Research On Controllability And Stability Of Linear Time-varying Impulsive Systems

Impulsive system is a hybrid dynamic system which combines continuous process and jumping in the state. It has been found many applications in physics, pharmacodynamics, ecosystem, automatic control etc. The impulsive controllability and exponential stability of the linear time-varying impulsive system are studied in this thesis.The thesis is organized as follows:In chapter1, the background and the recent development of time-varying impulsive system are introduced. Then the main research work of the thesis is summarized.In chapter2, the impulsive controllability of a class of linear time-varying system is considered. First, based on the state transition matrix of the time-varying system, necessary and sufficient conditions for impulsive controllability of linear time-varying system are obtained. Then, the frozen coefficient method combined with the contraction mapping principle is applied to investigate the impulsive controllability of a class of bounded time-varying system. An algebraic criterion for impulsive controllability is established. Finally, an example is provided to illustrate the effectiveness of our results.In chapter3, the exponential stability of a class of impulsive time-varying systems is studied. The frozen coefficient method combined with the Gronwall inequality is applied to obtain an algebraic criterion for exponential stability of a class of impulsive time-varying system. Finally, numerical example shows that the obtained results can be applied to stability analysis of linear time-varying impulsive in system which both of continuous and discrete subsystems are not stable.In chapter4, the exponential stability of a class of discrete-time time-varying impulsive system is discussed. The frozen coefficient method combined with the discrete Gronwall sum-difference inequality is applied to investigate the stability problem of discrete-time impulsive time-varying systems. Finally, the validity of our results is illustrated in a numerical example.Finally, some summaries and several problems that are valuable for further studied are given.