Singularly impulsive dynamical systems with time delay: Mathematical model and stability

In this paper we introduce new class of system, so called singularly impulsive or generalized impulsive dynamical systems with time delay. Dynamics of this system is characterized by the set of differential and difference equations with time delay, and algebraic equations. They represent the class of hybrid systems, where algebraic equations represent constraints that differential and difference equations with time delay need to satisfy. In this paper we present model, assumptions on the model, two classes of singularly impulsive dynamical systems with delay - time dependent and state dependent. Further, we present Lyapunov and asymptotic stability theorems for nonlinear time-dependent and state-dependent singularly impulsive dynamical systems with time delay.