On the principle of comparison and estimates of the Lyapunov functions for nonlinear systems

ЗаголовокOn the principle of comparison and estimates of the Lyapunov functions for nonlinear systems
Тип публікаціїJournal Article
Рік публікації2018
АвториMartynyuk, AA
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2018.09.003
Номер видання9
РозділMathematics
Нумерація сторінок3-11
Дата публікації9/2018
МоваRussian
Анотація

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.
Ключові словаestimate of the norm of solutions, Lyapunov function, nonlinear system of a general form, stability of motion
Посилання: 
  1. Corduneanu, C. (2009). The contribution of R. Conti to the comparison method in differential equations. Libertas Math., 29, pp. 113-115.
  2. 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.
  3. 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.
  4. Lyapunov, A. M. (1935). The general problem of the stability of motion. Leningrad, Moscow: ONTI (in Russian).
  5. Melnikov, G. I. (1956). Some questions of the direct Lyapunov method. Dokl. AN. SSSR, 110, No. 3, pp. 326-329 (in Russian).
  6. 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).
  7. 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
  8. 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
  9. 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
  10. 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
  11. Wintner, A. (1946). An Abelian lemma of asymptotic equilibria. Amer. J. Math., 78, pp. 451-454. doi: https://doi.org/10.2307/2371826
  12. Langenhop, C. E. (1960). Bounds on the norm of a solution of a general differential equation. Proc. Amer. Math. Soc., 11, pp. 795-799.
  13. 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
  14. Lakshmikantham, V., Leela, S. & Martynyuk, A. A. (1990). Practical stability of nonlinear systems. Singa pore: World Scientific. doi: https://doi.org/10.1142/1192
  15. Zubov, V. I. (1959). Mathematical methods for the automatic regulation systems analysis. Leningrad: Sudpromgiz (in Russian).