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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 |
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.
References:
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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. |
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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. |
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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. |
[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. |
show all references
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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