Stability in terms of two measures of hybrid systems with partially visible solutions

A smooth functioning system with some stability properties may break down abruptly due to unexpected disturbances and the entire mechanism may collapse. The system, in general, is made operable in a phased manner and after a finite time may again run smoothly, with some other stability properties. We explore this whole mechanism as a hybrid system and study the stability properties in terms of two measures so as to cater to a very general set up.