Impulsive differential equations describe processes with dynamics of differential equations combined with quick impacts caused by indefinite events. This circle of problems plays an important role in real life since in any design, there are unexpected events. Stochastic differential equations discuss these problems among others in modelling statistical mechanics. It was Barnov and his associates who first formulated the deterministic model of the impact of impulse effects on systems.

Barnov’s initiatives had a strong influence on the research direction in all the countries that embraced the concept. Consequently, some fundamental aspects of research areas which play important role in such applications as the treatment of delay equations and some other fields missed certain valuable points. There is need to develop a new formulation of the problem that contains the whole Barnovian theory and addresses the circle of problems missing in Barnov’s approach. These changes among others include:

 

·         The addition of impulse effects on the dynamics of the system like impacts of impulses on the engine;

·         Uniform treatment for delay and non-delay impulsive differential equations;

·         Extension of Barnov’s coordinate space model of impulses to phase state model where impulse may affect both the coordinates and the impulse momentum.

 

It is worthy of note that the framework of functions of bounded variation gives a chance to build together the coordinate space impulses with the impulses on the system dynamics into a consistent system. The approach is expected to offer a unified frame to handle delay/non-delay problems with the ease similar to what happens in the theory of conventional differential equations. My study aims at the following:

 

·         Precise formulation of the impulsive model with space state and system impulses;

·         Formulation of condition for the existence of solutions and basic qualitative analysis of the solutions; and finally

·         The study of asymptotic behaviour of delay and non-delay systems.