Geodesic
On a Method for Studying the Norm and the Stability of Solutions
Matematičeskie zametki, Tome 81 (2007) no. 4, pp. 540-546.

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We present a new method (the method of unitary transformations), which differs from the existing ones, for studying the stability and the norm of solutions of regular and singularly perturbed initial-value problems for nonautonomous linear and quasilinear systems of ODE with normal and “almost normal” matrices. Our results generalize similar theorems for the corresponding systems with constant matrices. This method allows one to avoid rather cumbersome traditional analysis, including the Lyapunov function method. For special classes of singularly perturbed problems, the method provides estimates for the norms of solutions in the presence of exponential or power boundary layers; these observations enrich the collection of known results in this field.
Keywords: Lyapunov function, initial-value problem, nonautonomous linear and quasilinear systems of ODE, boundary layer, stability and norm of solutions
Mots-clés : normal matrix. 
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Yu. A. Konyaev. On a Method for Studying the Norm and the Stability of Solutions. Matematičeskie zametki, Tome 81 (2007) no. 4, pp. 540-546. http://geodesic.mathdoc.fr/item/MZM_2007_81_4_a6/

[1] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR | Zbl

[2] M. Rozo, Nelineinye kolebaniya i teoriya ustoichivosti, Nauka, M., 1971

[3] N. Rush, P. Abets, M. Lalua, Pryamoi metod Lyapunova v teorii ustoichivosti, Mir, M., 1980 | MR | Zbl

[4] E. A. Barbashin, Funktsii Lyapunova, Nauka, M., 1970 | MR | Zbl

[5] D. R. Merkin, Vvedenie v teoriyu ustoichivosti dvizheniya, Nauka, M., 1987 | MR | Zbl

[6] B. F. Bylov, R. E. Vinograd, D. M. Grobman, V. V. Nemytskii, Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti, Nauka, M., 1966 | MR | Zbl

[7] Yu. A. Konyaev, “Dostatochnye usloviya ustoichivosti reshenii nekotorykh klassov ODU v kriticheskikh sluchayakh”, Differents. uravneniya, 26:4 (1990), 709–712 | MR | Zbl

[8] Yu. A. Konyaev, “Nachalnye i kraevye zadachi s osobennostyami”, Differents. uravneniya, 32:3 (1996), 419–421 | MR | Zbl

[9] Yu. A. Konyaev, Yu. S. Fedorov, “Asimptoticheskii analiz nekotorykh klassov singulyarno vozmuschennykh zadach na poluosi”, Matem. zametki, 62:1 (1997), 111–117 | MR | Zbl

[10] Yu. A. Konyaev, “Metod rasschepleniya v teorii regulyarnykh i singulyarnykh vozmuschenii”, Izv. vuzov. Ser. matem., 2000, no. 6, 10–15 | MR | Zbl

[11] V. V. Voevodin, Vychislitelnye osnovy lineinoi algebry, Nauka, M., 1977 | MR | Zbl

[12] P. Lankaster, Teoriya matrits, Nauka, M., 1978 | MR | Zbl