-
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] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[2] |
Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021035 |
[3] |
Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023 |
[4] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021006 |
[5] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[6] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[7] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[8] |
Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91 |
[9] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[10] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[11] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
[12] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[13] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[14] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[15] |
Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021040 |
[16] |
John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 |
[17] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[18] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[19] |
Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 |
[20] |
Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]