Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods

Titelangaben

Dashkovskiy, Sergey ; Kosmykov, Michael ; Mironchenko, Andrii ; Naujok, Lars:
Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods.
In: Nonlinear Analysis : Hybrid Systems. Bd. 6 (2012) Heft 3 . - S. 899-915.
ISSN 1878-7460
DOI: https://doi.org/10.1016/j.nahs.2012.02.001

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Abstract

In this paper, we consider the input-to-state stability (ISS) of impulsive control systems with and without time delays. We prove that, if the time-delay system possesses an exponential Lyapunov–Razumikhin function or an exponential Lyapunov–Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time delays and prove that the whole network is uniformly ISS under the small-gain and the average dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems, a Lyapunov–Krasovskii functional or a Lyapunov–Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples.