Skip to main content

Part of the book series: Communications and Control Engineering ((CCE))

  • 1892 Accesses

Abstract

This chapter is devoted to the statement of some open problems in Mathematical Control Theory. It is the view of the authors that solving these open problems will require novel mathematical tools, and more importantly, contribute greatly to the further development of modern nonlinear control theory.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Aeyels, D., Peuteman, J.: A new asymptotic stability criterion for nonlinear time-variant differential equations. IEEE Transactions on Automatic Control 43(7), 968–971 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angeli, D., Sontag, E.D., Wang, Y.: A characterization of integral input-to-state stability. IEEE Transactions on Automatic Control 45(6), 1082–1097 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blondel, V.D., Megretski, A. (eds.): Unsolved Problems in Mathematical Systems and Control Theory. Princeton University Press, Princeton (2004)

    MATH  Google Scholar 

  4. Burton, T.A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover, Mineola (2006)

    MATH  Google Scholar 

  5. Coron, J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. AMS, Providence (2007)

    MATH  Google Scholar 

  6. Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003)

    Book  MATH  Google Scholar 

  7. Jiang, Z.P.: Decentralized control for large-scale nonlinear systems: A review of recent results. Dynamics of Continuous, Discrete and Impulsive Systems 11, 537–552 (2004). Special Issue in honor of Prof. Siljak’s 70th birthday

    MATH  Google Scholar 

  8. Jiang, Z.P.: Control of interconnected nonlinear systems: a small-gain viewpoint. In: de Queiroz, M., Malisoff, M., Wolenski, P. (eds.) Optimal Control, Stabilization, and Nonsmooth Analysis. Lecture Notes in Control and Information Sciences, vol. 301, pp. 183–195. Springer, Heidelberg (2004)

    Google Scholar 

  9. Jiang, Z.P., Mareels, I.M.Y.: A small-gain control method for nonlinear cascaded systems with dynamic uncertainties. IEEE Transactions on Automatic Control 42, 292–308 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiang, Z.P., Teel, A., Praly, L.: Small-gain theorems for ISS systems and applications. Mathematics of Control, Signals, and Systems 7, 95–120 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Karafyllis, I.: Stabilization by means of approximate predictors for systems with delayed input. To appear in SIAM Journal on Control and Optimization

    Google Scholar 

  12. Karafyllis, I.: Can we prove stability by using a positive definite function with non sign-definite derivative? Submitted to Nonlinear Analysis Theory, Methods and Applications

    Google Scholar 

  13. Karafyllis, I., Jiang, Z.P.: Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization. ESAIM: Control, Optimization and Calculus of Variations (2009). doi:10.1051/cocv/2009029, pp. 1–42, August 2009

    Google Scholar 

  14. Krstić, M.: Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston (2009)

    Book  Google Scholar 

  15. Krstić, M.: Input delay compensation for forward complete and feedforward nonlinear systems. IEEE Transactions on Automatic Control 55, 287–303 (2010)

    Article  Google Scholar 

  16. Krstić, M.: Lyapunov stability of linear predictor feedback for time-varying input delay. IEEE Transactions on Automatic Control 55, 554–559 (2010)

    Article  Google Scholar 

  17. Peuteman, J., Aeyels, D.: Exponential stability of slowly time-varying nonlinear systems. Mathematics of Control, Signals and Systems 15, 42–70 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Peuteman, J., Aeyels, D.: Exponential stability of nonlinear time-varying differential equations and partial averaging. Mathematics of Control, Signals and Systems 15, 202–228 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Siljak, D.: Decentralized Control of Complex Systems. Academic Press, New York (1991)

    Google Scholar 

  20. Sontag, E.D.: Comments on integral variants of ISS. Systems Control Letters 3(1–2), 93–100 (1998)

    Article  MathSciNet  Google Scholar 

  21. Sontag, E.D., Wang, Y.: On characterizations of the input-to-state stability property. Systems and Control Letters 24, 351–359 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sontag, E.D., Wang, Y.: New characterizations of the input-to-state stability. IEEE Transactions on Automatic Control 41, 1283–1294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sontag, E.D., Wang, Y.: Lyapunov characterizations of input to output stability. SIAM Journal on Control and Optimization 39, 226–249 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Iasson Karafyllis .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag London Limited

About this chapter

Cite this chapter

Karafyllis, I., Jiang, ZP. (2011). Open Problems. In: Stability and Stabilization of Nonlinear Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-513-2_8

Download citation

  • DOI: https://doi.org/10.1007/978-0-85729-513-2_8

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-512-5

  • Online ISBN: 978-0-85729-513-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics