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On M. G. Krein's work in the theory of linear periodic Hamiltonian systems

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Abstract

We present basic ideas of the theory of linear periodic Hamiltonian systems.

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References

  1. M. G. Krein, “Basic principles of the theory of λ-regions of stability for a canonical system of linear differential equations with periodic coefficients,” in:A Collection of Papers Dedicated to the Memory of A. A. Andronov [in Russian], Izd. Akad. Nauk SSSR, Moscow (1955), pp. 413–498.

    Google Scholar 

  2. M. G. Krein, “A generalization of some results of A. M. Lyapunov to linear differential equations with periodic coefficients,”Dokl. Akad. Nauk SSSR,23, No. 3. 445–448 (1950).

    Google Scholar 

  3. M. G. Krein, “On tests for the stable boundedness of solutions of periodic canonical systems,”Prikl. Mat. Mekh.,19, Issue 6, 641–680 (1955).

    Google Scholar 

  4. I. Ts. Gokhberg and M. G. Krein,Theory of Volterra Operators in Hilbert Spaces and Applications [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  5. I. M. Gel'fand and V. B. Lidskii, “Structure of stability regions for linear canonical systems of differential equations with periodic coefficients,”Usp. Mat. Nauk,10, Issue 1 (63), 3–40 (1955).

    Google Scholar 

  6. M. G. Krein, “To the theory of entire matrix functions of exponential type,”Ukr. Mat. Zh.,3, No. 2, 164–173 (1951).

    Google Scholar 

  7. M. G. Krein, “Principal concepts of the theory of representations of Hermitian operators with deficiency index (m, m),”Ukr. Mat. Zh.,1, No. 2, 3–66 (1949).

    Google Scholar 

  8. M. G. Krein, “On an application of an algebraic assertion in the theory of monodromy matrices,”Usp. Mat. Nauk,6, Issue 1, 171–177 (1951).

    Google Scholar 

  9. M. G. Krein, “On the problem of extension of Hermite positive functions,”Dokl. Akad. Nauk SSSR,26, No. 1, 17–22 (1940).

    Google Scholar 

  10. M. G. Krein and G. Ya. Lyubarskii, “On analytic properties of multipliers of periodic canonical differential systems of positive type,”Izv. Akad. Nauk SSSR, Ser. Mat.,26, 549–572 (1962).

    Google Scholar 

  11. V. V. Bolotin,Dynamic Stability of Elastic Systems [in Russian], Gostekhteoretizdat, Moscow (1956).

    Google Scholar 

  12. M. G. Krein and V. A. Yakubovich, “Hamiltonian systems of linear differential equations with periodic coefficients,” in:Proceedings of Internat. Symp. on Nonlinear Oscillations. [in Russian], Vol. 1 (1963), pp. 277–305. (For a more detailed presentation, see Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1961), pp. 1–54.)

    Google Scholar 

  13. V. A. Yakubovich, “On dynamic stability of elastic systems,”Dokl. Akad. Nauk SSSR,121, No. 4, 602–605 (1958).

    Google Scholar 

  14. V. A. Yakubovich and V. M. Starzhinskii,Parametric Resonance in Linear Systems [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  15. Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  16. M. G. Krein, “An introduction to the geometry of indefiniteJ-spaces and the theory of operators in these spaces,” in:Proceedings of the 2nd Summer Math. School [in Russian], Vol. 1, Kiev (1965), pp. 15–92.

    Google Scholar 

  17. I. S. Iokhvidov and M. G. Krein, “Spectral theory of operators in spaces with an indefinite metric,”Trudy Mosk. Mat. Obshch.,5, 367–432 (1956).

    Google Scholar 

  18. M. G. Krein, “Solution of the inverse Sturm-Liouville problem,”Dokl. Akad. Nauk SSSR,26, No. 1, 21–24 (1951).

    Google Scholar 

  19. M. G. Krein and I. E. Ovcharenko, “To the theory of inverse problems for a canonical differential equation,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 14–18 (1982).

    Google Scholar 

  20. M. G. Krein, “On inverse problems in filter theory and λ-regions of stability,”Dokl. Akad. Nauk SSSR,93, No. 5, 767–770 (1953).

    Google Scholar 

  21. M. G. Krein and G. Ya. Lyubarskii, “To the theory of transmission bands of periodic waveguides,”Prikl. Mat. Mekh.,25, Issue 1, 24–37 (1961).

    Google Scholar 

  22. M. G. Krein and H. Langer, “On some mathematical principles in the linear theory of damped oscillations of continua, I, II,”Integr. Equat. Oper. Theory,1, Nos. 3, 4 (1978).

    Google Scholar 

  23. M. G. Krein, “On some maximum and minimum problems for characteristic numbers and on Lyapunov stability regions,”Prikl. Mat. Mekh.,15, Issue 3, 323–348 (1951).

