Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points

SIAM Journal on Control and Optimization

Volumn 52, Issue 6, 2014, Pages 4082-4121

  Sun, Jingrui ^a   Yong, Jiongmin ^b  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

Closed loop; Linear quadratic differential game; Open loop; Riccati differential equation; Saddle point; Stochastic differential equation; Two person; Zero sum

Indexed keywords

DIFFERENTIAL EQUATIONS; EQUATIONS OF STATE; GAME THEORY; OPTIMAL CONTROL SYSTEMS; QUADRATIC PROGRAMMING;

CLOSED LOOPS; LINEAR QUADRATIC DIFFERENTIAL GAMES; OPEN-LOOP; RICCATI DIFFERENTIAL EQUATION; SADDLE POINT; STOCHASTIC DIFFERENTIAL EQUATIONS; TWO-PERSON; ZERO SUMS;

STOCHASTIC SYSTEMS;

EID: 84920200956     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/140953642     Document Type: Article

References (26)

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), pp. 1296-1311.
2. ndd ed., Birkhäauser Boston, Boston, 1995.
3. R. BELLMAN, I. GLICKSBERG, AND O. GROSS, Some Aspects of the Mathematical Theory of Control Processes, RAND Corporation, Santa Monica, CA, 1958.
4. L. D. BERKOVITZ, Lectures on Differential Games, Differential Games and Related Topics, H. W. Kuhn and G. P. Szego, eds., North-Holland, Amsterdam, The Netherlands, 1971, pp. 3-45.
5. P. BERNHARD, Linear-quadratic, two-person, zero-sum differential games: Necessary and sufficient conditions, J. Optim. Theory Appl., 27 (1979), pp. 51-69.
6. S. CHEN AND J. YONG, Stochastic linear quadratic optimal control problems with random coefficients, Chin. Ann. Math., 21 B (2000), pp. 323-338.
7. S. CHEN AND J. YONG, Stochastic linear quadratic optimal control problems, Appl. Math. Optim., 43 (2001), pp. 21-45.
8. M. C. DELFOUR, Linear quadratic differential games: Saddle point and Riccati differential equations, SIAM J. Control Optim., 46 (2007), pp. 750-774.
9. M. C. DELFOUR AND O. D. SBARBA, Linear quadratic differential games: Closed loop saddle points, SIAM J. Control Optim., 47 (2009), pp. 3138-3166.
10. Y. C. HO, A. E. BRYSON, AND S. BARON, Differential games and optimal pursuit-evasion strategies, IEEE Trans. Automat. Control, 10 (1965), pp. 385-389.
11. Y. HU AND X. Y. ZHOU, Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), pp. 123-137.
12. R. E. KALMAN, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 5 (1960), pp. 102-119.
13. I. KARATZAS AND S. E. SHREVE, Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, New York, 1991.
14. A. M. LETOV, The analytical design of control systems, Automat. Remote Control, 22 (1961), pp. 363-372.
15. J. MA, P. PROTTER, AND J. YONG, Solving forward-backward stochastic differential equations explicitly - A four-step scheme, Probab. Theory Related Fields, 98 (1994), pp. 339-359.
16. J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math. 1702, Springer-Verlag, New York, 1999.
17. M. MCASEY AND L. MOU, Generalized Riccati equations arising in stochastic games, Linear Algebra Appl., 416 (2006), pp. 710-723.
18. L. MOU AND J. YONG, Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method, J. Industrial Management Optim., 2 (2006), pp. 95-117.
19. R. PENROSE, A generalized inverse of matrices, Proc. Cambridge Philos. Soc., 52 (1955), pp. 17-19.
20. Z. QIAN AND X. Y. ZHOU, Existence of solutions to a class of indefinite stochastic Riccati equations, SIAM J. Control Optim., 51 (2013), pp. 221-229.
21. W. E. SCHMITENDORF, Existence of optimal open-loop strategies for a class of differential games, J. Optim. Theory Appl., 5 (1970), pp. 363-375.
22. 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), pp. 53-75.
23. J. YONG, Linear forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), pp. 93-119.
24. J. YONG, Linear forward-backward stochastic differential equations with random coefficients, Probab. Theory Related Fields, 135 (2006), pp. 53-83.
25. J. YONG AND X. Y. ZHOU, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
26. P. ZHANG, Some results on two-person zero-sum linear quadratic differential games, SIAM J. Control Optim., 43 (2005), pp. 2157-2165.