06. Kozlov M. V., Shchennikov V. N. Asymptotic Stability of Homogeneous Singular Systems

LATEST ISSUE

For readers

For authors

FOR REVIEWERS

SOCIAL NETWORKS

Engineering Technologies and Systems is an open access journal, which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author in accordance with the BOAI definition of open access.

All the materials of the “Engineering Technologies and Systems” journal are made available under a Creative Commons «Attribution» 4.0 license.

Useful Materials

To download article.

UDC 517.9

DOI: 10.15507/0236-2910.027.201704.546-554

Asymptotic Stability of Homogeneous Singular Systems

Mikhail V. Kozlov
Lecturer of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0001-7681-8931, This email address is being protected from spambots. You need JavaScript enabled to view it.

Vladimir N. Shchеnnikov
Professor of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), Dr.Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-5230-3482, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction: The paper provides an overview of singularly perturbed systems of ordinary differential equations with a homogeneous right-hand side of rational degree. The subject of the study is the asymptotic stability of the zero solution of these systems for sufficiently small values of the parameter.
Materials and Methods: Decomposition of the perturbed system into a reduced and a boundary system of smaller dimension is used as the main method of investigation. For the stability analysis, Zubov's theorems on the stability of homogeneous systems are applied to Lyapunov second method.
Results: In the course of research, the authors have obtained the conditions under which the asymptotic stability of the zero solution of a singularly perturbed system is a consequence of the analogous property of the reduced and boundary systems. This conclusion is valid for sufficiently small values of the perturbing parameter. To verify the hypothesis of the theorem, it is required to construct homogeneous Lyapunov functions.
Discussion and Conclusions: The paper gives a numerical example showing the class of systems satisfying the obtained theorem is not empty. An upper bound for the variation of a small parameter has been obtained, within which the zero solution is guaranteed to be asymptotically stable.

Keywords: singularity, small parameter, stability, decomposition, homogeneous function

For citation: Kozlov M. V., Shchennikov V. N. Asymptotic Stability of Homogeneous Singular Systems. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2017; 27(4):546–554. DOI: 10.15507/0236-2910.027.201704.546-554

This work is licensed under a Creative Commons Attribution 4.0 License.