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

  • Mikhail GusevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


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.


Control system Controllability Gramian Small parameter Reachable set Integral constraints 


  1. 1.
    Anan’ev, B.I.: Motion correction of a statistically uncertain system under communication constraints. Autom. Remote Control 71(3), 367–378 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baier, R., Gerdts, M., Xausa, I.: Approximation of reachable sets using optimal control algorithms. Numer. Algebra Control Optim. 3(3), 519–548 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dar’in, A.N., Kurzhanskii, A.B.: Control under indeterminacy and double constraints. Differ. Equ. 39(11), 1554–1567 (2003)MathSciNetCrossRefGoogle Scholar
  4. 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. 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)MathSciNetCrossRefGoogle Scholar
  6. 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. 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). Scholar
  8. 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). 20th IFAC WORLD CONGRESS, 2017CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lee, E.B., Marcus, L.: Foundations of Optimal Control Theory. Willey, Hoboken (1967)Google Scholar
  13. 13.
    Horn, R.A., Jonson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1986)Google Scholar
  14. 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)MathSciNetzbMATHGoogle Scholar
  15. 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Walter, W.: Differential and Integral Inequalities. Springer, Berlin (1970). Scholar
  17. 17.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Press, Boston (1988)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.N.N. Krasovskii Institute of Mathematics and MechanicsEkaterinburgRussia
  2. 2.Ural Federal UniversityEkaterinburgRussia

Personalised recommendations