Nonlinear impulsive dynamical systems. I. Stability and dissipativity

We develop Lyapunov and invariant set stability theorems for nonlinear impulsive dynamical systems. Furthermore, we generalize the dissipativity theory to nonlinear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics, and dissipated energy over the resetting instances. Furthermore, extended Kalman-Yakubovich-Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness in terms of system storage functions are derived. Finally, the framework is specialized to passive and nonexpansive impulsive systems to provide a generalization of the classical notions of passivity and nonexpansivity.