Skip to main content
SpringerLink
Log in
Book cover

International Conference on Mathematical Optimization Theory and Operations Research

MOTOR 2019: Mathematical Optimization Theory and Operations Research pp 461–473Cite as

Estimates of the Minimal Eigenvalue of the Controllability Gramian for a System Containing a Small Parameter

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

We consider a linear time-invariant control system with right-hand side depending on a small parameter. Assuming that the system is controllable, we study the asymptotics of the minimal eigenvalue of a system’s controllability Gramian and provide some bounds for the eigenvalue. These estimates are applied to the study of convexity properties of reachable sets for nonlinear control systems with integral constraints on control variables.

This is a preview of subscription content, 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   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
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

Learn about institutional subscriptions

References

  1. Anan’ev, B.I.: Motion correction of a statistically uncertain system under communication constraints. Autom. Remote Control 71(3), 367–378 (2010)

    Article  MathSciNet  Google Scholar 

  2. Baier, R., Gerdts, M., Xausa, I.: Approximation of reachable sets using optimal control algorithms. Numer. Algebra Control Optim. 3(3), 519–548 (2013)

    Article  MathSciNet  Google Scholar 

  3. Dar’in, A.N., Kurzhanskii, A.B.: Control under indeterminacy and double constraints. Differ. Equ. 39(11), 1554–1567 (2003)

    Article  MathSciNet  Google Scholar 

  4. Filippova, T.F.: Estimates of reachable sets of impulsive control problems with special nonlinearity. In: AIP Conference Proceedings Application of Mathematics in Technical and Natural Sciences, 2016, vol. 1773, Article number 100004, pp. 1–10 (2016)

    Google Scholar 

  5. Guseinov, K.G., Ozer, O., Akyar, E., Ushakov, V.N.: The approximation of reachable sets of control systems with integral constraint on controls. Nonlinear Differ. Equ. Appl. 14(1–2), 57–73 (2007)

    Article  MathSciNet  Google Scholar 

  6. Guseinov, Kh. G., Nazlipinar, A.S.: Attainable sets of the control system with limited resources.Trudy Inst. Mat. i Mekh. Uro RAN 16(5), 261–268 (2010)

    Google Scholar 

  7. Gusev, M.: On reachability analysis of nonlinear systems with joint integral constraints. In: Lirkov, I., Margenov, S. (eds.) LSSC 2017. LNCS, vol. 10665, pp. 219–227. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73441-5_23

    Chapter  Google Scholar 

  8. Gusev, M.I., Zykov, I.V.: On extremal properties of boundary points of reachable sets for a system with integrally constrained control. IFAC-PapersOnLine 50(1), 4082–4087 (2017). https://doi.org/10.1016/j.ifacol.2017.08.792. 20th IFAC WORLD CONGRESS, 2017

    Article  Google Scholar 

  9. Gusev, M.I.: Internal approximations of reachable sets of control systems with state constraints. Proc. Steklov Inst. Math. 287(Suppl. 1), S77–S92 (2014)

    Article  Google Scholar 

  10. Huseyin, N., Huseyin, A.: Compactness of the set of trajectories of the controllable system described by an affineintegral equation. Appl. Math. Comput. 219, 8416–8424 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)

    Article  MathSciNet  Google Scholar 

  12. Lee, E.B., Marcus, L.: Foundations of Optimal Control Theory. Willey, Hoboken (1967)

    Google Scholar 

  13. Horn, R.A., Jonson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  14. Patsko, V.S., Pyatko, S.G., Fedotov, A.: Three-dimensional reachability set for a nonlinear control system. J. Comput. Syst. Sci. Int. 42(3), 320–328 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Polyak, B.T.: Convexity of the reachable set of nonlinear systems under l2 bounded controls. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11, 255–267 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Walter, W.: Differential and Integral Inequalities. Springer, Berlin (1970). https://doi.org/10.1007/978-3-642-86405-6

    Book  Google Scholar 

  17. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Press, Boston (1988)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. N.N. Krasovskii Institute of Mathematics and Mechanics, 16 S.Kovalevskaya str., 620108, Ekaterinburg, Russia

    Mikhail Gusev

  2. Ural Federal University, 19 Mira street, 620002, Ekaterinburg, Russia

    Mikhail Gusev

Authors
  1. Mikhail Gusev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Gusev .

Editor information

Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gusev, M. (2019). Estimates of the Minimal Eigenvalue of the Controllability Gramian for a System Containing a Small Parameter. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22629-9_32

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22628-2

  • Online ISBN: 978-3-030-22629-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation