-
Previous Article
Nonzero-sum differential game of backward doubly stochastic systems with delay and applications
- MCRF Home
- This Issue
-
Next Article
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.
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. |
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. |
| [1] |
Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3 |
| [2] |
Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2415-2433. doi: 10.3934/jimo.2021074 |
| [3] |
Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117 |
| [4] |
Yu Li, Kok Lay Teo, Shuhua Zhang. A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022105 |
| [5] |
Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 |
| [6] |
Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95 |
| [7] |
Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026 |
| [8] |
Ishak Alia. Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021053 |
| [9] |
Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 |
| [10] |
Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9 |
| [11] |
René Carmona, Kenza Hamidouche, Mathieu Laurière, Zongjun Tan. Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization. Journal of Dynamics and Games, 2021, 8 (4) : 403-443. doi: 10.3934/jdg.2021023 |
| [12] |
Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial and Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 |
| [13] |
Liming Zhang, Rongming Wang, Jiaqin Wei. Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021140 |
| [14] |
Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030 |
| [15] |
Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153 |
| [16] |
Michael Schönlein. Computation of open-loop inputs for uniformly ensemble controllable systems. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021046 |
| [17] |
Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455 |
| [18] |
Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037 |
| [19] |
Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355 |
| [20] |
Huaqing Cao, Xiaofen Ji. Optimal recycling price strategy of clothing enterprises based on closed-loop supply chain. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2021232 |
2021 Impact Factor: 1.141
Tools
Metrics
Other articles
by authors
[Back to Top]