Invariance Pressure for Control Systems

  • Fritz ColoniusEmail author
  • Alexandre J. Santana
  • João A. N. Cossich


Notions of invariance pressure for control systems are introduced based on weights for the control values. The equivalence is shown between inner invariance pressure based on spanning sets of controls and on invariant open covers, respectively. Furthermore, a number of properties of invariance pressure are derived and it is computed for a class of linear systems.


Invariance pressure Invariance entropy Control systems Invariant covers Feedbacks 


  1. 1.
    Colonius, F., Fukuoka, R., Santana, A.: Invariance entropy for topological semigroup actions. Proc. Am. Math. Soc. 141, 4411–4423 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Colonius, F., Kawan, C., Nair, G.: A note on topological feedback entropy and invariance entropy. Syst. Control Lett. 62, 377–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    da Silva, A.: Outer invariance entropy for linear systems on Lie groups. SIAM J. Control Optim. 52, 3917–3934 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    da Silva, A., Kawan, C.: Invariance entropy of hyperbolic control sets. Discret. Contin. Dyn. Syst. A 36, 97–136 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huang, Y., Zhong, X.: Carathéodory–Pesin structures associated with control systems. Syst. Control Lett. 112, 36–41 (2018)CrossRefzbMATHGoogle Scholar
  6. 6.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kawan, C.: Invariance Entropy for Deterministic Control Systems. An Introduction, vol. 2089 of Lecture Notes in Mathematics, Springer-Verlag (2013)Google Scholar
  8. 8.
    Liberzon, D., Mitra, S.: Entropy and minimal bit rates for state estimation and model detection, IEEE Trans. Automatic Control., Date of Publication: 11 December (2017)
  9. 9.
    Nair, G., Evans, R.J., Mareels, I., Moran, W.: Topological feedback entropy and nonlinear stabilization. IEEE Trans. Autom. Control 49, 1585–1597 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nair, G.N., Fagnani, F., Zampieri, S., Evans, R.J.: Feedback control under data rate constraints: an overview. Proc. IEEE 95, 108–137 (2007)CrossRefGoogle Scholar
  11. 11.
    Pesin, Y.B., Pitskel, B.S.: Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl. 18(4), 307–318 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Matveev, A., Pogromsky, A.Y.: Observation of nonlinear systems via finite capacity channels: constructive data rate limits. Automatica 70, 217–229 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Savkin, A.V.: Analysis and synthesis of networked control systems: topological entropy, observability, robustness and optimal control. Automatica 42, 51–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Viana, M., Oliveira, K.: Foundations of Ergodic Theory. Cambridge University Press, Cambridge (2016)CrossRefzbMATHGoogle Scholar
  15. 15.
    Walters, P.: An Introduction to Ergodic Theory. Springer-Verlag, Berlin (1982)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität AugsburgAugsburgGermany
  2. 2.Departamento de MatemáticaUniversidade Estadual de MaringáMaringáBrazil

Personalised recommendations