Skip to main content
SpringerLink
Log in

A Criterion for Existence of Global-in-Time Trajectories of Non-deterministic Markovian Systems

  • Conference paper
ICT in Education, Research, and Industrial Applications (ICTERI 2012)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 347))

Abstract

We consider the following question: given a continuous-time non-deterministic (not necessarily time-invariant) dynamical system, is it true that for each initial condition there exists a global-in-time trajectory. We study this question for a large class of systems, namely the class of complete non-deterministic Markovian systems. We show that for this class our question can be answered using analysis of existence of locally defined trajectories in a neighborhood of each time moment.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Coddington, E., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  2. Fillipov, A.: Differential equations with discontinuous right-hand sides. AMS Trans. 42, 199–231 (1964)

    Google Scholar 

  3. Gliklikh, Y.: Necessary and sufficient conditions for global-in-time existence of solutions of ordinary, stochastic, and parabolic differential equations. Abstract and Applied Analysis 2006, 1–17 (2006)

    Google Scholar 

  4. Tangiguchi, T.: Global existence of solutions of differential inclusions. Journal of Mathematical Analysis and Applications 166, 41–51 (1992)

    Article  MathSciNet  Google Scholar 

  5. Aubin, A., Cellina, A.: Differential inclusions. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  6. Seah, S.W.: Existence of solutions and asymptotic equilibrium of multivalued differential systems. J. Math. Anal. Appl. 89, 648–663 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Heemels, W., Camlibel, M., Van der Schaft, A.J., Schumacher, J.M.: On the existence and uniqueness of solution trajectories to hybrid dynamical systems. In: Johansson, R., Rantzer, A. (eds.) Nonlinear and Hybrid Control in Automotive Applications, pp. 391–422. Springer, Berlin (2003)

    Google Scholar 

  8. Goebel, R., Sanfelice, R., Teel, R.: Hybrid dynamical systems. IEEE Control Systems Magazine 29, 29–93 (2009)

    MathSciNet  Google Scholar 

  9. Henzinger, T.: The theory of hybrid automata. In: IEEE Symposium on Logic in Computer Science, pp. 278–292 (1996)

    Google Scholar 

  10. Doob, J.B.: Stochastic processes. Wiley-Interscience (1990)

    Google Scholar 

  11. Willems, J.: Paradigms and Puzzles in the Theory of Dynamical Systems. IEEE Transactions on Automatic Control 36, 259–294 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  13. Constantin, A.: Global existence of solutions for perturbed differential equations. Annali di Matematica Pura ed Applicata 168, 237–299 (1995)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Ukraine

    Ievgen Ivanov

  2. Paul Sabatier University, Toulouse, France

    Ievgen Ivanov

Authors
  1. Ievgen Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information Technologies, Zaporozhye National University, 66 Zhukovskogo st, 69600, Zaporozhye, Ukraine

    Vadim Ermolayev

  2. Mayr Application Engineering, Alpen-Adria University Klagenfurt, Universitätsstrasse 65, 9020, Klagenfurt, Austria

    Heinrich C. Mayr

  3. Department of Theory and Technology of Programming, National Taras Shevchenko University of Kyiv, 64, Volodymyrska Street, 01033, Kyiv, Ukraine

    Mykola Nikitchenko

  4. Kherson State University, 40-rokiv, Zhovtnya st. 27, 73000, Kherson, Ukraine

    Aleksander Spivakovsky

  5. School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody Sqr, 61022, Kharkov, Ukraine

    Grygoriy Zholtkevych

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ivanov, I. (2013). A Criterion for Existence of Global-in-Time Trajectories of Non-deterministic Markovian Systems. In: Ermolayev, V., Mayr, H.C., Nikitchenko, M., Spivakovsky, A., Zholtkevych, G. (eds) ICT in Education, Research, and Industrial Applications. ICTERI 2012. Communications in Computer and Information Science, vol 347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35737-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35737-4_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35736-7

  • Online ISBN: 978-3-642-35737-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation