Google Scholar

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J. SunX. 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.  Google Scholar

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[11]

H. WangJ. 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.  Google Scholar

[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.  Google Scholar

[13]

W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697.  doi: 10.1137/0306044.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[2]

S. ChenX. 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.  Google Scholar

[3]

S. Chen and J. Yong, Stochastic linear quadratic optimal control problems, Appl. Math. Optim., 43 (2001), 21-45.  doi: 10.1007/s002450010016.  Google Scholar

[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.  Google Scholar

[5]

J. HuangX. 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.  Google Scholar

[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.  Google Scholar

[7]

M. A. RamiJ. 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.  Google Scholar

[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.  Google Scholar

[9]

J. SunX. 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.  Google Scholar

[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.  Google Scholar

[11]

H. WangJ. 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.  Google Scholar

[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.  Google Scholar

[13]

W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697.  doi: 10.1137/0306044.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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