ABSTRACT
The theory of impulsive systems has been emerging as an important area of investigation in recent years [2], An impulsive system is, in essence, a combination of a differential system and a difference system, which describes both continuous evolutions and discrete events occurring in the model of a physical system. Stability theory employing discontinuous Lyapunov functions has recently been developed for such systems [3, 4, 5]. These stability results do not require a Lyapunov function, v, to have a negative definite derivative along trajectories of the system, but allow for cases where, for example, v increases during the continuous portion of the trajectory and then experiences a jump decrease at the impulses. Nonetheless, a condition on v is required to ensure that its growth is not too rapid. Such conditions are essentially placed in [5] on the discrete and continuous portions separately. In this paper, we shall seek conditions that are expressed in terms of the combined continuous and discrete portions of the system. The price to pay for this type of estimate is the necessity of placing a condition on the generalized second derivative of the Lyapunov function, v. However, this new approach is advantageous for the stability analysis of large scale impulsive systems.
