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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 Rami, J. 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. [2] R. Bellman, I. Glicksberg and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, RAND Corporation, Santa Monica, CA, 1958. [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. [4] J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444. doi: 10.1137/0314028. [5] 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. [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. [7] S. Chen and J. Yong, Stochastic linear quadratic optimal control problems, Appl. Math. Optim., 43 (2001), 21-45. doi: 10.1007/s002450010016. [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. [9] M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall, London, 1977. [10] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc., Mat. Mexicana, 5 (1960), 102-119. [11] A. M. Letov, The analytical design of control systems, Automat. Remote Control, 22 (1961), 363-372. [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. [13] 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. [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. [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. [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. [17] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697. doi: 10.1137/0306044. [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.

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##### References:
 [1] M. Ait Rami, J. 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. [2] R. Bellman, I. Glicksberg and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, RAND Corporation, Santa Monica, CA, 1958. [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. [4] J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444. doi: 10.1137/0314028. [5] 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. [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. [7] S. Chen and J. Yong, Stochastic linear quadratic optimal control problems, Appl. Math. Optim., 43 (2001), 21-45. doi: 10.1007/s002450010016. [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. [9] M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall, London, 1977. [10] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc., Mat. Mexicana, 5 (1960), 102-119. [11] A. M. Letov, The analytical design of control systems, Automat. Remote Control, 22 (1961), 363-372. [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. [13] 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. [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. [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. [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. [17] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control, 6 (1968), 681-697. doi: 10.1137/0306044. [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.
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