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doi: 10.3934/mcrf.2020026

Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability

Jingrui Sun 1, and  Hanxiao Wang 2,,

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.

Keywords: Mean-field stochastic differential equation, linear-quadratic optimal control, open-loop solvability, weak closed-loop solvability, state feedback.
Mathematics Subject Classification: Primary: 93E20; Secondary: 49N10, 49N35.
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. 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.  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. 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.  Google Scholar

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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

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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.  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. 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.  Google Scholar

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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.  doi: 10.1137/140953642.  Google Scholar

[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.  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

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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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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