Qualitative properties of coupled parabolic systems of evolution equations
ODE–热方程级联系统的事件触发控制
第40卷第8期2023年8月控制理论与应用Control Theory&ApplicationsV ol.40No.8Aug.2023 ODE–热方程级联系统的事件触发控制杨辉1,宗西举1,2†,郑泽阳1,徐秀妮3(1.济南大学自动化与电气工程学院,山东济南250022;2.济南大学信息科学与工程学院,山东济南250022;3.陇东学院电气工程学院,甘肃庆阳745000)摘要:本文针对常微分方程(ODE)耦合偏微分方程(PDE)建模的分布式参数多智能体系统进行研究,针对一致性同步问题,提出了事件触发的网络化ODE–热方程级联系统多智能体一致性边界交互协议.本文考虑的热方程左边界为Neumann边界条件,并且与ODE系统耦合,右边界为绝热边界条件.假设网络化多智能体系统的连接方式为全联通有向拓扑图,给出ODE–热方程级联系统的多智能体的一致性控制协议.另外针对现有数字式控制器,设计了事件触发的一致性控制协议,并利用李雅普诺夫函数验证了在事件触发条件下ODE–热方程级联系统的稳定性.最后给出了由5个ODE–热方程级联的多智能体系统的仿真结果,验证了事件触发控制器的有效性.关键词:ODE–热方程级联系统;有向拓扑图;多智能体系统;李雅普诺夫函数;事件触发控制器引用格式:杨辉,宗西举,郑泽阳,等.ODE–热方程级联系统的事件触发控制.控制理论与应用,2023,40(8): 1349–1356DOI:10.7641/CTA.2022.20285Event-triggered control of ode-heat equation cascade systemYANG Hui1,ZONG Xi-ju1,2†,ZHENG Ze-yang1,XU Xiu-ni3(1.School of Electrical Engineering,University of Jinan,Jinan Shandong250022,China;2.School of Information Science and Engineering,University of Jinan,Jinan Shandong250022,China;3.School of Electrical Engineering,Longdong University,Qingyang Gansu745000,China)Abstract:In this paper,a distributed parameter multi-agent system modeled by the ordinary differential equation(ODE) coupled with the partial differential equation(PDE)is studied,and an event-triggered networked ODE-heat equation cas-cade system with multi-agent consensus boundary interaction protocol are proposed.The left boundary of the heat equation considered in this paper is the Neumann boundary condition,which is coupled with the ODE system,and the right bound-ary is the adiabatic boundary condition.It is assumed that the connection mode of the networked multi-agent system is fully connected and directed the topology diagram,the multi-agent consensus control protocol of the ODE-heat equation cascade system is given.In addition,for the existing digital controller,event-triggered-based consensus control protocol is designed,and the Lyapunov function is used.The