Abstract
We present basic ideas of the theory of linear periodic Hamiltonian systems.
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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.
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).
M. G. Krein, “On tests for the stable boundedness of solutions of periodic canonical systems,”Prikl. Mat. Mekh.,19, Issue 6, 641–680 (1955).
I. Ts. Gokhberg and M. G. Krein,Theory of Volterra Operators in Hilbert Spaces and Applications [in Russian], Nauka, Moscow (1967).
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).
M. G. Krein, “To the theory of entire matrix functions of exponential type,”Ukr. Mat. Zh.,3, No. 2, 164–173 (1951).
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).
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).
M. G. Krein, “On the problem of extension of Hermite positive functions,”Dokl. Akad. Nauk SSSR,26, No. 1, 17–22 (1940).
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).
V. V. Bolotin,Dynamic Stability of Elastic Systems [in Russian], Gostekhteoretizdat, Moscow (1956).
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.)
V. A. Yakubovich, “On dynamic stability of elastic systems,”Dokl. Akad. Nauk SSSR,121, No. 4, 602–605 (1958).
V. A. Yakubovich and V. M. Starzhinskii,Parametric Resonance in Linear Systems [in Russian], Nauka, Moscow (1987).
Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
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.
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).
M. G. Krein, “Solution of the inverse Sturm-Liouville problem,”Dokl. Akad. Nauk SSSR,26, No. 1, 21–24 (1951).
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).
M. G. Krein, “On inverse problems in filter theory and λ-regions of stability,”Dokl. Akad. Nauk SSSR,93, No. 5, 767–770 (1953).
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).
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).
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).
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).
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).
M. G. Krein, “On a hypothesis of A. M. Lyapunov,”Funkts. Anal. Prilozhen.,7, Issue 3, 45–54 (1973).
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).
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).
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).
V. A. Yakubovich and V. M. Starzhinskii,Linear Differential Equations with Periodic Coefficients and Applications [in Russian], Nauka, Moscow (1972).
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).
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.
V. A. Yakubovich, “Linear-quadratic optimization problem and the frequency theorem for periodic systems. II,”Sib. Mat. Zh.,31, No. 6, 176–191 (1990).
V. I. Arnol'd, “On a characteristic class in the quantization conditions,”Funkts. Anal. Prilozhen.,1, Issue 1, 1–14 (1967).
I. A. Polunin and V. A. Yakubovich, “Orbital stability of periodic Laplacian motions,”Vestn. Leningrad. Univ., Ser. 1, Issue 3, 106–108 (1988).
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.
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).
G. Borg, “Über die Stabilität gewisser Klassen von linearen Differentialgleichungen,”Arch. Mat., Astr. Fys.,31A, No. 1, 1–30 (1944).
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).
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).
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).
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).
V. I. Derguzov, “On the stability of solutions of Hamiltonian equations with unbounded operator coefficients,”Mat. Sb.,66(105), Issue 4, 591–619 (1964).
V. N. Fomin,Mathematical Theory of Parametric Resonance in Linear Distributed Systems [in Russian], Leningrad University, Leningrad (1972).
V. A. Yakubovich, “Nonoscillating of linear periodic Hamiltonian equations and related questions,”Algebra Anal.,3, No. 5 (1991).
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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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DOI: https://doi.org/10.1007/BF01057005