抽象空间的线性微分方程及线性算子半群无界扰动的若干问题  

Some Problems of Linear Differential Equations on Abstract Spaces and Unbounded Perturbations of Linear Operator Semigroup

作　　者：许跟起 XU Genqi(School of Mathematics,Tianjin University,Tianjin,300350,P.R.China)

机构地区：[1]天津大学数学学院

出　　处：《数学进展》2019年第6期641-666,共26页Advances in Mathematics（China）

基　　金：国家自然科学基金(No.61773277)

摘　　要：本文对线性分布参数系统的研究进展进行综述.首先,我们列举带有无界控制算子和无界观测算子的L^p线性系统研究前沿中产生的若干问题,例如和镇定问题与状态反馈算子密切相关的观测理论.然后,我们简述一般Banach空间中线性微分方程可解性和半群扰动有关问题的若干结果,并提出系统(A,B)的可容许状态反馈算子的一般性概念.最后,我们对L^p适定系统给出可容许状态反馈算子的一个存在性结果,其中包括半群生成以及可容许状态反馈算子的等价条件.This paper is a survey for development of linear distributed parameter system.At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator,such as stabilization problem and observer theory that are closely relevant to state feedback operator.After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations.As a principle,we propose a concept of admissible state feedback operator for system(A,B).Finally we give an exis tence result of admissible state feedback opera tors,including semigroup generation and the equivalent conditions of admissibility of state feedback operators,for an L^p well-posed system.

关 键 词：线性微分方程 半群扰动 L^p系统 可容许状态反馈算子 

分 类 号：O231.4[理学—运筹学与控制论]