stability of the ODE-heat equation cascade system is verified under the event-triggered condition.Finally,the simulation results of the multi-agent system composed offive ODE-heat equation cascades are given,which verifies the effectiveness of the event-triggered controller.Key words:ODE-heat equation cascade system;directed topology;multi-agent system;Lyapunov function;event-triggered controllerCitation:YANG Hui,ZONG Xiju,ZHENG Zeyang,et al.Event-triggered control of ode-heat equation cascade system. Control Theory&Applications,2023,40(8):1349–13561引言随着传感器技术、人工智能技术和分布式网络的迅速崛起,多智能体系统(multi-agent systems,MAS)成为控制科学和人工智能等领域的研究热点,引起了广大学者的关注[1].智能体之间的信息交互通过网络拓扑结构进行,相对于单个个体(智能体)而言,多智能体系统具有更强的控制能力和协调能力,从而能够解决单个个体无法实现的功能,因此被广泛应用于电网经济调度优化、无人机编队联合侦查与搜索、卫星集群通信、智能交通与物流等领域.多智能体系统的协调控制研究包括多个方面,如编队控制、同步稳定、蜂拥控制等,而其中最基本的问题是多智能体系统的一收稿日期:2022−04−17;录用日期:2022−11−18.†通信作者.E-mail:cse**************.cn;Tel.:+86531-89736515.本文责任编委:郭宝珠.山东省高等学校青年创新科技计划项目(2019KJN029),国家自然科学基金项目(12026215)资助.Supported by the Youth Innovation and Technology Program of Shandong Province(2019KJN029)and the National Natural Science Foundation of China(12026215).1350控制理论与应用第40卷致性问题[2–6].一致性问题的研究成为各类企业提高生产率的关键,蓝宝石生产工艺便是其中之一.工业蓝宝石是氧化铝的晶体,是国民经济、国防工业和科学技术发展必不可少的基础材料和重要的战略物资[7].在蓝宝石结晶初期,融融状态的蓝宝石引晶点处温度检测和控制对后续整个结晶过程起到至关重要的作用,由于工业蓝宝石的熔点温度为2050◦C,所以目前不存在有效的自动温度检测手段与检测设备来获得如此高温,工程中现行的检测方法是利用一根细长的、外部包裹了透明合金材料的晶体深入加热炉内,并置细晶体棒的一端于熔融状态的蓝宝石中心上方2∼3cm,通过人眼不定期地观察晶体棒端点来近似估计熔融状态蓝宝石中心的温度.很多蓝宝石企业,为了提高成产效率,引入了多智能体系统.在工业生产中,将多台加热炉联接成网络,通过控制其中部分加热,利用加热炉之间的信息传递,可以实现自动化与网络化同步控制,既能节约成本又能提高生产效率,这就是本文要研究的网络化系统的状态一致性控制问题.目前,一方面由于实际工程中许多系统模型均不能质点化,并且一、二阶常微分方程(ordinary differen-tial equation,ODE)建模的多智能体系统相对完善,但基于偏微分方程(partial different equation,PDE)建模的多智能体系统的研究没有得到同样程度的重视,所以在这方面存在很大的挖掘空间[8–10].一般来说,利用ODE或PDE对一些系统建模都不能完美的接近实际的模型,但ODE与PDE相互耦合的级联系统却能完美的逼近实际系统[11–13].其中,由于大部分实际工程的需要,如水利、钻井、电加热炉等常见工业生产模型,其分布参数控制系统往往需要在边界耦合集中参数控制系统.并且大部分都是通过对边界进行控制.