EXISTENCE AND CONSTRUCTION OF A PERIODIC SOLUTION OF AUTONOMOUS DYNAMICAL SYSTEMS

Authors

M.M. Pashan Al-Farabi Kazakh National University
S.A. Aysagaliev Al-Farabi Kazakh National University

Keywords:

periodic solution, existence of a solution, integral equation, minimizing sequences, optimal control.

Abstract

A new method for investigating periodic solutions of autonomous dynamical systems described by ordinary differential equations is proposed. Necessary and sufficient conditions for the existence of a periodic solution are obtained and an algorithm for constructing a periodic solution based on the limit points of minimizing sequences is developed. By introducing a control function, the original problem is reduced to the Fredholm integral equation of the first kind. Based on the construction of a general solution of the Fredholm integral equation of the first kind, the construction of a periodic solution is reduced to solving a special initial optimal control problem.

References

Пуанкаре А. О кривых, спределяемых дифференциальними уравнениями. - М., Гостехиздат, 1947. -256c.

Смирнов В.И. Курс высшей математики. Т.4. 4.2, -6 изд. -М.: Наука. 1981. -550c.

Ляпунов А.М. Общая задача, об устойчивости движения. -М., Физматгиз,

-403с.

Моисеев Н.Н. Асимптотические методы нелинейной механики. -М.: Наука, 1969.

-305с.

Крылов Н.Н. Введение в нелинейную меленику / Н.Н. Крылов, Н.Н. Боголюбов. - Киев, 1937. -280c.

Боголюбов Н.Н. Асимптотические методы в теории нелинейных колебаний /H.H. Боголюбов, Ю.А. Митронольский. - М.: Наука, 1974. - 309с.

Биркгоф Г.Д. (Birhhoff G.D), furface Transformations and their Dinamicae Applications. Acta, Matt. V.43.-1920, p. 5-20.

Неймарк Ю.И. Метод точечных отображений в теории нелинейных Колебаний. Изв.вузов Радиофизика. т.1, 1958. p. 7-20

Андронов А.А. Теория колебаний / А.А. Андронов, А.А. Витт, С.Э. Хайкин. - М. Физматгиз, 1959. - 120c.

Нелепин P.A. Об исследовании нелинейных автоматических систем высокого порядка точными математическими методами. Доклады АН CССР. -1965. T.161, textnumero 4. C. 111-116.

Петровский И.Г. Лекции по теории обыкновенных дифференциальных уравнений. - М.: Наука, 1964. - 320 с.

Понтрягин Л.С. Обыкновенные дифференциальные уравнение. -М.: Наука, 1970. -495с.

Пoпов Е.П. Прикладная теория процессов управления в нелинейных системах. - M. Наука, 1973. -500с.

Aisagaliev S.A. Constructive method for solvability of Fredholm equation of the first kind || Electronic Journal of Qualitative Theory of Differential Equations this (EJQTDE), 2017., No, 72, p. 1-11, DOI: 10.14232/ejqtde.2017.

Aisagaliev S.A. To the boundary value problem of ordinary differential Equations || Electronic Journal of Qualitative Theory of Differential Equationsthis (EJQTDE), 2015, No. 57, p. 1-17 DOI: 10.14232/ejqtde.2015

Aisagaliev S.A. On Periodic solutions of Autonomous systems || Journal of Mathematical Sciences (United States), Volume 229, Issue 4, March, 2018, P. 335-353.

Aisagaliev S.A. Aizerman's problem in absolute seability theory for regulated systems || Sbornic Mathematics ( SB MATH), 2018, 209, DOI: 10.1070/SM 8998, p. 780-801

Aisagaliev S.A. Controllability of differential equation systems || Differential Equations, vol. 27. No. 9. 1991. p. 1037-1047.

Aisagaliev S.A. Aisagalieva S.S. A constructive method for solving a controllability problem for ordinary differential equations || Differential Equations, vol. 29, textnumero 4. 1993, p. 471-482.

Aisagaliev S.A., Belogurov A.P. Controllability and speed of the process described by a parabolic equation with bounded control || Siberian Mathematical Journal, 2012. vol. 53, textnumero 1, p. 13-28.

Aisagaliev S.A. Optimal control of linear systems with fixed trajectory end points and bounded control || Differential Equations, -1996. -vol. 32, No.6, p. 1017- 1023.

Aisagaliev S.A., Kabidoldanova A.A. On the Optimal Control of linear systems with linear performance criterion and constraints || Differential Equations, 2012. - vol. 48, No. 6, p. 826-836.

Aisagaliev S.A. Certain problems of synchronization theory || Journal

inverse III Posed Problems - textnumero 21, 2013. p. 159-175.

Aisagaliev S.A., Aipanov Sh.A. A remark of the global asymptotic stability theory of phase system // Differential Equations (35), 8. 1999. p. 1019-1026.

Aisagaliev S.A. Absolute stadility in controlled systems // Differential Equations, vol. 80, textnumero 5, 1994. p. 687-695.

Aisagaliev S.A. To the solution of a boundary value problem with a parameter for ordinary differential equations // ISSN 2074-1863. Ufa Mathematical Journal, vol. 8, No. 3 (2016). p. 3-13.

Айсагалиев С.А. Теория краевых задач динамических систем. - Алматы: 2021-565с.

Айсагалиев С.А. Кечественная теория интегро-дифференциальных уравнение - Алматы: 2022. - 272с. Казахский национальный университет имени Аль-Фараби

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