Stability of linear impulsive differential equations

Milev, N. V.; Байнов, Д. Д.

doi:10.1080/00207729008910543

Cited by 7 publications

(5 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Problems of preservation of stability of solutions under perturbation of a linear homogeneous pulse system of the form (2) were also studied in [24]. Considering along with system (2) the perturbed system…”

Section: Theorem 5 [6] Let the Matrices A And B Be Commuting Let Thmentioning

confidence: 99%

“…(2) Related to the results of [24] are results of the paper [26] where relations between the notions of exponential stability of a system of the form (2) and equiasymptotic stability, uniform asymptotic stability and uniform exponential stability, weak exponential stability of system (2) with respect to a space of solutions Lk, and exponential stability of system (2) were investigated.…”

Section: Dy = (A(t)+ A(t))y T~t I Drmentioning

confidence: 99%

“…where ,~(t) is a piecewise continuous matrix, Bi are constant matrices, i = 1, 2 ..... and using Propositions 2,3 and an analogue of the GronwalI-Bellman lemma for piecewise continuous functions [4][5][6], the authors of [24] obtained the following conditions of preservation of stability for the linear homogeneous pulse differential system under perturbations. (2) Related to the results of [24] are results of the paper [26] where relations between the notions of exponential stability of a system of the form (2) and equiasymptotic stability, uniform asymptotic stability and uniform exponential stability, weak exponential stability of system (2) with respect to a space of solutions Lk, and exponential stability of system (2) were investigated.…”

Section: Dy = (A(t)+ A(t))y T~t I Drmentioning

confidence: 99%

“…The definitions for the pulse system (2) do not differ from corresponding dx definitions for the system --= A (t)x. For the study of stability of pulse linear systems of differential equations, the dt following definitions in another form are also useful [24,25]:…”

Section: Stability Of Linear Systems Of Differential Equations With Pmentioning

confidence: 99%

See 3 more Smart Citations

Stability of solutions of pulsed systems

Perestyuk

Chernikova²

1997

Ukr Math J

View full text Add to dashboard Cite

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

show abstract

Section: Theorem 5 [6] Let the Matrices A And B Be Commuting Let Thmentioning

confidence: 99%

Section: Dy = (A(t)+ A(t))y T~t I Drmentioning

confidence: 99%

Section: Dy = (A(t)+ A(t))y T~t I Drmentioning

confidence: 99%

Section: Stability Of Linear Systems Of Differential Equations With Pmentioning

confidence: 99%

See 2 more Smart Citations

Stability of solutions of pulsed systems

Perestyuk

Chernikova²

1997

Ukr Math J

View full text Add to dashboard Cite

show abstract

“…We finally get the required integral equation ( 4) In [3] a precise definition of stability for systerns of linear dilfercnt.ial equations with impulses al. fixed tintes is given.…”

Section: U(t) =U( A)+ [ 1/-l (S)f(s¡f;(s)u(s))dsmentioning

confidence: 99%