正如文献[14]研究了麦克斯韦方程与热方程的耦合;文献[15]用一类ODE与KDV(korteweg-de vries equa-tion)方程边界级联的形式,来简化存在机械振动的潜水波模型;文献[16]利用反映–扩散–对流方程对冶金固定床内部颗粒发生化学反应的动力学行为进行建模.所以本文选择ODE–热方程级联的方式对蓝宝石加热炉进行建模.另一方面,由于实际传感器测量速度的限制,网络带宽限制以及系统处理速度等各类因素的限制,显然连续时间控制器很难满足控制要求,而离散信号是很容易获取的,并且连续时间控制器的离散和采样已成为有限维多智能体系统的研究热点[5,9].虽然出现了许多优秀的控制方案,但触发频率较高,仍然可能会造成计算资源和能量的不必要损耗.为了解决上述问题,受到文献[10,17–18]等文章的启发,针对ODE–热方程建模的加热炉多智能体系统,为降低控制器的要求,本文提出了事件触发控制方案,并利用Lyapunov 函数验证在事件触发条件下该系统的稳定性.本文所考虑的热方程左边界为Neumann边界条件,与ODE系统耦合,右边界为绝热边界条件,采用这种方式对系统进行建模,更加符合实际工程,并且所有智能体通过全联通有向拓扑进行通信.设计了ODE–热方程级联系统的同步控制器与事件触发的同步控制器,与之比较,事件触发控制值只在触发时刻进行更新,很大程度上,节约了计算资源,降低了控制器的负载.通过Lyapunov稳定性分析,证明了误差系统可以收敛到一致的值,选取5个智能体进行仿真验证,证明了事件触发控制器的有效性.2问题描述2.1代数图论为了便于研究多智能体系统,研究学者引入了图论知识G=(V,E,A),用于分析所有智能体之间的信息交换,其中节点集V={1,2,···,N},代表系统中的每个智能体;边集E=V×V,表示智能体之间的通信连接.N i={j∈V:(i,j)∈V},代表智能体i 的邻点集.邻接矩阵A=[a ij]∈R n×n表示节点之间的连接关系,其中a ij=1表示节点j能够获取节点i 的信息,否则a ij=0.度矩阵D用来描述与之连接的所有节点的连接度.定义节点V i的入度为deginV i= n∑j=1a ji,出度为degoutV i=n∑j=1a ij,那么在图G的度矩阵D=[D ij]中,D ij=0,∀i=j且D ii=deg out V i[8,18–19].2.2模型描述在竞争激烈的今天,每个细节都能决定企业的成功与否.其中制药工程、晶体结晶、发酵工艺等都对温度有严格的要求.正如蓝宝石工艺,在加热的过程中,会形成固体与液体交接的情况.蓝宝石的温度分布应该保证固体与液体交接点的变化满足工程实际需要,即固体与液体交接点的变化速度不能过快或者过慢,否则会导致蓝宝石结晶过程中晶体因受热不均而发生炸裂,或者与加热炉粘连,从而影响晶体质量.然而蓝宝石加热炉是通过给外部的钨棒网通电进行加热的,所以最终目的是通过改变钨棒网的电压来实现温度控制,从而使固液交接点处的温度分布均匀,其加热炉如图1所示.图1中:1表示CCD摄像机;2表示目镜;3表示旋转挡板驱动;4表示旋转挡板把手;5表示挡板;6表示隔热层;7表示孔洞;8表示提拉杆;9表示坩埚壁;10表示籽晶;11表示钨加热器;12表示炉壁;13表示熔融状蓝宝石.利用各向同性性质,归一化处理后,可以将熔融状态的蓝宝石温度分布用抽象的一维反应扩散方程近第8期杨辉等:ODE–热方程级联系统的事件触发控制1351似描述,因此系统模型可表示为˙z i(t)=u i(t),i=1,2,···,N,ωit(x,t)=ωixx(x,t),ωix(0,t)=0,ωix(1,t)=cz i(t),z i(0)=z i,ωi(x,0)=ωi(x),(1)其中:N为智能体的个数,i表示第i个智能体,z(t)表示电源控制器内部的状态变量和加热炉内钨棒网的温度,ω(x,t)为蓝宝石溶液在x位置t时刻的温度分布,其状态空间为H:=L2(0,1),x表示空间坐标.通过对电源z(t)的控制改变蓝宝石温度分布ω(x,t)以保证固体与液体交接点x的变化满足工程实际需要.钨棒网与溶液边界直接耦合,由于位置x的对称性,交接点两侧的温度近似相同,所以交接点不存在热交换,因此可以将另一侧边界假设为绝热条件.图1蓝宝石加热炉Fig.1Sapphire heating furnace根据图论知识,针对具有N个智能体的多智能体系统,第i个智能体的状态为ωi(x,t),若系统中每个智能体的状态都达到一致,需要满足limt→∞∥ωi(x,t)−ωj(x,t)∥H=0,i=j.本文目的是通过控制电源,最终使所有状态能够自发的达成一致,也就是说,液体热量最终能够在各个位置相同.接下来将设计通信交互协议,并证明该通信协议能够使系统稳定,并给出了事件触发控制方案,从而能够节约成本.