Skip to main content
SpringerLink
Log in

Topological and Invariance Entropy for Infinite-Dimensional Linear Systems

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

For linear (control) systems on infinite-dimensional state spaces with finite-dimensional unstable subspace, this paper introduces the concepts of topological entropy and invariance entropy. For linear dynamical systems on Banach spaces, described by a strongly continuous semigroup, the topological entropy is given by the sum of the real parts of the unstable eigenvalues of the infinitesimal generator. An application is provided by computing the topological entropy of delay equations and of a parabolic partial differential equation. Furthermore, the invariance entropy for infinite-dimensional linear control systems is equal to the topological entropy of the homogeneous equation and so it is also described by the eigenvalues of the infinitesimal generator.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arino O, et al., (eds). Delay differential equations and applications. In: Chapter 2: some general results and remarks on delay differential equations. NATO Science Series II: Mathematics, Physics and Chemistry. Niederlande: Springer-Verlag; 2006.

  2. Bowen R. Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc. 1971;153:401–14.

    Article  MATH  MathSciNet  Google Scholar 

  3. Colonius F, Kawan C. Invariance entropy for control systems. SIAM J Control Optim. 2009;48(3):1701–721.

    Article  MATH  MathSciNet  Google Scholar 

  4. Colonius F. Minimal data rates and invariance entropy. In: Electronic proceedings of the conference on mathematical theory of networks and systems (MTNS), July 5–9. Budapest: 2010.

  5. Colonius F, Kawan C. Invariance entropy for outputs. Math Control Signals Syst. 2011;22(3):203–27.

    Article  MATH  MathSciNet  Google Scholar 

  6. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE. On the LambertW-function. Adv Comput Math. 1996;5:329–59.

    Article  MATH  MathSciNet  Google Scholar 

  7. Curtain RF, Zwart HJ. An introduction to infinite-dimensional linear systems theory. Texts in applied mathematics, vol. 21. New York: Springer-Verlag; 1995.

    Google Scholar 

  8. Damm T, Jarlebring E. The LambertWfunction and the spectrum of some multidimensional time-delay systems. Automatica 2007;43(12):2124–128.

    Article  MATH  MathSciNet  Google Scholar 

  9. Jacob B, Zwart H. Linear port-Hamiltonian systems on infinite-dimensional spaces. Operator theory: advances and applications/linear operators and linear systems (Buch 223). Basel: Birkh¨auser (Springer- Verlag); 2012.

  10. Kawan C. Upper and lower estimates for invariance entropy. Discret Contin Dyn Syst. 2011;30(1):169–86.

    Article  MATH  MathSciNet  Google Scholar 

  11. Kawan C. Invariance entropy of control sets. SIAM J Control Optim. 2011;49(2):732–51.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kawan C. Lower bounds for the strict invariance entropy. Nonlinearity 2011;24(7):1910–936.

    Article  MathSciNet  Google Scholar 

  13. Nair GN, Evans RJ, Mareels IMY, Moran W. Topological feedback entropy and nonlinear stabilization. IEEE Trans Autom Control. 2004;49(9):1585–97.

    Article  MathSciNet  Google Scholar 

  14. Pazy A. Semigroups of linear operators and applications to partial differential equations. Applied mathematical sciences, vol. 44. New York: Springer-Verlag; 1983.

  15. Temam R, Foias C, Eden A. Local and global Lyapunov exponents. J Dyn Diff Equat. 1991;3(1):133–77.

    Article  MATH  MathSciNet  Google Scholar 

  16. Walters P. An introduction to ergodic theory, graduate texts in mathematics. New York: Springer-Verlag; 1982.

Download references

Acknowledgments

The aut hor is grateful to the anonymous reviewer and Professor Fritz Colonius.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, 86159, Augsburg, Germany

    Anne-Marie Hoock

Authors
  1. Anne-Marie Hoock
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anne-Marie Hoock.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoock, AM. Topological and Invariance Entropy for Infinite-Dimensional Linear Systems. J Dyn Control Syst 20, 19–31 (2014). https://doi.org/10.1007/s10883-013-9203-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-013-9203-6

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation