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Zbl 0989.34061
Liu, Xinzhi; Ballinger, G.
Uniform asymptotic stability of impulsive delay differential equations.
(English)
[J] Comput. Math. Appl. 41, No.7-8, 903-915 (2001). ISSN 0898-1221

The paper has been written by famous scientists in the area of differental equations with impulsive effect. The system $$\frac{dx(t)}{dt}=f(t,x_t), \qquad t\ne\tau_k,\tag a$$ $$\Delta x(t)=I(t,x_{t^-}),\qquad t=\tau_k,\tag b$$ is considered. Here, $t\in \bbfR_+$, $0=\tau_0<\tau_1<\tau_2<\dots$, $\lim_{k\to\infty}\tau_k=+\infty$, $\Delta x(t)=x(t)-x(t^-)$, $x(t^-)=\lim_{s\to t^-}x(s)$; (a) is a system of functional-differential equations with delay. It is assumed that $f(t,0)\equiv 0$, $I(\tau_k,0)=0$ for all $\tau_k\in \bbfR_+$, and the system (a), (b) possesses a trivial (zero) solution $x(t)\equiv 0$. Two theorems on the uniform asymptotic stability of the trivial solution to (a), (b) are proved by means of Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the system (a) does not enjoy any stability behavior. The theorems are illustrated by some examples.
[Alexander Olegovich Ignatyev (Donetsk)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K45 Equations with impulses
34A37 Differential equations with impulses

Keywords: impulsive delay differential equations; stability

Cited in: Zbl 1136.93037

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