On stability in terms of two measures of a hybrid system having partly invisible solutions

We consider three different dynamic systems. The first runs “smoothly” during a certain finite time interval, undergoes an abrupt change in the dynamics during the next (finite) time interval and is governed by the second system. The solution of this second system lies on a surface for a finite amount of time and becomes invisible. At the beginning of the third phase, the system is subjected to an impulse which causes the solution to leave the surface and we have the new hybrid impulsive system. In this paper, we employ two measures to find suitable conditions so that the new system again runs as smoothly as the first.