On the principle of comparison and estimates of the Lyapunov functions for nonlinear systems
DOI:
https://doi.org/10.15407/dopovidi2018.09.003Keywords:
estimate of the norm of solutions, Lyapunov function, nonlinear system of a general form, stability of motionAbstract
Some new estimates of the Lyapunov function for a nonlinear system and conditions of Lyapunov stability and stability on a finite interval are established. The above conditions are based on estimates of the norms of solutions of a nonlinear system of equations of perturbed motion.
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Corduneanu, C. (2009). The contribution of R. Conti to the comparison method in differential equations. Libertas Math., 29, pp. 113-115.
Conti, R. (1956). Limitazione "in ampiezza"delle soluzioni di un sistema di equazioni diferenziali ordinarie e applicazini. Boll. Unione Mat. Ital. Ser. 3, 11, No. 3, pp. 344-349.
Conti, R. (1956). Sulla prolungabilita delle soluzioni di un sistema di equazioni diferenziali ordinarie. Boll. Unione Mat. Ital. Ser. 3, 11, No. 4, pp. 510-514.
Lyapunov, A. M. (1935). The general problem of the stability of motion. Leningrad, Moscow: ONTI (in Russian).
Melnikov, G. I. (1956). Some questions of the direct Lyapunov method. Dokl. AN. SSSR, 110, No. 3, pp. 326-329 (in Russian).
Martynyuk, A. A. (2011). Asymptotic stability criterion for nonlinear monotonic systems and its applications (review). Int. Appl. Mech., 47, Iss. 5, pp. 475-534; Martynyuk, A.A. (2015). A criterion for the asymptotic stability of non-linear monotonic si- and its application. In Modern problems of mechanics, Vol. 1 (pp. 276-339). Kiev: LITERA LTD (in Russian).
Martynyuk, A. A. (2017). Constructive estimates of Lyapunov V-functions for the equations of perturbed motion. Prikl. Mekhanika, 53 (63), Iss. 5, pp. 122-128 (in Russian). doi: https://doi.org/10.1007/s10778-017-0840-4
Bihari, I. (1956). A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Hung., 7, pp. 81-94. doi: https://doi.org/10.1007/BF02022967
Brauer, F. (1963). Bounds for solution of ordinary differential equations. Proc. Amer. Math. Soc. 14, No. 1, pp. 36-43. doi: https://doi.org/10.1090/S0002-9939-1963-0142829-0
Cooke, K. L. (1955). A non-local existence theorem for systems of ordinary differential equations. Rend. Circ. Mat. Palermo, 4, pp. 301-308. doi: https://doi.org/10.1007/BF02854201
Wintner, A. (1946). An Abelian lemma of asymptotic equilibria. Amer. J. Math., 78, pp. 451-454. doi: https://doi.org/10.2307/2371826
Langenhop, C. E. (1960). Bounds on the norm of a solution of a general differential equation. Proc. Amer. Math. Soc., 11, pp. 795-799.
Martynyuk, A. A. (2015). Novel bounds for solutions of nonlinear differential equations. Appl. Math., 6, pp. 182-194. doi: https://doi.org/10.4236/am.2015.61018
Lakshmikantham, V., Leela, S. & Martynyuk, A. A. (1990). Practical stability of nonlinear systems. Singa pore: World Scientific. doi: https://doi.org/10.1142/1192
Zubov, V. I. (1959). Mathematical methods for the automatic regulation systems analysis. Leningrad: Sudpromgiz (in Russian).
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