UDK 517.91

DOI: 10.15507/0236-2910.028.201803.344-351

** Stability with Respect to a Part of Variables under Constant Perturbations of the Partial Equilibrium Position of Differential Equation Nonlinear Systems **

**Pavel P. Lipasov **Маster Degree Student of Chair of Fundamental Informatics and Information Technologies, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ResearcherID: N-7266-2018, ORCID: https://orcid.org/0000-0001-9524-7353, This email address is being protected from spambots. You need JavaScript enabled to view it.

** Vladimir N. Shchennikov **Professor of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), D. Sc. (Physics and Mathemetics), ResearcherID: Q-3912-2018, ORCID: http://orcid.org/0000-0001-5230-3482, This email address is being protected from spambots. You need JavaScript enabled to view it.

**Introduction.** It is impossible to take into account all the forces acting in the process of mathematical modeling of dynamic processes. In order that mathematical models the most accurately describe the dynamic processes, they must include the terms that correspond the constant perturbations. These problems arise in applied tasks. In this paper we consider the case when the system allows for the partial equilibrium position. The aim of this work is to prove the stability theorem for the partial equilibrium position at constant perturbations, which are small at every instant.

** Materials and Methods.** The research objects are nonlinear systems of differential equations that allow for a partial equilibrium position. Using the second Lyapunov method, there are proved the stability theorems for the constant perturbations of the partial equilibrium position, which are small at every instant.

** Results.** Together with the introduction of stability for a part of the variables, it has become necessary to introduce stability for the part of phase variables under constant perturbations. The first stability theorem of the part of phase variables under constant perturbations was obtained by A. S. Oziraner. In this work, we prove a theorem of the stability of the constant perturbations of the partial equilibrium position, small at every instant. It should be noted that there is no stability theorems of constant perturbations for the partial equilibrium position. Thus, the theorem proved in this work is of a pioneer nature. ** Conclusions.** The theorem 3 proved in the work is the development of the mathematical theory of stability. The results of this work are applicable in the mechanics of controlled motion, nonlinear system.

**Keywords:** constantly acting disturbances, stability at constantly acting disturbances, partial equilibrium position, differential equation

**For citation: **Lipasov P. P., Schennikov V. N. Stability with Respect to a Part of Variables under Constant Perturbations of the Partial Equilibrium Position of Differential Equation Nonlinear Systems. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(3):344‒351. DOI: https://doi.org/10.15507/0236-2910.028.201803.344-351

**Authors’ contribution: **P. P. Lipasov – proof of theorem; V. N. Shchennikov – introduction of the research task, the definition of research methods.

All authors have read and approved the final version of the paper.

Received 18.04.2018; revised 07.06.2018; published online 20.09.2018

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