ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 6, pp. 21–26.
Allerton Press, Inc., 2018.
Original Russian Text
V.I. Kachalov, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 6, pp. 25–30.
On One Method of Solving Singularly Perturbed Systems
of Tikhonov’s Type
V. I. Kacha lov
National Research University “Moscow Power Engineering Insitute”
ul. Krasnokazarmennaya 14, Moscow, 111250 Russia
Received April 7, 2017
Abstract—To investigate Tikhonov systems arising in problems of control theory we apply the
method of holomorphic regularization, which allows one to obtain solutions to singularly perturbed
problems in the form of series converging in the usual sense in powers of a small parameter.
Keywords: Tikhonov system, holomorphic regularization, homomorphism, pseudoholomor-
Singularly perturbed system arise with constructing mathematical models of many applied problems
of Mechatronics, Theory of Control, Hydrodynamics, Chemical Kinetics, Economical and Social
Dynamics etc. [1–5]. A particular place among them is occupied by systems of the Tikhonov type
[2, 6], which contain both slow and quick variables. Firstly, it is connected with the fact that for
such systems the Tikhonov theorems about limit transition take place, and there are well elaborated
methods of their asymptotic integrating [6, 7]. Here we propose an approach based on an assertion
that any singularly perturbed system has holomorphic by a small parameter integrals, under suﬃciently
general assumptions [5, 8]. In these conditions, using the theorem about an implicit function, one can
obtain so-called pseudo-holomorphic solutions of these systems [9, 10]. The elaborated method can be
applied, using symbol calculations on computers, for constructing approximate analytic descriptions of
families of solutions to nonlinear singularly perturbed problems, that allows one to realize procedures
of parametric synthesis of motion trajectories in real time, to conduct much more economically the
investigation of parametric stability and robustness of obtained solutions in comparison with traditional
1. INTEGRALS OF SINGULARLY PERTURBED SYSTEMS HOLOMORPHIC WITH
RESPECT TO A SMALL PARAMETER
Let us consider on a segment [0,T] an initial problem for the Tikhonov system with one quick variable
= f(t, y, v),ε
= F (t, y, v), (1)
Here ε>0 is a small parameter; y =(y
), f =(f
); F , v are scalar
functions. We assume that y
are independent of ε.
With ε =0we obtain a degenerate system
y, v), 0=F(t, y, v). (3)