  In this paper, tools to study forward invariance properties with robustness
to dis- turbances, referred to as robust forward invariance, are proposed for
hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are
given in terms of dif- ferential and difference inclusions with state and
disturbance constraints, for whose definition only four objects are required.
The proposed robust forward invariance notions allow for the diverse type of
solutions to such systems (with and without dis- turbances), including
solutions that have persistent flows and jumps, that are Zeno, and that stop to
exist after finite amount of (hybrid) time. Sufficient conditions for sets to
enjoy such properties are presented. These conditions are given in terms of the
objects defining the hybrid inclusions and the set to be rendered robust
forward invariant. In addition, as special cases, these conditions are
exploited to state results on nominal forward invariance for hybrid systems
without disturbances. Furthermore, results that provide conditions to render
the sublevel sets of Lyapunov-like functions forward invariant are established.
Analysis of a controlled inverter system is presented as an application of our
results. Academic examples are given throughout the paper to illustrate the
main ideas.
