-
Previous Article
Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity
- DCDS Home
- This Issue
-
Next Article
Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $
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:
| [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. |
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. |
| [1] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control and Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
| [2] |
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 |
| [3] |
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, 2021 doi: 10.3934/jimo.2021074 |
| [4] |
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 |
| [5] |
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 |
| [6] |
Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030 |
| [7] |
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 |
| [8] |
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 |
| [9] |
Michael Schönlein. Computation of open-loop inputs for uniformly ensemble controllable systems. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021046 |
| [10] |
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 |
| [11] |
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 |
| [12] |
Shuaishuai Fu, Weida Chen, Junfei Ding, Dandan Wang. Optimal financing strategy in a closed-loop supply chain for construction machinery remanufacturing with emissions abatement. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022002 |
| [13] |
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 |
| [14] |
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 |
| [15] |
Qi Lü, Tianxiao Wang, Xu Zhang. Characterization of optimal feedback for stochastic linear quadratic control problems. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 11-. doi: 10.1186/s41546-017-0022-7 |
| [16] |
Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267 |
| [17] |
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 |
| [18] |
Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1583-1602. doi: 10.3934/jimo.2021034 |
| [19] |
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 |
| [20] |
Robert S. Anderssen, Martin Kružík. Modelling of wheat-flour dough mixing as an open-loop hysteretic process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 283-293. doi: 10.3934/dcdsb.2013.18.283 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]