Stability of solutions of pulsed systems

Abstract

We present the principai results in the theory of stability of pulse differential equations obtained by mathematicians of the Kiev scientific school of nonlinear mechanics. We also present some results of foreign authors.In the last I5-20 years, the theory of differential equations with pulse influence has been extensively developed. Pulse differential equations appear in various problems of nonlinear mechanics; pulse differential equations are equations that provide an adequate description for real processes in systems subjected to the action of instantaneous perturbations. The theory of pulse differential equations is basically founded and developed by scientists from the Kiev school of nonlinear mechanics which is associated with the names of such prominent scientists as N. M. Ka-ylov, N. N. Bogolyubov, and Yu. A. Mitropol'skii. As early as in 1937, by using the averaging method, oscillations of a pendulum under pulsed influence were studied in [1]. Later, Yu. A. Mitropol'skii, A. M. Samoilenko, and N. A. Perestyuk developed the ideas of [1] and applied these ideas to a larger class of systems under pulsed influence (a detailed list of the investigated problems can be found in the survey [2]). The investigations of scientists from the Kiev school stimulated comprehensive systematic studies of pulse differential equations in many other scientific centers (see [3,4] for more details). As a separate direction of the theory of pulse differential equations, the theory of stability of pulse systems was formed. Fundamental results, associated with the investigation of the stability of solutions of differential equations with pulsed influence, were established by A. M. Samoilenko and N. A. Perestyuk. In [3][4][5][6][7][8][9][10][11][12], the classical theory of the first and second Lyapunov methods was extended to the case of pulse systems. Later, various results in the theory of stability of pulse systems were established by foreign scientists.It turned out that the method of integral manifolds, which was developed in the works of Yu. A. Mitropol'skii and his disciples for various classes of differential equations (see, e.g., [ 13]), is quite efficient for the investigation of stability of systems under pulsed influence.Below, we present results which, from our point of view, describe the contemporary state of the theory of stability of systems with pulsed influence.Let the following system of differential equations with pulsed influence be given dx --= f(t, x), t 7~ zi(x), dt

Some modern aspects of the theory of impulsive differential equations

“…The first deep systematic investigations in this field were carried out by mathematicians of the Kiev school of nonlinear mechanics. The results obtained in 1970-1980 by scientists of this school are well known and widely used by domestic and foreign mathematicians (for a brief survey of the corresponding works, see [3]). The method proposed in [1,2] for the investigation of the stability and asymptotic behavior of solutions of systems of differential equations turned out to be also efficient for studying problems related to the stability of sets.…”

“…and I x ( , ) ϕ satisfy condition (3) with sufficiently small a, then, for sufficiently small ε > 0, the trivial invariant torus x = 0, ϕ ∈ℑ m of system (2) is exponentially stable.…”

Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems

“…The results of the first articles on impulsive differential equations are gathered in the monograph [11] which presents the basics of this theory. Lately, we observe a significant increase of the number of mathematical articles on the various aspects of the theory of impulse systems [12][13][14][15][16][17] which is due to the demands of the modern technology. Simeonov and Bainov [18] stated the stability problem for solutions to impulsive systems with respect to part of variables and proved a series of theorems.…”

Gradient vector fields with pulse action on manifolds

“…The foundations of this theory were laid in the works of Krylov, Bogolyubov, and Mitropol'skii. A decisive contribution to the development of this theory was made by Samoilenko and his disciples [1,2]. At the same time, the investigation of vector fields with pulse action on smooth manifolds is at its initial stage.…”