Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A leader-follower stochastic linear quadratic differential game with time delay

  • 180 Accesses

  • 7 Citations


In this paper, we are concerned with the leader-follower stochastic differential game of Itô type with time delay appearing in the leader’s control. The open-loop solution is explicitly given in the form of the conditional expectation with respect to several symmetric Riccati equations. The key technique is to establish the nonhomogeneous relationship between the forward variables and the backward ones obtained in the optimization problems of both the follower and the leader.

This is a preview of subscription content, log in to check access.


  1. 1

    Basar T, Olsder G J. Dynamic Noncooperative Game Theory. New York: Academic Press, 1995

  2. 2

    Isaacs R. Differential Games: A Mathematical Theory With Applicaitons to Warfare and Pursuit, Control and Optimization. Hoboken: John Wiley and Sons, 1999

  3. 3

    Starr A W, Ho Y C. Nonzero-sum differential games. J Optim Theory Appl, 1969, 3: 184–206

  4. 4

    Shi J T, Wang G C, Xiong J. Leader-follower stochastic differential game with asymmetric information and applications. Automatica, 2016, 63: 60–73

  5. 5

    Mu Y F, Guo L. How cooperation arises from rational players? Sci China Inf Sci, 2013, 56: 112201

  6. 6

    Mukaidani H. Dynamic games for stochastic systems with delay. Asian J Control, 2013, 3: 1251–1260

  7. 7

    Zhang H S, Xu J J. Control for Itô stochastic systems with input delay. IEEE Trans Autom Control, 2017, 62: 350–365

  8. 8

    Wang T X, Shi Y F. Linear quadratic stochastic integral games and related topics. Sci China Math, 2015, 58: 2405–2420

  9. 9

    Freiling G, Jank G, Lee S R. Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games. J Optim Theory Appl, 2001, 110: 515–544

  10. 10

    Papavassilopoulos G P, Cruz J B. Nonclassical control problems and Stackelberg games. IEEE Trans Autom Control, 1979, 24: 155–166

  11. 11

    Simaan M, Cruz J B. On the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl, 1973, 11: 533–555

  12. 12

    Basar T. Stochastic stagewise Stackelberg strategies for linear quadratic systems. In: Stochastic Control Theory and Stochastic Differential Systems. Berlin: Springer, 1979

  13. 13

    Bensoussan A, Chen S, Sethi S P. The maximum principle for global solutions of stochastic Stackelberg differential games. SIAM J Control Optim, 2015, 53: 1956–1981

  14. 14

    Hamadéne S. Nonzero sum linear-quadratic stochastic differential games and backward-forward equations. Stoch Anal Appl, 1999, 17: 117–130

  15. 15

    Hu Y. N-person differential games governed by semilinear stochastic evolution systems. Appl Math Optim, 1991, 24: 257–271

  16. 16

    Wu Z. Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games. J Syst Sci Complex, 2005, 18: 179–192

  17. 17

    Yong J M. A leader-follower stochastic linear quadratic differential game. SIAM J Control Optim, 2002, 41: 1015–1041

  18. 18

    Shi J T, Wang G C, Xiong J. Linear-quadratic stochastic Stackelberg differential game with asymmetric information. Sci China Inf Sci, 2017, 60: 092202

  19. 19

    Dong X W, Li Q D, Ren Z, et al. Formation-containment control for high-order linear time-invariant multi-agent systems with time delays. J Franklin Inst, 2015, 352: 3564–3584

  20. 20

    Dong X W, Han L, Li Q D, et al. Containment analysis and design for general linear multi-agent systems with time-varying delays. Neurocomputing, 2016, 173: 2062–2068

  21. 21

    Mukaidani H, Unno M, Yamamoto T, et al. Nash strategy for Markov jump stochastic delay systems. In: Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, 2013. 1198–1203

  22. 22

    Chen L, Wu Z. Maximum principle for the stochastic optimal control problem with delay and application. Automatica, 2010, 46: 1074–1080

  23. 23

    Huang J H, Li X, Shi J T. Forward-backward linear quadratic stochastic optimal control problem with delay. Syst Control Lett, 2012, 61: 623–630

  24. 24

    Øksendal B, Sulem A. A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In: Optimal Control and Partial Differential Equations — Innovations and Applications. Amsterdam: IOS Press, 2000

  25. 25

    Wang H X, Zhang H S. LQ control for Itô-type stochastic systems with input delays. Automatica, 2013, 49: 3538–3549

  26. 26

    Yu Z Y. The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls. Automatica, 2012, 48: 2420–2432

  27. 27

    Xu J J, Zhang H S. Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay. IEEE Trans Cybern, 2016, 46: 438–449

  28. 28

    Yong J M, Zhou X Y. Stochatic Controls: Hamiltonian Systems and HJB Equations. Berlin: Springer, 1999

  29. 29

    Rami M A, Chen X, Moore J B, et al. Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls. IEEE Trans Autom Control, 2001, 46: 428–440

  30. 30

    Tadmor G, Mirkin L. H1 control and estimation with preview-part I: matrix ARE solutions in continuous time. IEEE Trans Autom Control, 2005, 50: 19–28

Download references


This work was supported by Taishan Scholar Construction Engineering by Shandong Government, National Natural Science Foundation of China (Grant Nos. 61403235, 61104050, 11201264, 61573221, 61633014), Natural Science Foundation of Shandong Province (Grant Nos. ZR2011AQ012, ZR2014FQ011).

Author information

Correspondence to Huanshui Zhang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xu, J., Shi, J. & Zhang, H. A leader-follower stochastic linear quadratic differential game with time delay. Sci. China Inf. Sci. 61, 112202 (2018). https://doi.org/10.1007/s11432-017-9293-4

Download citation


  • time delay
  • leader-follower differential game
  • open-loop strategy
  • Riccati equation