Journal of Dynamical and Control Systems

, Volume 11, Issue 3, pp 329–351 | Cite as

On Stability Cones for Quadratic Systems of Differential Equations

  • V. Ye. Belozyorov
Original Article


New sufficient conditions of conditional stability of the trivial solution of a system of quadratic ordinary differential equations are obtained. For the homogeneous system the domain of unlocal stability is also found (it is a cone). This domain is then used for design of piecewise linear (discontinuous) control laws on output for a bilinear control system of any order. Thus, a system closed by such discontinuous feedback becomes globally stable in the Lyapunov sense on the whole state space. Examples are given.

Key words and phrases.

System of quadratic ordinary differential equations bilinear control system linear control law stability cone feedback 


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  1. 1.
    1. V. Ye. Belozyorov, On an invariant design of feedback for bilinear control systems of second order. Int. J. Appl. Math. Comput. Sci. 11 (2001), No. 2, 377–389.zbMATHMathSciNetGoogle Scholar
  2. 2.
    2. ——, Design of linear feedback for bilinear control systems. Int. J. Appl. Math. Comput. Sci. 12 (2002), No. 4, 493–511.zbMATHMathSciNetGoogle Scholar
  3. 3.
    3. B. P. Demidovich, Lectures on the mathematical theory of stability. Nauka, Moscow (1967) (in Russian).Google Scholar
  4. 4.
    4. F. R. Gantmacher, The theory of matrices. Chelsea Publ. (1990).Google Scholar
  5. 5.
    5. H. Haken, Synergetics. Springer-Verlag, Berlin–Heidelberg–New York (1978).Google Scholar
  6. 6.
    6. A. Isidory, Nonlinear control systems. Springer-Verlag, Berlin–Heidelberg–New York (1995).Google Scholar
  7. 7.
    7. S. Lang, Algebra. Addison-Wesley Publ. (1965).Google Scholar
  8. 8.
    8. Yu. L. Sachkov, On invariant orthants of bilinear systems. J. Dynam. Control Systems 4 (1998), No. 1, 137–147.zbMATHMathSciNetGoogle Scholar
  9. 9.
    9. H. Wang, Feedback stabilization of bilinear control systems. SIAM J. Control Optim. 36 (1998), No. 5, 1669–1684.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Dnepropetrovsk National UniversityPhysical and Technical InstituteDnepropetrovskUkraine

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