The paper studies control problem of nonlinear dynamical systems discribed by difference equation x k +1 = f k (x k ,u k ), k = 0,1,…, N − 1 with phase constraints x k M k . Basing on new results in multivalued Lipschitzian analysis we obtain local controllability and reachability conditions for linear control systems of this kind. Some corollaries of controllability criteria for linear and nonlinear nondifferentiable discrete-time systems are also given.


Discrete-time systems Controllability nonsmooth analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kalman R., On the general theory of control systems, Proc. IFAC, London, (1960), 481–492.Google Scholar
  2. [2]
    Faradzev R.G., Phat V.N. and Shapiro A., Controllability theory of discretetime dynamical systems (survey), [in Russian], Avtomatika i Telemekhanika 1 (1986), 5–25.Google Scholar
  3. [3]
    Conti R., Linear controllability in finite-dimension,Mathematishe (Catania) 1 (1986), p. 36.Google Scholar
  4. [4]
    Vu Ngoc Phat, Controllability of nonlinear discrete-time delay systems,Acta Math. Vietnamica 2 (1980), 63–72.Google Scholar
  5. [5]
    Vu Ngoc Phat, Controllability of discrete-time systems with nonconvex constrained controls,Optimization 3 (1983), 371–375.Google Scholar
  6. [6]
    Faradzev R.G. and Vu Ngoc Phat, On the controllability of nonlinear two-parametric discrete-time systems with restrained controls,Optimization 6 (1985), 869–876.Google Scholar
  7. [7]
    Yen N.D., Local controllability for Lipschitzian discrete-time systems,Acta Math. Vietnamica 2 (1986), 172–179.Google Scholar
  8. [8]
    Clarke F.H., A new approach to Lagrange multipliers,Math. Oper. Reas. 2 (1976), 247–262.Google Scholar
  9. [9]
    Ekeland I.E., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–362.CrossRefGoogle Scholar
  10. [10]
    Dien F.H., Locally Lipschitzian set-valued maps and generalized extremal problems with inclusion constraints,Acta Math. Vietnamica 2 (1983), 102–122.Google Scholar
  11. [11]
    Sach F.H., Les point reguliers des applications multivoques et la commadabilite dans les systems discretes,Universite de Bordeaux, Analyse Appl. et Inf. 8022 (1980), 1–47.Google Scholar
  12. [12]
    Vu Ngoc Phat, Controllability of discrete-time systems with multiple delays on control and states,Int.I. of Control 5 (1989), 1645–1654.Google Scholar

Copyright information

© Springer Basel AG 1991

Authors and Affiliations

  • Vu Ngoc Phat
    • 1
  1. 1.Institute of MathematicsBo Ho HanoiVietnam

Personalised recommendations