Journal of Global Optimization

, Volume 23, Issue 3–4, pp 245–265 | Cite as

Discrete-time Indefinite LQ Control with State and Control Dependent Noises

  • M. Ait Rami
  • X. Chen
  • X.Y. Zhou


This paper deals with the discrete-time stochastic LQ problem involving state and control dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. In this general setting, it is shown that the well-posedness and the attainability of the LQ problem are equivalent. Moreover, a generalized difference Riccati equation is introduced and it is proved that its solvability is necessary and sufficient for the existence of an optimal control which can be either of state feedback or open-loop form. Furthermore, the set of all optimal controls is identified in terms of the solution to the proposed difference Riccati equation.

Indefinite stochastic LQ control Discrete time Multiplicative noise Generalized difference Riccati equation Linear matrix inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Ait Rami, X. Chen, J.B. Moore, and X.Y. Zhou. Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls. IEEE Trans. Autom. Contr. AC-46 (2001), 428–440.Google Scholar
  2. 2.
    M. Ait Rami and X. Y. Zhou. Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control. IEEE Trans. Autom. Contr. AC-45 (2000), 1131–1143.Google Scholar
  3. 3.
    A. Albert. Conditions for positive and nonnegative definiteness in terms of pseudo-inverse. SIAM J. Appl. Math. 17 (1969), 434–440.Google Scholar
  4. 4.
    M. Athans. The matrix minimum principle. Inform. and Contr. 11 (1968), 592–606.Google Scholar
  5. 5.
    M. Athans. Special issues on linear-quadratic-Gaussian problem, IEEE Trans. Autom. Contr. AC-16 (1971), 527–869.Google Scholar
  6. 6.
    A. Beghi and D. D'Alessandro. Discrete-time optimal control with control-dependent noise and generalized Riccati difference equations. Automatica 34(8) (1998), 1031–1034.Google Scholar
  7. 7.
    S. Chen, X. Li and X.Y. Zhou. Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Contr. Optim. 36 (1998), 1685–1702.Google Scholar
  8. 8.
    R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1990.Google Scholar
  9. 9.
    R. E. Kalman. Contribution to the theory of optimal control, Bol. Soc. Mat. Mex. 5 (1960), 102–119.Google Scholar
  10. 10.
    M. Kohlmann and X.Y. Zhou. Relations hip between backward stochastic differential equations and stochastic controls: a linear-quadratic approach. SIAM J. Contr. Optim. 38 (2000), 1392–1407.Google Scholar
  11. 11.
    R.T. Ku and M. Athans. Further results on the uncertainty threshold principle. IEEE Trans. Autom. Contr. AC-22 (1977), 866–868.Google Scholar
  12. 12.
    J.B. Moore, X.Y. Zhou and A.E.B. Lim. Discrete time LQG controls with control dependent noise. Syst. Contr. Lett. 36 (1999), 199–206.Google Scholar
  13. 13.
    R. Penrose. A generalized inverse of matrices. Proc. Cambridge Philos. Soc. 52 (1955), 17–19.Google Scholar
  14. 14.
    L. Vandenerghe and S. Boyd. Semidefinite programming. SIAM Rev. 38 (1996), 49–95.Google Scholar
  15. 15.
    J. Yong and X.Y. Zhou. Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, Berlin, 1999.Google Scholar
  16. 16.
    X.Y. Zhou and D. Li. Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl. Math. Optim. 42 (2000), 19–33.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • M. Ait Rami
    • 1
  • X. Chen
    • 1
  • X.Y. Zhou
    • 1
  1. 1.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong KongShatinHong Kong

Personalised recommendations