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Limiting equations and stability of non-stationary motions. (English) Zbl 0651.34047

“The purpose of the present paper is to review results obtained on the basis of limiting equations in the process of investigating systms with and without small parameters but with a non-asymptotically stable zero solution of a shortened system”. This is a quotation from the Introduction. Two theorems, one on Hadamard stability and one on asymptotic stability, based on an asymptotic form (as \(t\to \infty)\) of the nonautonomous system are given. No proofs are given and no reference to the source of the reported results is included. Not all works references in the paper are listed in the references.
Reviewer: J.F.Toland

MSC:

34C29 Averaging method for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:

[1] Anashkin, O. V., On asymptotic stability in non-linear systems, Differential Equations, 14, No. 3, 1490-1493 (1978) · Zbl 0398.34039
[2] A. Karimzhanov; A. Karimzhanov
[3] Karimzhanov, A., Limiting systems in the problem on stability of non-autonomous systems, Appl. Mech., 21, No. 5, 110-117 (1985) · Zbl 0603.70025
[4] Karimzhanov, A.; Kosolapov, V. I., To the stability of systems of small parameter, (Stability of Motion (1985), Nauka: Nauka Novosibirsk), 40-43 · Zbl 0544.34046
[5] Kosolapov, V. I., Investigation of asymptotic stability and instability of non-linear systems with integrating approximation, Math. Phys., 29, 44-48 (1981) · Zbl 0464.70022
[6] Krassovsky, N. N., Problems of stabilization of control motions, (Malkin, I. G., Stability of Motions Theory (1966), Nauka: Nauka Moscow), 475-514
[7] Martynyuk, A. A.; Karimzhanov, A., Limiting equations and optimal stabilization of non-stationary motions, \((V\) All-Union Conference on Control in Mechanical Systems. \(V\) All-Union Conference on Control in Mechanical Systems, 12-14 June, \(1985. V\) All-Union Conference on Control in Mechanical Systems. \(V\) All-Union Conference on Control in Mechanical Systems, 12-14 June, 1985, Theses of reports (1985)), 77, Kasan · Zbl 0651.34047
[8] Martynyuk, A. A.; Kosolapov, V. I., Principle of Comparison and Average Method in Problem of Stability of Non-asymptotically Stable Motions under Persistent Disturbances, Preprint, USSR Ac. of Sci. Inst, of Math., 78.33, 24 (1978), Kiev
[9] Hapaev, M. M., On one theorem of Lyapunov’s type, Dokl. Akad. Nauk. USSR, 176, No. 6, 1262-1265 (1967)
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