Research On Existence And Controllability Of Mild Solutions For Impulsive Differential Inclusions With Delay

Impulsive differential equation describe the system which occurrs a rapid change in some stage. The rapid process spends very little, some effects the whole syetem. Since the 1980s of the 20th century, Impulsive differential system became familiar by some experts in applied mathematics, control theory, engineering technology, information technology, medical, have developed rapidly in theory and applications. Impulsive differential equation is a typical hybrid system, which combines the features of continuous and discrete systems, but goes beyond the scope of continuous and discrete systems. So, it is the challenge and attractive field of research. Such hybrid system has important applications in space technology, information science, control systems, communications, life sciences, medicine, the economy.When the system is impossible to describe precisely, or has diversity, not accurately described by differential equation model, differential inclusions depicts a differential system. Since the 50-60s of the 20th century, the research of control system caused it rapid development. Differential inclusions is playing an increasingly important role in diffential equations whose non-contiunous right cases, the non-unique value of the non-linear control systems, adaptive control theory, economic dynamic systems.Impulsive differential equation and differential inclusions are two new areas of reasearch. They have differential practical applications in different backgrounds. The research combining impulsive differential equation and differential inclusions will be necessary and very meaningful. In recent years, the researches on impulsive differential inclusions are only some explorative work. Impulsive differential inclusions can not only describe the process which rapid changes have taken place, but also itself be uncertainty, diversity and complexity, and contains the characteristics of these two systems. Therefore, impulsive differential inclusions contains much more complex than these two systems.The existence of solutions of impulsive differential inclusions is the primary basis of other property of the system. All kinds of problems containing stability and attractive of the system are discussed throught the existence of solutions. Meanwhile, Controllablity of impulsive differential inclusions in the research of control system is the fundamental issues. The main ideas and significance of this thesis is that here.Based on the above considerations, the existence and controllablity of solutions for impulsive differential inclusions with delay will be discussed in this thesis. This thesis focuses on research of non-resonance problems of n-order impulsive differential inclusions, the existence of solutions for impulsive neutral integrodifferential inlcusions, the existence of solutions for impulsive differential inlcusions on half-line, the controllability for impulsive stochastic evolution differential inclusions, the approximate controllability for impulsive stochastic differential inclusions. Main results are follows:The non-resonance problems of n-order impulsive differential inclusions. When the multivalued functions satisfy Lipschitz and Carathéodory conditions, the existence of mild solutions for n-order impulsive differential inclusions with delay is obtained by using recent mixed multivalued alternative theorem due to Dhage. A simple impulsive differential equation and its simulation is gave to illustrate the result.The existence of mild solutions for impulsive evolution neutral, second-order neutral integrodifferential inclusions with infinity delay is discussed. At first, their phase space are established by using axiomatic definition method. Secondly, the compactness of the operator is dropped out. At last, the results of the existence are proved basing on mixed multivalued alternative theorems of contration and completely continuous functions. These results improve the corresponding ones existed.The existence of mild solutions for impulsive evolution, neutral differential inclusions with infinity delay on half-line is discussed. At first, their phase space are established by using axiomatic definition method. To impulsive evolution differential inclusions, the compactness of the evolution system is dropped out. The multivalued functions are condensing operators by using non-compactness technique. To impulsive neutral differential inclusions, when linear operator is analytic semigroup, The multivalued function is condensing operators when it is the sum of the contration and completely continuous functions. The existence is proved under Leray-Schauder multivalued alternative theorems in Fréchet space. These method is better than the corresponding ones existed. The controllability of mild solutions for impulsive stochastic evolution neutral differential inclusions with infinity delay is considered. At first, its phase space is established by using axiomatic definition method in Hilbert space. Secondly, the compactness of evolution system is denied. The function which is the sum of the contration and completely continuous functions is condensing operator. At last, the result of the controllability is proved throught Leray-Schauder multivalued alternative theorems in Hilbert space. The result generanizes the conditions of the corresponding ones existed.The approximate controllability of mild solutions for non-densing defined impulsive stochastic differential inclusions with infinity delay is investigated. At first, its phase space is established by using axiomatic definition method in Hilbert space. Secondly, The function is divided the contration and completely continuous functions. At last, the result of the approximate controllability is obtained throught mixed multivalued alternative theorems in Hilbert space. The result makes corrections some existing ones in relevant literatures.