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Input-to-state stability of impulsive and switching hybrid systems with time-delay. (English) Zbl 1233.93083
Summary: This paper investigates Input-to-State Stability (ISS) and Integral Input-to-State stability (IISS) of impulsive and switching hybrid systems with time-delay, using the method of multiple Lyapunov-Krasovskii functionals. It is shown that, even if all the subsystems governing the continuous dynamics, in the absence of impulses, are not ISS/IISS, impulses can successfully stabilize the system in the ISS/iISS sense, provided that there are no overly long intervals between impulses, i.e., the impulsive and switching signal satisfies a dwell-time upper bound condition. Moreover, these impulsive ISS/IISS stabilization results can be applied to systems with arbitrarily large time-delays. Conversely, in the case when all the subsystems governing the continuous dynamics are ISS/IISS in the absence of impulses, the ISS/IISS properties can be retained if the impulses and switching do not occur too frequently, i.e., the impulsive and switching signal satisfies a dwell-time lower bound condition. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.
##### MSC:
 93D25 Input-output approaches to stability of control systems 93C30 Control systems governed by other functional relations 93C15 Control systems governed by ODE