The Optimal Control Problem Of A Class Of Semi-linear Differential Systems And The Disturbance Of The Outer Inverse Of Linear Operators

This paper is divided into two parts:the first part is the optimal control problem of a class of semilinear differential systems;the second part is the simplest expression on the perturbation of outer inverse of linear operators in Banach spaces.First of all,the theory of nonlinear differential equations in Banach space is one of the important branch of functional analysis,which is applied to many fields,such as control management,modern physics,engineering and optimization theory.For example,when we deal with the evolutional partial differential equations,we usually discuss their related problems indirectly by transforming them into differential equations in abstract space with the relevant properties and the theroy of differential equations.At the same time,in the theory of control,the controllability and the problem of optimal control are usually solved by converting them into differential equations.Therefore,this article has the great theoretical significance and research value to discuss the existence and optimal control about the solutions of nonlinear differential equations.Secondly,the stability of perturbation and the simplest expression for outer inverses of linear operators expand and deepen the research theory of generalized inverse,which were founded its extensive applications in optimization problems,mathematical statistics,differential equations and applied mathematics.In Banach spaces,every outer inverse of nonzero bounded linear operator always exists.Furthermore,the pertubation of outer inverses is stable.This article mainly discusses the stability of generalized inverse,Moore-Penrose inverse,Drazin inverse and group inverse of the bounded linear operator after perturbation,then give the simplest expressions,which are based on the stable perturbation of outer inverse.In the first part of paper,we discuss the class of impulsive evolution equations with control items in Banach space,we use the theory of operator semigroup,the measure of noncompactness,approximation and fixed point theory to discuss the related consequences.Firstly,we deal with the existence of the nonlinear impulsive equations,then we choose an optimal strategy by constructing two kinds of minimizing sequences of approximating solutions to obtain the optimal solution.In the end,an example demonstrate the feasibility of our results.In the second part,according to the stable pertubation of outer inverses of the linear bounded operator and the space decomposition,we study the necessary and suffcient conditions of the simplest expression B=T{2}(1+?TT{2})-1 for the perturbation of operator T=T+?T to be the generalized inverse,Moore-Penrose inverse,group inverse and Drazin inverse.The results in this paper extend and improve the conclusions in[32,36,40,41,42,54]...