GE-半群法探讨奇异分布参数系统的弱稳定性与渐近指数稳定性

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本文主要探讨了奇异分布参数系统的稳定性问题，特别关注于GE-半群（Generalized Operator Semigroup，简称GE-半群）的方法。GE-半群是一种广义的、在希尔伯特空间中的连续泛函分析和算子理论工具，它与经典的C0-半群（Strongly Continuous C0-semigroup）有显著区别，对于研究奇异型分布参数系统具有重要意义。奇异分布参数系统通常涉及非线性行为和无穷维动态，因此对它们的稳定性分析更为复杂。作者Ge Zhaoqiang来自西安交通大学应用数学系，通过GE-半群理论，探讨了弱稳定性、渐近稳定性以及指数稳定性这三个关键的概念。弱稳定性关注的是系统在小扰动下的表现，渐近稳定性则考察长时间行为是否收敛到一个不变的状态，而指数稳定性则是指系统响应随着时间的指数衰减，是控制理论中的理想状态。文章的核心内容围绕着如何利用GE-半群的性质给出关于奇异分布参数系统稳定性的一般性和必要条件，以及充分条件。这些条件不仅包括系统的内在结构，如算子的谱分析，还包括可能的边界条件和初始数据的影响。通过这些理论框架，研究者能够设计有效的控制策略并评估系统的稳健性。文中引用了Ge在2009年的研究成果，其中对奇异分布参数系统的指数稳定性进行了深入研究，进一步扩展了GE-半群方法在该领域的应用。此外，还可能提及了与Zhu合作的部分，表明这是一项基于团队协作的前沿研究，旨在推动奇异分布参数系统稳定性理论的深化和实践应用。这篇研究论文是奇异分布参数系统稳定性研究的重要文献，提供了GE-半群这一新颖工具在复杂系统稳定性分析中的应用基础，并可能为未来的控制系统设计和分析提供理论指导。

GE-Semigroup Methods for the Stability of the Singular

Distributed Parameter Systems

GE Zhaoqiang

Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, China

E-mail: gezqjd@mail.xjtu.edu.cn

Abstract: Weak, asymptotic and exponential stability of the singular distributed parameter system are discussed in the light of

GE–semigroup (i.e., generalized operator semigroup ), functional analysis and operator theory in Hilbert space. The necessary

and sufficient conditions concerning the weak, asymptotic and exponential stability of the singular distributed parameter system

are given.

Key Words: Singular distributed parameter system, stability, GE–semigroup, Hilbert space



1 Introduction

There are essential distinctions between GE

–semigroup

(i.e., strongly continuously generalized operator semigroup)

and C

-semigroup [1-21]. GE

–semigroup is very important

for studying the singular distributed parameter systems

[14-31]. In 2009, Ge [14] discussed the exponential stability

of the singular distributed parameter system by using the

–semigroup in Hilbert space under some conditions; Ge,

Zhu and Feng [15] studied the exact controllability of

singular distributed parameter control system by using the

theory of GE

–semigroup in Hilbert space. In 2010, Ge, Zhu

and Feng [16] studied the well-posedness of the singular

distributed parameter system via the theory of

–semigroup in Hilbert space; Ge [18] discussed the mild

solution and exponential stability of the time varying

singular distributed parameter system via functional

analysis and GE

– semigroup in Hilbert space; Li and Wang

[19] studied the exponential stabilizability for a class of the

singular distributed parameter control systems by using the

theory of GE

–semigroup in Hilbert space; Liu and Shi 20]

discussed the exponential stability of the time varying

singular distributed parameter system via functional

analysis and GE

–semigroup in Hilbert space; Li and Wang

[21] studied the linear quadratic opertimal control problem

for singular distributed parameter systems by using the

functional analysis and GE

–semigroup in Hilbert space. On

the bases of the above papers we study the stability of the

following singular distributed parameter system in Hilbert

space:

()

(), ( ) , 0

dx t

EAxtxsxts

, (1)

where are linear operators in Hilbert space , and

,EA

is bounded; .

(),xt x H

In section 2, first of all, preliminaries of the basic

concepts concerning the GE–semigroup are given; In

section 3, the weak and asymptotic stability of the

distributed parameter system are discussed via the GE

–

semigroup, functional analysis and operator theory in

This work is supported by National Natural Science Foundation (NNSF)

of China under Grant 61174081.

Hilbert space; In section 4, the exponentially stability

concerning the distributed parameter system are discussed

in the light of the GE

– semigroup, functional analysis and

operator theory in Hilbert space. This research is

theoretically important for studying the GE

–semigroup and

the stability of the singular distributed parameter control

systems.

In the following,

denotes the set of all complex

numbers, the set of all real numbers,

the separable

and infinite dimensional complex Hilbert space,

()

H the

set of all bounded linear operators on , the inner

H ,!

product on ,



the norm. For 1 ,pdf

([0, ); )

LHf {and(): () , [0, ),xt xt Htf

()

xt dt

f

2 Preliminaries

In this section, some basic concepts concerning the

GE–semigroup are given, which will be used in the next

sections.

We assume that the set

(,) { : ( ,) ( )

is a bounded operator on }

EA C R EA E A

UOOO



 

is not empty. This set is called the -resolvent set of

operator and the operator

(,)REA

is called

the -resolvent of operator .

Definition 1 [14,15,16] A one parameter family of

bounded linear operators

0:)( tttU on is called a

generalized operator semigroup, or GE–semigroup in short

induced by if the following condition holds

()

()()(),,Ut h UtEUh th 0 t

;

If is strongly continuous with respect to , i.e.,

)(tU

0tt

lim || ( ) (0) || 0,

Utx U x





for any

H ,

then is called a GE

)(tU

–semigroup.

According to [14,15,16], if is a GE

0:)( tttU

–

semigroup, then there exist contents

0and

!

such

that

() , 0

Ut Me t

3URFHHGLQJVRIWKHVW&KLQHVH&RQWURO&RQIHUHQFH

-XO\+HIHL&KLQD

1234

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