假设1假设系统(1)的连接方式由全联通有向拓扑图G=(V,E)来描述.图G的拉普拉斯矩阵L可以表示为L=D−A,或常用L(A)=L(G)来表示图G的拉普拉斯矩阵.定义1设X是线性赋范空间,X上的单参数强连续有界线性算子族T(t)称为算子半群,简称C0半群[20].对任何t>0,T(t)都是线性有界算子,且满足T(0)=I,T(t+s)=T(t)T(s),limt→0∥T(t)x−x∥=0,∀x∈X.定义2在线性赋范空间X上满足∥T(t)∥ 1,∀t 0的C0半群称为压缩C0半群.定义3线性赋范空间X上一线性算子A称为耗散的,如果对任意x∈D(A),∃x∗∈D′(A),使Re⟨A x,x∗⟩ 0,若A还满足R(λ−A)=X,∀λ>0,则称A为耗散的[21].定理1设A是线性赋范空间X中的一稠定算子,则以下条件等价:若A是耗散的,且∃λ0>0使得R(λ0−A)=X,则A生成X上的压缩C0半群.若A生成X上的压缩C0半群,则A是耗散的,且R(λ−A)=X,∀λ>0,此外Re⟨A x,x∗⟩ 0,∀x∈D(A),x∗∈D′(A). 2.3一致性控制协议设计由于级联系统存在耦合的边界条件,传统的ODE 或PDE的多智能体一致性控制已不能满足需求,若继续使用传统的一阶多智能体控制器的设计形式,稳定性证明时则会产生不稳定的耦合项,因此本文根据级联系统特性重新设计一致性同步协议并予以证明.对于型如系统(1)的多智能体ODE–热方程级联系统,设计控制器u(t)=−c Lω(1,t)−kIz(t),(2)其中:c>0,k>0均是常数,L是拉普拉斯矩阵,I表示单位矩阵,ω(x,t)和z(t)均为N维向量,将控制器(2)代入系统(1),可以得到级联系统如下:˙z(t)=−c Lω(1,t)−kIz(t),ωt(x,t)=ωxx(x,t),ωx(0,t)=0,ωx(1,t)=cz(t),z(0)=z0,ω(x,0)=ω0(x).(3)引理1(文献[8]引理3.1)对于任意的z(t)∈L2loc(0,∞),ω0(x)∈H2(0,1),系统(3)存在唯一解ω(x,t),使ω(x,t)∈C1(0,∞;L2(0,1))∩C(0,∞; H2(0,1)).此外,通过H2(0,1)⊂C[0,1],那么ω(1,t)对于时间t>0是连续的、有界的.考虑状态空间H=(L2(0,1))N×R N.(4)1352控制理论与应用第40卷定义内积如下:⟨ϕ1,ϕ2⟩H =1ωT1(x,t )L ω2(x,t )d x +12z T1(t )z 2(t ),(5)其中∀ϕi =(ωTi (x,t ),z i (t )),i =1,2.定义算子A :D (A )→H :A (ω(x,t ),z (t ))=(ωxx (x,t ),−c L ω(1,t )−kz (t )),D (A )={(ω,z )∈(H 2(0,1))N ×R N |ωx (0)=0,ωx (1)=cz }.(6)将方程(3)写为发展方程形式如下:˙ϕ(t )=A ϕ(t ).(7)定理2算子A 在空间H 上生成压缩C 0半群.在式(6)中,计算可以得出Re ⟨A ϕ,ϕ⟩=−kz T(t )z (t )−1ωTx (x,t )ωx (x,t )d x 0.(8)因此得出算子A 在空间H 上是能量耗散的,由于初值条件给定,其具有唯一解,因此可以得出A −1存在.根据Sobolev 嵌入定理得出A −1在H 中是紧的.利用定理1(Lumer-Phillips)可以得出,算子A 在空间H 上生成压缩C 0半群.本文的目标是使系统同步,分析可知同步的一个充分条件是使边界无能量流动,同时使得所有状态趋近于0.又由于边界绝热条件的充分条件是使ODE 部分收敛到0,因此初步分析ODE 部分特征值需要为负,而热方程部分系统的特征值所在空间仅有一个零特征值,其余为负.根据以上思路设计的控制器(2).接下来证明在控制器(2)作用下系统的稳定性,定义Lyapunov 函数如下:V =12 10ωT (x,t )L ω(x,t )d x +12z T (t )z (t ).(9)对Lyapunov 函数求导得˙V = 10ωT (x,t )L ωt (x,t )d x +z T (t )z t (t )=10ωT (x,t )L ωxx (x,t )d x +z T(t )(−c L ω(1,t )−kz (t ))=ωT (x,t )L ωx (1,t )− 1ωT x (x,t )L ωx (x,t )d x +z T(t )(−c L ω(1,t )−kz (t ))=cωT (1,t )L z (t )− 1ωT x (x,t )L ωx (x,t )d x −cω(1,t )L z T (t )−kz T(t )z (t )=− 1ωTx (x,t )L ωx (x,t )d x −kz T (t )z (t ) 0.