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May  2019, 39(5): 2785-2805. doi: 10.3934/dcds.2019117

Weak closed-loop solvability of stochastic linear-quadratic optimal control problems

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China

3. 

Department of Mathematics, University of Central Florida, Orlando FL 32816, USA

* Corresponding author: Jingrui Sun

Received  June 2018 Published  January 2019

Fund Project: The first author is supported in part by the China Scholarship Council, while visiting University of Central Florida. The third author is supported in part by NSF DMS-1812921.

Recently it has been found that for a stochastic linear-quadratic optimal control problem (LQ problem, for short) in a finite horizon, open-loop solvability is strictly weaker than closed-loop solvability which is equivalent to the regular solvability of the corresponding Riccati equation. Therefore, when an LQ problem is merely open-loop solvable not closed-loop solvable, which is possible, the usual Riccati equation approach will fail to produce a state feedback representation of open-loop optimal controls. The objective of this paper is to introduce and investigate the notion of weak closed-loop optimal strategy for LQ problems so that its existence is equivalent to the open-loop solvability of the LQ problem. Moreover, there is at least one open-loop optimal control admitting a state feedback representation. Finally, we present an example to illustrate the procedure for finding weak closed-loop optimal strategies.

Citation: Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
References:
[1]

M. Ait RamiJ. B. Moore and X. Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., 40 (2001), 1296-1311.  doi: 10.1137/S0363012900371083.  Google Scholar

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A. Bensoussan, Lectures on stochstic control, part Ⅰ, in Nonlinear Filtering and Stochastic Control, Lecture Notes in Math., Springer-Verlag, Berlin, 972 (1982), 1–62.  Google Scholar

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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

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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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R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc., Mat. Mexicana, 5 (1960), 102-119.   Google Scholar

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A. M. Letov, The analytical design of control systems, Automat. Remote Control, 22 (1961), 363-372.   Google Scholar

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A. E. B. Lim and X. Y. Zhou, Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights, IEEE Trans. Automat. Control, 44 (1999), 1359-1369.  doi: 10.1109/9.774108.  Google Scholar

[13]

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

[14]

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

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S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 53-75.  doi: 10.1137/S0363012901387550.  Google Scholar

[16]

S. Tang, Dynamic programming for general linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 53 (2015), 1082-1106.  doi: 10.1137/140979940.  Google Scholar

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

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J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

show all references

References:
[1]

M. Ait RamiJ. B. Moore and X. Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., 40 (2001), 1296-1311.  doi: 10.1137/S0363012900371083.  Google Scholar

[2]

R. Bellman, I. Glicksberg and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, RAND Corporation, Santa Monica, CA, 1958.  Google Scholar

[3]

A. Bensoussan, Lectures on stochstic control, part Ⅰ, in Nonlinear Filtering and Stochastic Control, Lecture Notes in Math., Springer-Verlag, Berlin, 972 (1982), 1–62.  Google Scholar

[4]

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

[5]

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

[6]

S. Chen and J. Yong, Stochastic linear quadratic optimal control problems with random coefficients, Chin. Ann. Math., 21 B (2000), 323-338.  doi: 10.1142/S0252959900000339.  Google Scholar

[7]

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

[8]

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

[9]

M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall, London, 1977.  Google Scholar

[10]

R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc., Mat. Mexicana, 5 (1960), 102-119.   Google Scholar

[11]

A. M. Letov, The analytical design of control systems, Automat. Remote Control, 22 (1961), 363-372.   Google Scholar

[12]

A. E. B. Lim and X. Y. Zhou, Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights, IEEE Trans. Automat. Control, 44 (1999), 1359-1369.  doi: 10.1109/9.774108.  Google Scholar

[13]

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

[14]

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

[15]

S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 53-75.  doi: 10.1137/S0363012901387550.  Google Scholar

[16]

S. Tang, Dynamic programming for general linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 53 (2015), 1082-1106.  doi: 10.1137/140979940.  Google Scholar

[17]

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

[18]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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