Impulsive Functional Differential Stability Theory

Such systems are very complicated than the systerms with only impulses or delays ,we will use Lyapunov-Razurninkhin functions and Lypunov functionals to obtain some results of the systerm,The paper are divided into three parts in the first part ,we obtain the sufficient condions for the stability in terms of two measures of such systerm by means of Lyapunov-Razuminkhin functions and Lypunov functionals. In the second part, sufficient condions for the boundedness theorems are considered.In the last part,a new approach to stability theory of such systerm is proposed. Instead of putting all components of state variable x in one Pazumikhin-Lyapunov functions,several functions of partial components of x ,which can be much easier constructed,are used so that the conditions ensuring that stability are simple and less restrictive .Also, an example is given to illustrate the advantages of the obtained results.