Qualitative theory of impulsive delay differential equations

Abstract

Systems of impulsive delay differential equations are considered and the qualitative theory of these equations are developed. Delay differential equations (without impulses) and impulsive differential equations (without delays) are first discussed and these are then combined to yield impulsive delay differential equations. These more general systems can suitably model evolutionary processes that exhibit both delay and impulse characteristics. After formulating the initial value problem for these systems and defining the notion of a solution, theorems establishing certain fundamental properties of solutions are developed. Specifically, theorems on local and global existence, uniqueness, continuability, and continuous dependence of solutions are presented. Subsequently, a variety of stability and boundedness results are obtained for the case when impulses occur at fixed times. Lyapunov functionals and functions are the main tools used to obtain stability and boundedness results. Finally, a number of examples and applications are presented to help motivate the study of these equations.