  We prove that impulsive systems, which possess an ISS Lyapunov function, are
ISS for time sequences satisfying the fixed dwell-time condition. If an ISS
Lyapunov function is the exponential one, we provide a stronger result, which
guarantees uniform ISS of the whole system over sequences satisfying the
generalized average dwell-time condition. Then we prove two small-gain theorems
that provide a construction of an ISS Lyapunov function for an interconnection
of impulsive systems, if the ISS-Lyapunov functions for subsystems are known.
The construction of local ISS Lyapunov functions via linearization method is
provided. Relations between small-gain and dwell-time conditions as well as
between different types of dwell-time conditions are also investigated.
Although our results are novel already in the context of finite-dimensional
systems, we prove them for systems based on differential equations in Banach
spaces that makes obtained results considerably more general.
