\(\mu\)-stability of infinite delay functional differential systems with impulsive effects. (English) Zbl 1266.34122
Consider the impulsive functional differential systems
\[
\begin{cases} x'(t) = f(t,x(\cdot)),\quad t\geq \tau,\;t\neq t_k ,\\ \Delta x |_{t=t_k} = x(t_k) -x(t{_k^-})= I_k (t_k, x_{t_k^-}),\quad k \in \mathbb{Z}_+,\\ x_{\sigma}= \phi(s), \qquad \alpha\leq s\leq 0, \end{cases}\tag{1}
\]
The authors prove criteria for \( \mu \)-stability by using the improved Razumikhin technique and Lyapunov functions for a class of impulsive differential systems with infinite delays under certain impulsive perturbations. It is shown that the new results impose weaker requirements on the Razumikhin condition and are less conservative than those presented in the literature. An interesting example is provided to illustrate the effectiveness and advantages of the proposed results.
The authors prove criteria for \( \mu \)-stability by using the improved Razumikhin technique and Lyapunov functions for a class of impulsive differential systems with infinite delays under certain impulsive perturbations. It is shown that the new results impose weaker requirements on the Razumikhin condition and are less conservative than those presented in the literature. An interesting example is provided to illustrate the effectiveness and advantages of the proposed results.
Reviewer: Haydar Akca (Abu Dhabi)
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
Keywords:
impulsive functional differential systems; infinite delays; \(\mu\)-stability; improved Razumikhin technique; Lyapunov functionsReferences:
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