Dynamics Analysis Of Impulsive Functional Differential Systems With P-delay With Impulses At Variable Times

In this paper, we study the stability and boundedness for impulsive functional differential systems with p-delay with impulses at variable times as follows The hysteresis phenomena and impulse phenomena, exist in widespread area of the modern science and technology. The mathematical models of these practical problems often come down to the delay impulsive functional differential systems. The most dis-tinguishing feature of delay impulsive functional differential systems is able to reflect the laws of things more accurately and deeply by considering the influence of the phenom-ena of hysteresis and transient process. These systems have been widely applied into control field, aerospace technology, information science, communication, biotechnology,medical science, economics etc. Impulsive functional differential systems with p-delay,as one of delay impulsive functional differential systems, are worthy of studying whether these have important theoretical value or practical value. As far as I know,the results of impulsive functional differential systems with p-delay mostly focus on the functional differential systems with fixed moments of impulsive effects and few take the functional differential systems with p-delay with variable times of impulsive effects into account. In this paper, we use the tool for Lyapunov functions coupled with Razumikhin technique to investigate the dynamic properties for system (?) with pulse, such as stability and boundedness. This paper is divided into three parts.In chapter one, we mainly study the stability of system (I). In section 3, we estab-lish a new comparison principle between impulsive functional differential systems with p-delay and ordinary differential systems by vector Lyapunov functions and the corre-sponding differential inequalities, from which we get the comparison criteria on stability in terms of two measures of system (?). It should be noticed that finite collision phe-nomena can be allowed in the comparison principle of this section. In section 4, we first use the tool for Lyapunov functions coupled with Razumikhin technique to get the direct results of uniform stability in terms of two measures of system (?). Then we gain the direct results of uniform stability of zero solution of system (I) by the method of several Lyapunov functions of partial components coupled with Razumithin technique. Finally,we give an example to illustrate the effectiveness of our results. It should be noticed that the pulse phenomena can be allowed in the paper,but the beating number should be finite.In chapter two, we mainly study the boundedness of system (?). In section 3, we get the comparison criteria on boundedness in terms of two measures of system (I) ac-cording to the comparison principle in section 3 of chapter one. In section 4, we use the tool for Lyapunov functions coupled with Razumikhin technique to get the direct results of uniform boundedness in terms of two measures of system (?). The derivative of V function along trajectories of system (I) is no longer required to be nonpositive or negative definite and can be weakened to be positive in Theorem 2.4.1 and Theorem 2.4.2. V function can be weakened to be nondecreasing or increasing at impulsive times in Theorem 2.4.3 and Theorem 2.4.4. Finally we give an example to illustrate the prac-ticability of our results.It should be noticed that the pulse phenomena can be allowed in the paper,but the beating number should be finite.In chapter three, we mainly study the Lagrange stability of system (?). In section 3,we get the comparison criteria on Lagrange stability in terms of two measures of system(?) according to the comparison principle in section 3 of chapter one. In section 4, we gain the direct results of uniform Lagrange stability of zero solution of system (?) by the method of several Lyapunov functions of partial components coupled with Razumithin technique. Some Lyapunov functions are adopted, where every Lyapunov function sat-isfies weaker conditions and is easier to be constructed. Finally we give an example to illustrate the applicability of our results.It should be noticed that the pulse phenomena can be allowed in the paper,but the beating number should be finite.