Pulse Cycle System And Its Control. Infinite Dimensional Space

In engineering, physics, biology, auto-control and signal processing, there are many phenomena where period mingles with impulse. Usually, many of these situations can be described by impulsive periodic systems, thus, it is quite necessary to do research on impulsive periodic systems. Problem of single variable in above situations can be described by impulsive periodic systems in finite dimensional spaces, however, as for problem of multivariable, impulsive periodic systems in infinite dimensional spaces can be adopted to describe. While doing research on impulsive periodic systems, we always expect to maintain the periodic systems moving state through relative fast additional methods, or they may use impulsive perturbation methods to adjust periodic systems to reach expected aims. Hence, it is rather necessary to investigate the control of impulsive periodic systems.In this thesis, operator semigroup theory, optimal control theory of distributed parameter systems, and nonlinear functional analysis are adopted to systematically research impulsive periodic systems in infinite dimensional spaces, including linear impulsive periodic system, semi-linear impulsive periodic system, Voltterra type nonlinear integrodifferential impulsive periodic system, the corresponding impulsive periodic system with time-varying generating operators, as well as the robust control and optimal control of some impulsive periodic systems. The main results of this thesis are as follows:First, we construct impulsive evolution operator corresponding to homogeneous linear impulsive periodic system, discuss properties of the impulsive evolution operator, and introduce mild solution of the homogeneous linear impulsive periodic system. And the equivalence theorem, that is, the relationship between the existence of periodic solutions of the homogeneous linear impulsive periodic system and the fixed point of impulsive evolution operator is given. The compactness and exponential stability of the impulsive evolution operator are applied to study nonhomogeneous linear impulsive periodic system to prove the existence and stability of periodic solutions.Further, we discuss the semilinear impulsive periodic system and Voltterra type nonlinear integrodifferential impulsive periodic system. By constructing the proper Pioncare operator, we change the question of existence of periodic solutions into the question of fixed point of the Pioncare operator. In order to obtain the prior estimation, some new generalized Gronwall inequalities with mixed type integral operators are established. Banach fixed point theorem, Horn fixed point theorem, and Leary-Schauder fixed point theorem are adopted to prove the existence of periodic solutions.On the basis of systematical analysis, robust control of linear impulsive periodic system with parameter perturbations is discussed. Meanwhile, optimal control problems arising in systems governed by a class of impulsive periodic system is also discussed. The existence of periodic optimal controls is proved by virtue of analysis of compactness, exponential stability, and exponential stabilizability of semigroup.At last, we also discuss the corresponding impulsive periodic systems with time-varying generating operators, including (periodic variance) impulsive periodic system with time-varying generating operators, semilinear impulsive periodic system with time-varying generating operators, Voltterra type nonlinear integrodifferential impulsive periodic system with time-varying generating operators, and mixed type nonlinear integrodifferential impulsive periodic system with time-varying generating operators. As a result, some existence and stability of periodic solutions are obtained.This thesis builds up a foundation for the further research on impulsive periodic systems in infinite dimensional spaces and its control.