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Dissipativity theory and stability of feedback interconnections for hybrid dynamical systems

We develop a unified framework for a general class of dynamical systems possessing left-continuous flows; that is, left-continuous dynamical systems. These systems are shown to generalize virtually all existing notions of dynamical systems and include hybrid and impulsive dynamical systems as special cases. Furthermore, we generalize dissipativity theory to left-continuous dynamical systems. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback interconnections of left-continuous systems. These results generalize the positivity and small gain theorems to the case of left-continuous, hybrid, and impulsive dynamical systems.