Preasymptotic Stability and Homogeneous Approximations of Hybrid Dynamical Systems

Hybrid dynamical systems are systems that combine features of continuous-time dynamical systems and discrete-time dynamical systems, and can be modeled by a combination of differential equations or inclusions, difference equations or inclusions, and constraints. Preasymptotic stability is a concept that results from separating the conditions that asymptotic stability places on the behavior of solutions from issues related to existence of solutions. In this paper, techniques for approximating hybrid dynamical systems that generalize classical linearization techniques are proposed. The approximation techniques involve linearization, tangent cones, homogeneous approximations of functions and set-valued mappings, and tangent homogeneous cones, where homogeneity is considered with respect to general dilations. The main results deduce preasymptotic stability of an equilibrium point for a hybrid dynamical system from preasymptotic stability of the equilibrium point for an approximate system. Further results relate the degree of homogeneity of a hybrid system to the Zeno phenomenon that can appear in the solutions of the system.