    Google Scholar 

  24. M. G. Krein,Lectures in the Theory of Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1964).

    Google Scholar 

  25. K. R. Kovalenko and M. G. Krein, “On some researches of A. M. Lyapunov into differential equations with periodic coefficients.”Dokl. Akad. Nauk SSSR,25, No. 4, 495–498 (1950).

    Google Scholar 

  26. M. G. Krein, “On a hypothesis of A. M. Lyapunov,”Funkts. Anal. Prilozhen.,7, Issue 3, 45–54 (1973).

    Google Scholar 

  27. M. G. Krein, “On some problems related to Lyapunov's ideas and in the theory of stability,”Usp. Mat. Nauk,3, No. 3, 166–169 (1948).

    Google Scholar 

  28. I. Ts. Gokhberg and M. G. Krein, “On the influence of certain transformations of the kernels of integral equations on the spectra of these equations,”Ukr. Mat. Zh.,13, No. 3, 12–38 (1961).

    Google Scholar 

  29. M. G. Krein, “On a characteristic function of a linear canonical system of second-order differential equations with periodic coefficients,”Prikl. Mat. Mekh.,21, Issue 3, 320–329 (1957).

    Google Scholar 

  30. V. A. Yakubovich and V. M. Starzhinskii,Linear Differential Equations with Periodic Coefficients and Applications [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  31. V. A. Yakubovich, “Structure of a group of symplectic matrices and a set of unstable canonical systems with periodic coefficients,”Mat. Sb.,44(86), No. 3, 313–352 (1958).

    Google Scholar 

  32. A. M. Lyapunov, “On the stability of motion in a particular case of the three-body problem,” in:Selected Works [in Russian]. Vol. 1, Izd. Akad. Nauk SSSR, Moscow, Leningrad (1954), pp. 327–401.

    Google Scholar 

  33. V. A. Yakubovich, “Linear-quadratic optimization problem and the frequency theorem for periodic systems. II,”Sib. Mat. Zh.,31, No. 6, 176–191 (1990).

    Google Scholar 

  34. V. I. Arnol'd, “On a characteristic class in the quantization conditions,”Funkts. Anal. Prilozhen.,1, Issue 1, 1–14 (1967).

    Google Scholar 

  35. I. A. Polunin and V. A. Yakubovich, “Orbital stability of periodic Laplacian motions,”Vestn. Leningrad. Univ., Ser. 1, Issue 3, 106–108 (1988).

    Google Scholar 

  36. A. M. Lyapunov, “On the series emerging in the theory of linear second-order differential equations with periodic coefficients,” in:Selected Works [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Moscow, Leningrad (1956), pp. 410–472.

    Google Scholar 

  37. M. G. Neigauz and V. B. Lidskii, “On the boundedness of solutions of linear systems of differential equations with periodic coefficients,”Dokl. Akad. Nauk SSSR,77, No. 1, 25–28 (1951).

    Google Scholar 

  38. G. Borg, “Über die Stabilität gewisser Klassen von linearen Differentialgleichungen,”Arch. Mat., Astr. Fys.,31A, No. 1, 1–30 (1944).

    Google Scholar 

  39. Yu. V. Komlenko, “On some criteria of nonoscillation and boundedness for solutions of linear differential equations,”Dokl. Akad. Nauk SSSR,164, No. 2, 270–272 (1965).

    Google Scholar 

  40. W. A. Coppel and A. Howe, “On the stability of linear canonical systems with periodic coefficients,”J. Austral. Math. Soc.,5, No. 2, 169–195 (1965).

    Google Scholar 

  41. V. A. Yakubovich, “Structure of the function space of complex canonical equations with periodic coefficients,”Dokl. Akad. Nauk SSSR,139, No. 1, 54–57 (1961).

    Google Scholar 

  42. V. B. Lidskii and V. A. Frolov, “Structure of stability regions for self-adjoint systems of differential equations with periodic coefficients,”Mat. Sb.,71(113), Issue 1, 48–64 (1966).

    Google Scholar 

  43. V. I. Derguzov, “On the stability of solutions of Hamiltonian equations with unbounded operator coefficients,”Mat. Sb.,66(105), Issue 4, 591–619 (1964).

    Google Scholar 

  44. V. N. Fomin,Mathematical Theory of Parametric Resonance in Linear Distributed Systems [in Russian], Leningrad University, Leningrad (1972).

    Google Scholar 

  45. V. A. Yakubovich, “Nonoscillating of linear periodic Hamiltonian equations and related questions,”Algebra Anal.,3, No. 5 (1991).

    Google Scholar 

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Authors and Affiliations

  1. St-Petersburg University, St-Petersburg

    V. A. Yakubovich

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  1. V. A. Yakubovich
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 128–144, January–February, 1994.

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Yakubovich, V.A. On M. G. Krein's work in the theory of linear periodic Hamiltonian systems. Ukr Math J 46, 133–148 (1994). https://doi.org/10.1007/BF01057005

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