(10)通过LaSalle 不变集原理,可以找出有且仅有一个稳定解收敛,即系统状态最终可以达成同步∥(ω(·,t )),z (t ))∥H →0,t →∞.(11)以上是连续系统ODE–热方程级联系统的控制器及闭环系统的证明.下文将设计事件触发控制器的控制规则,并证明该系统依然保持收敛性能.3事件触发控制器设计上文提到的控制器是时间连续形式的,但是连续控制需要在时间尺度上,不间断的采集并获取信息,同时需要实时不间断的对信息进行处理.由于目前广泛应用的数字式执行器执行测量、运算或输出等操作均有最小时钟周期限制,因此数字式执行器无法做到无限小时刻采样并对信息进行处理.为解决数字式控制难以处理连续性控制的缺点,所以本文提出事件触发的ODE–热方程级联系统一致性控制协议,给出具体的触发条件,在达成触发条件时改变控制输入的值,而其余时刻则保持之前状态不变.这样不仅仅能够解决连续性控制的缺点,同时还可以节约控制器的计算资源.下面介绍事件触发控制器的设计,以及触发条件的确认.根据第2.3节得出的一致性同步协议可以假设ODE–热方程系统事件触发控制器如下:{u (t )=−c L ω(1,t )−kz (t ),t =t k ,u (t )=u (t k ),(12)其中:u (t )为ODE–热方程级联系统事件触发控制器,t ∈[t k ,t k +1),∀k =0,1,2,···且t 0为初始时刻,触发间隔时间点t k 满足t k =arg min t>t k −1{∥e (1,t )∥=σ1∥z (t )∥|c |∥L∥,∥ϵ(t )∥=σ2∥z (t )∥k},(13)其中:argmin 表示∥e (1,t )∥和∥ϵ(t )∥达到最小值时t 的取值,∀σ1,σ2∈R +满足条件∥e (1,t )∥ σ1∥z (t )∥|c |∥L∥,∥ε(t )∥ σ2∥z (t )∥k ,k −σ1−σ2<0.(14)为便于证明在事件触发控制器作用下系统仍保持稳定,首先引入误差系统{e (x,t )=ω(x,t k )−ω(x,t ),ϵ(t )=z (t i )−z (t ),(15)将误差状态(15)代入式(12)中,可以得到控制器在触发时刻的误差表示形式u (t )=−c L e (1,t )−c L ω(1,t )−kϵ(t )−kz (t ).(16)第8期杨辉等:ODE–热方程级联系统的事件触发控制1353为了证明控制器(16)能够使系统(1)的每个子系统保持稳定,并能够使各个子系统最终状态达成一致,利用Lyapunov 稳定性判据对事件触发系统进行稳定性判别.设计Lyapunov 函数如下:V (t )=12(z T(t )z (t )+ 10ωT (x,t )L ω(x,t )d x ).(17)对上式左右两边关于时间变量求导,可得˙V(t )=d d t 12(z T(t )z (t )+ 10ωT (x,t )L ω(x,t )d x )=z T (t )u (t )+ 10ωTt (x,t )L ω(x,t )d x =z T (t )u (t )+ 10ωT xx (x,t )L ω(x,t )d x =z T (t )u (t )+ωT x (x,t )L ω(x,t )|10−10ωT x (x,t )L ωx (x,t )d x =z T(t )u (t )+ωTx (1,t )L ω(1,t )− 1ωT x (x,t )L ωx (x,t )d x.(18)结合耦合边界条件cz (t )=ωx (1,t ),可以得出˙V(t )=−cz T (t )L [e (1,t )+ω(1,t )]−kz T (t )[ϵ(t )+z (t )]+ωT x (1,t )L ω(1,t )−1ωTx (x,t )L ωx (x,t )d x =−ωTx (1,t )L e (1,t )−kz T (t )ϵ(t )−kz t(t )z (t )− 1ω1x ωT x (x,t )L ωx (x,t )d x∥ωx (1,t )∥∥L∥∥e (1,t )∥+k ∥z (t )∥∥ϵ(t )∥−k ∥z (t )∥2−1ωT x (x,t )L ωx (x,t )d x.(19)设∃σ1,σ2>0,满足以下不等式组条件:∥e (1,t )∥ σ1∥z (t )∥|c |∥L∥,∥ε(t )∥ σ2∥z (t )∥k ,k −σ1−σ2<0.