A new development for the Tikhonov Theorem in nonlinear singular perturbation systems. (English) Zbl 1247.34104
Contrary to the title, the paper doesn’t contain a new development of the Tikhonov theorem. The authors use the exponential stability of a boundary-layer system while in A. N. Tikhonov’s original work for finite time interval [Mat. Sb., Nov. Ser. 31 (73), 575–586 (1952; Zbl 0048.07101)] the asymptotic stability is supposed only (see also [W. Wasow, Asymptotic expansions for ordinary differential equations. New York-London-Sydney: Interscience Publishers, a division of John Wiley & Sons, Inc. (1965; Zbl 0133.35301)]). Concerning an infinite time interval, the authors’ theorem is an immediate corollary of well known results, which are based on the integral manifolds theory (see, for example, [V. A. Sobolev, “Integral manifolds and decomposition of singularly perturbed systems”, Syst. Control Lett. 5, 169–179 (1984; Zbl 0552.93017)]). It is obvious that the authors aren’t familiar with the classical Klimushev-Krasovsky theorem, either (A. N. Klimushev and N. N. Krasovskij [J. Appl. Math. Mech. 25 (1961), 1011–1025 (1962); translation from Prikl. Mat. Mekh. 25, No. 4, 680–690 (1961; Zbl 0106.29302)]).
Reviewer: Vladimir Sobolev (Samara)
MSC:
| 34E15 | Singular perturbations for ordinary differential equations |
| 34D15 | Singular perturbations of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
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