Optimal Control Problem Of Pulse System Based On Logical Judgmen

The object changing is often disturbed by the outside world,which makes the change greatly in a short time.It is called the phenomenon of pulse.This kind of mutation can be described by impulsive differential system.Sometimes people observe and judge the condition of objects at certain fixed time,and impose different interference according to different situations.For example,people decide whether to fish by observing whether the number of fish in the fish pond has reached a certain number.In mathematics,it can be described as impulsive differential system with logic judgment,which is of great significance to the study of optimal control of impulsive system with logic judgment with application background.In this paper,a kind of optimal control problems of impulsive differential controlled systems with given impulse time and logical judgment is studied.Existence theories and optimality conditions are established.Based on the matrix semi-tensor product method,the impulse expression with logic judgment is transferred into algebraic expression,so that the optimal control problem can be studied under a unified framework.The three kinds of impulsive differential controlled systems,namely linear controlled systems,non-linear controlled systems with linear control and nonlinear controlled systems,and these corresponding optimal control problems are considered.At first,existence theorems of solutions for controlled systems are established.Then,sufficient conditions of optimal control under the condition are given for the set of control values is compact.Further,optimality conditions are derived by using the variational method for the control value set U=R~m or U being open and by introducing needle variational for the control value setbeing closed set.Finally,application examples are given to demonstrate the theoretical results.