Advertisement

Applied Mathematics and Optimization

, Volume 14, Issue 1, pp 173–185 | Cite as

Linear periodic control systems: Controllability with restrained controls

  • Nguyen Khoa Son
  • Nguyen Van Su
Article

Abstract

Consider the problem of null-controllability for a linear non-autonomous system of the form\(\dot x(t) = A(t)x(t) + B(t)u(t),x(t)\varepsilon X,u(t)\varepsilon \Omega \subset U\), whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht ≥ 0, and Ω is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.

Keywords

Control System Banach Space Continuous Function System Theory Mathematical Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brammer R (1972) Controllability in linear autonomous systems with positive controllers. SIAM J Control Optim 10:339–353Google Scholar
  2. 2.
    Conti R (1974) Teoria del Controllo e del Controllo Ottimo. UTET, TorinoGoogle Scholar
  3. 3.
    Conti R (1975) On global controllability. In: Antosiewicz H (ed) International Conference on Differential Equations. Academic Press, New York, pp 203–228Google Scholar
  4. 4.
    Curtain RF, Pritchard AJ (1978) Infinite Dimensional Linear Systems Theory. Springer-Verlag, New York, Heidelberg, BerlinGoogle Scholar
  5. 5.
    Fuhrmann PA (1972) On weak and strong reachability and controllability of infinite-dimensional systems. J Optim Theory Appl 9:77–89Google Scholar
  6. 6.
    Kerimov AK (1975) Global controllability of linear periodic systems with constraints on the controls. Differencial'nye Urav 11:1575–1583Google Scholar
  7. 7.
    Korobov VI, Nguyen Khoa Son (1980) Controllability of linear systems in Banach space with constraints on the controls. Differencial'nye Urav 16:806–817, 1012–1022Google Scholar
  8. 8.
    Korobov VI, Rabah R (1979) Exact controllability in Banach space. Differencial'nye Urav 15:2142–2150Google Scholar
  9. 9.
    Krein MG, Rutman MA (1958) Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat Nauk 3:3–95Google Scholar
  10. 10.
    Megan M, Hiris V (1975) On the space of linear controllable systems in Hilbert spaces. Glas Mat Ser III 10:161–167Google Scholar
  11. 11.
    Nguyen Khoa Son (1980) Local controllability on linear systems with restrained controls in Banach space. Acta Math Vietnam 5:78–87Google Scholar
  12. 12.
    Nguyen Khoa Son (1982) Controllability of linear discrete-time systems with restrained controls in Bamach space. Contr Cybern 10:5–16Google Scholar
  13. 13.
    Nguyen Khoa Son A note on the null-controllability of linear discrete-time systems. J Optim Theory Appl (to appear)Google Scholar
  14. 14.
    Nguyen Khoa Son, Le Thanh On the null-controllability of infinite-dimensional discrete-time systems. Acta Math Vietnam (to appear)Google Scholar
  15. 15.
    Nguyen Khoa Son, Nguyen Van Chau, Nguyen Van Su (1984) On the global null-controllability of linear discrete-time systems with restrained controls in Banach spaces. Preprint Series 20/84, Institute of Mathematics, HanoiGoogle Scholar
  16. 16.
    Olbrot A, Sosnowski A (1981) Duality theorems on control and observation of discrete-time infinite-dimensional systems. Math Systems Theory 14:173–187Google Scholar
  17. 17.
    Pandolfi L (1976) Linear control systems: controllability with constrained controls. J Optim Theory Appl 19:577–585Google Scholar
  18. 18.
    Przyluski KM (1980) The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space. Appl Math Optim 6:97–112Google Scholar
  19. 19.
    Schmitendorf WE, Barmish BR (1980) Null controllability of linear systems with constrained controls. SIAM J Control Optim 18:327–345Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Nguyen Khoa Son
    • 1
  • Nguyen Van Su
    • 2
  1. 1.Center for Applied Systems AnalysisHanoiVietnam
  2. 2.Institute of MathematicsHanoiVietnam

Personalised recommendations