Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities

  • A. Halanay
  • V. L. Răsvan
Part of the Operator Theory: Advances and Applications book series (OT, volume 117)


One of the fields of Applied Mathematics that grew up from the pioneering papers of A.M. Liapunov and M.G. Krein is the theory of dynamical systems with periodic coefficients. The results of M.G. Krein and V.A. Yakubovich concerning stability of linear periodic Hamiltonian systems turned out to have applications in the so-called linear-quadratic theory of the controlled systems. The present paper deals with a “subset” of this theory: existence of nonlinear oscillations (periodic and almost periodic solutions) in systems with sector-restricted nonlinearities (the so-called absolutely stable systems). Since almost all results obtained for differential equations have their discrete-time (more or less) counterpart, both continuous time and discrete time periodic cases are presented here. The existence conditions are expressed in terms of an associated periodic Hamiltonian system that is required to be dichotomic and strongly disconjugate. This property may be checked in terms of the properties of an associated matrix Riccati equation or of some Linear Matrix Inequalities.


Hamiltonian System Linear Matrix Inequality Riccati Equation Invariant Manifold Exponential Dichotomy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Barbălat and A. Halanay, Conditions de comportement “presque lineaire” dans la théorie des oscillations, Rev. Roum. Sci. Techn-Electrotech. et Energ. vol. 29 (1974), pp. 321–341.Google Scholar
  2. [2]
    S. Bittanti, P. Colaneri and G. Guadabassi, Analysis of periodic Liapunov and Riccati equations via canonical decomposition, SIAM J. Control and Optimization, vol24 (1986), pp. 1138–1149.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    S. Bittanti, P. Colaneri and G. DeNicolao, A note on the Maximal Solution of the Periodic Riccati equation, IEEE Trans. Aut. Contr. vol. AC-34 (1989), pp. 1316–1319.MathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. and Stat. Comp. vol. 2 (1981), pp. 121–135.zbMATHCrossRefGoogle Scholar
  5. [5]
    A.Kh. Gelig, Dynamics of pulse systems and neural networks, Leningrad Univ. Publ. House, 1982 (in Russian).Google Scholar
  6. [6]
    A.Kh. Gelig and A.N. Churilov, Oscillations and stability of nonlinear pulse systems, St. Petersburg Univ. Publ. House, 1993 (in Russian).Google Scholar
  7. [7]
    I.Ts. Gohberg and M.G. Krein, Systems of integral equations on semi-axis,with kernel depending on argument difference,Usp. Mat. Nauk, 1958, no. 2, (in Russian).Google Scholar
  8. [8]
    A. Halanay, Invariant manifolds for systems with time lags, in Differential Equations and dynamical systems (Hale and La Salle eds.) pp. 199–213, Academic Press, 1967.Google Scholar
  9. [9]
    A. Halanay and D. Wexler, Qualitative Theory of pulse systems,Editura Academiei, Bucharest, 1968 (in Romanian, Russian Edition by Nauka, Moscow, 1971).Google Scholar
  10. [10]
    A. Halanay and V. Ionescu, Time-varying discrete linear systems, Birkhäuser, 1994.zbMATHCrossRefGoogle Scholar
  11. [11]
    M.G. Krein, Foundations of the theory of λ-zones of stability of a canonical system of linear differential equations with periodic coe f ficients, In memoriam: A.A. Andronov, Izd. Akad. Nauk SSSR, Moscow, 1955, pp. 413–98 (English version in AMS Transl. (2), vol. 120 (1983), pp. 1–70).Google Scholar
  12. [12]
    M.G. Krein, On Tests for Stable Boundness of Solutions of Periodic Canonical Systems, Prikl. Mat. Mekh. vol. 19 (1955), pp. 641–680 (English version in AMS Transl. (2), vol. 120 (1983), pp. 71–110).Google Scholar
  13. [13]
    M.G. Krein and G.Ia. Lyubarskii, About analytic properties of multipliers of periodic canonical systems of positive types, Izv. Akad. Nauk SSSR Ser. Mat. vol. 26 (1962, pp. 549–572) (English version in AMS Transl.(2) vol. 89 (1970), pp. 1–28).Google Scholar
  14. [14]
    M.G. Krein and V.A. Yakubovich, Hamiltonian Systems of Linear Differential Equations with periodic Coefficients, Proc. Intl Symp. on Nonlinear Vibrations vol. 1, pp. 277–305, Izd. Akad. Nauk Ukrain. SSR, Kiev, 1963 (English version in AMS Trans. (2) vol. 120 (1983), pp. 139–168).Google Scholar
  15. [15]
    V.M. Kuntsevich and Ju.N. Chekhovoy, Nonlinear Control Systems with Frequency and Pulse-Width Modulation, Naukova Dumka, Kiev, 1970 (in Russian).Google Scholar
  16. [16]
    A.I. Lurie and V.N. Postnikov, About the theory of stability for controlled systems,Prikl. Mat. Mekh. vol. 8, no. 3 (1944) (in Russian).Google Scholar
  17. [17]
    V.M. Popov, Hyperstability of Control Systems, Springer Verlag, 1973.zbMATHCrossRefGoogle Scholar
  18. [18]
    V.A. Yakubovich, Structure of the group of symplectic matrices and of the set of unstable canonical systems with periodic coefficients, Mat. Sbornik, vol. 44(86) (1958), pp. 313–352 (in Russian).Google Scholar
  19. [19]
    V.A. Yakubovich, Oscillatory properties of solutions of canonical systems,Mat. Sbornik, vol. 56(98) (1962), pp. 3–42 (in Russian).Google Scholar
  20. [20]
    V.A. Yakubovich and V.M. Starzhinskii, Linear differential equations with periodic coefficients,Nauka, Moscow, 1972 (English version, J. Wiley, 1975).Google Scholar
  21. [21]
    V.A. Yakubovich, Method of matrix inequalities in stability theory for nonlinear controlled systems. I. Absolute stability of forced oscillations, Avtom. i telemekh. vol. XXV (1964) pp. 1017–1029.Google Scholar
  22. [22]
    V.A. Yakubovich, Periodic and almost periodic limit regimes of controlled systems with several, generally speaking, discontinuous nonlinearities, Dokl. Akad. Nauk SSSR vol. 171, no. 3, pp. 533–536 (in Russian).Google Scholar
  23. [23]
    V.A. Yakubovich, Frequency domain conditions for self-sustained oscillations in nonlinear systems with a single time-invariant nonlinearity, Sib. Mat. Journal vol. 12 (1973), no. 5 (in Russian).Google Scholar
  24. [24]
    V.A. Yakubovich, Frequency domain methods for qualitative study of nonlinear controlled systems, VII Int. Konferenz ü. nichtlin. Schwingungen, Bd. I, 1, Akademie Verlag, Berlin, 1977 (in Russian).Google Scholar
  25. [25]
    V.A. Yakubovich, Linear-quadratic optimization problem and frequency domain theorem for periodic systems 1, Sib. Mat. Journal vol. 27 (1986), no. 4, pp. 181–200.MathSciNetzbMATHGoogle Scholar
  26. [26]
    G. Wade, Signal coding and processing, Cambridge Univ. Press, 1994.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2000

Authors and Affiliations

  • A. Halanay
    • 1
  • V. L. Răsvan
    • 2
  1. 1.Dept. of MathematicsBucharest UniversityRomania
  2. 2.Dept. of Automatic ControlCraiova UniversityCraiovaRomania

Personalised recommendations