Finite-gain LpLp stability for hybrid dynamical systems

We characterize finite gain Lp stability properties for hybrid dynamical systems. By defining a suitable concept of hybrid Lp norm, we introduce hybrid storage functions and provide sufficient Lyapunov conditions for Lp stability of hybrid systems, which cover the well-known continuous-time and discrete-time Lp stability notions as special cases. We then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin, global exponential stability, existence of a homogeneous Lyapunov function with suitable properties for the hybrid system with no inputs, and input-to-state stability, and we show how these properties all imply Lp stability. Finally we characterize systems with direct and reverse average dwell time properties and establish parallel results for this class of systems. We also make several connections to the existing results on dissipativity properties of hybrid dynamical systems.