On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems

Gladilina, R. I.; Ignat’ev, A. O.

doi:10.1023/b:ukma.0000010757.22157.1a

Cited by 13 publications

(9 citation statements)

References 5 publications

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“…Necessary conditions for the stability of solutions of systems with pulse action at fixed times were obtained in [6][7][8][9]. In the present paper, we establish necessary conditions for the stability of impulsive systems of a more general form, namely, systems with pulse action at nonfixed times.…”

Section: Introductionmentioning

confidence: 99%

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

Gladilina

2011

Nonlinear Oscill

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Section: Introductionmentioning

confidence: 99%

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

Gladilina

2011

Nonlinear Oscill

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“…The systematic description of the theory of difference equations one can find in books [2,7,16]. Difference equations are a convenient model for discrete dynamic systems description and for mathematical simulation of systems with impulse effect [8,12,14,15,19]. One of directions arising from applications of difference equations is linked with qualitative investigation of their solutions (stability, boundedness, controllability, observability, oscillation, robustness.…”

Section: Introductionmentioning

confidence: 99%

On the stability in periodic and almost periodic difference systems

Ignatyev

2006

Journal of Mathematical Analysis and Applications

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“…One should mention the results of [20], where, in addition to sufficient conditions, necessary conditions were established for the stability of solutions of an impulsive system.…”

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Some modern aspects of the theory of impulsive differential equations

Perestyuk

Chernikova

2008

Ukr Math J

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We give a brief survey of the main results obtained in recent years in the theory of impulsive differential equations.The theory of impulsive differential equations is one of the most important branches of the contemporary theory of differential equations. The formation and development of the theory of pulse systems are closely connected with studies of the Kyiv scientific school of nonlinear mechanics (see, e.g., [1,2] and the survey [3]). One of the first such studies was the investigation of oscillations of a pendulum subjected to a pulse influence [4]. The ideas of [4] were further developed by Yu. A. Mitropol'skii, A. M. Samoilenko, and their disciples [1, 2, 5 -7]. The results obtained in these works drew the attention of experts all over the world and stimulated the further comprehensive development of the theory of impulsive differential equations.In recent years, there has been the growth of interest in various aspects of the theory of pulse systems, and the expansion of the circle of problems considered has been observed. Many works published in recent years are devoted to the problems of stability and various "stability-like" properties of solutions and sets (integral, invariant, etc.) for various classes of impulsive systems. The objects of investigation are systems of ordinary differential equations with pulse influence, systems of equations with delay and pulse influence, singularly perturbed differential equations with pulse influence, equations with random pulse influence, impulsive differential equations in Banach spaces, etc. The main methods for the investigation of stability are the direct Lyapunov method extended to the case of impulsive systems [1, 2], the method of comparison [8], and a combination of the methods indicated. The analysis of the properties and behavior of solutions of impulsive systems often involves the method of integral inequalities, in particular, analogs of the well-known Gronwall -Bellman and Bihari inequalities for piecewise-continuous functions (see, e.g., [1, 2]). A separate series of works is devoted to the investigation of boundary-value problems for impulsive systems of differential equations; the results of these studies were published in [9,10].In the present work, we give a survey of some results obtained in recent years in the theory of impulsive systems. We preserve the notation accepted by the authors of the publications mentioned above.As noted above, numerous works in the theory of impulsive systems are devoted to problems related to the stability of solutions for various classes of impulsive systems. 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 equatio...

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On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems

Cited by 13 publications

References 5 publications

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

On the stability in periodic and almost periodic difference systems

Some modern aspects of the theory of impulsive differential equations

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