(20)事实上,对于∀σ1,σ2∈R +,总是至少存在着一组解满足不等式组(20),这是很容易得出的,因为前两个不等式组在出发时刻∥e (1,t )∥和∥ϵ(t )∥都等于0.所以σ1,σ2>0即可满足要求.并且由于设定参数k >0,所以必然存在σ1+σ2<k 满足第3个不等式.将假设的不等式组代入式(19),可以得出˙V(t ) (k −σ1−σ2)∥z (t )∥∥ϵ(t )∥−k ∥z (t )∥2− 1ωT x (x,t )L ωx (x,t )d x.(21)根据Lyapunov 稳定性判据可以得出,该系统是Ly-apunov 意义下稳定的,同时通过计算其稳定点所构成的不变集,容易得出最终有且仅有一个共同收敛的点,因此该ODE–热方程级联系统事件触发控制器最终能够使得系统(1)中的状态达成一致.根据不等式组(20)的条件,对于k =1,2,···,以及σ1,σ2满足式(20)条件,设触发时刻为t k =arg min t>t k −1{∥e (1,t )∥=σ1∥z (t )∥|c |∥L∥,∥ϵ(t )∥=σ2∥z (t )∥k}.(22)通过上述证明,可以得出控制器u (t )在式(12)的作用下能够使系统(1)最终状态达成一致.由于PDE 部分边界条件经由ODE 中的控制器引入了负实部极点,因此系统具有一定的鲁棒性,即存在一定的扰动的情况下,系统仍然可以按照预期的目标达成一致,下面针对上述事件触发一致性协议设计计算机仿真程序来进行验证.4数值仿真验证为验证ODE–热方程级联系统事件触发一致性控制协议的实际有效性,本文采用数字计算机平台进行仿真模拟验证.这里仿真离散格式中,对于空间采用中心差分法进行离散,时间采用前向差分方法进行离散化,ODE 部分采用前向差分离散化.选取仿真步长d t =0.001s,空间步长d x =0.05m,仿真总时长T =15s,PDE 选取长度为1m.设系统参数c =5,k =10/3.假设系统仿真的初值条件如下:ω(x,0)= 4+2sin(2πx +1)+0.7(x −0.5)−2+1.4sin(3πx +2)−0.4(x −0.7)9−6sin(1.5πx +3)−0.9(x −0.3)9−1.4sin(2πx +2)+0.9(x −0.4)4−2sin(1.5πx +1)+0.7(x −0.2),z (0)=[412−1−4]T ,考虑多智能体个数为5,且为全连通有向拓扑,假设权重均为1,其连接方式如图2.通过上述假设的参数进行仿真,能够得出仿真所用控制系统如下:˙z (t )=−5 1−100001−100001−100001−1−10001 ω(1,t )−103z (t ),ωt (x,t )=ωxx (x,t ),ωx (0,t )=0,ωx (1,t )=5z (t ).(23)1354控制理论与应用第40卷因此根据本文所给出的计算方法,设计控制器u(t)=˙z(t)=−51−100001−100001−100001−1−10001ω(1,t)−103z(t),u(t)=u(t k),t∈(t k,t k+1),t k=arg mint>t k−1{∥e(1,t)∥=σ1∥z(t)∥|c|∥L∥,∥ϵ(t)∥=σ2∥z(t)∥k}.(24)其中选取σ1=0.43,σ2=11.1.仿真结果如图3–10所示.图2多智能体连接拓扑图Fig.2Multi-agent connectiontopology图3多智能体系统ODE部分状态图Fig.3Partial ODE state diagram of multi-agentsystem图4多智能体系统PDE部分状态1Fig.4Multi-agent system PDE partial State1图5多智能体系统PDE部分状态2Fig.5Multi-agent system PDE partial State2图6多智能体系统PDE部分状态3Fig.6Multi-agent system PDE partial State3图7多智能体系统PDE部分状态4Fig.7Multi-agent system PDE partial State4图3描述了多智能体系统(24)中的ODE部分随时间趋向于0,其物理含义是为了最终使PDE系统能够稳定在一致温度下,热系统边界随着时间增加,其热流动逐渐停止.图4–8五个多智能体系统的PDE部分在边界耦合ODE系统的控制下,最终能够收敛到一致状态.图9–10表示了多智能体共同决策得出的触发条件,其中触发时刻用“*”着重标记.根据仿真结果可以看出,通过利用本文提出的基于事件触发的ODE–热方程级联系统多智能体一致性控制协议,能够使系统在大约12s左右通过较少的触发次数达到同步,表第8期杨辉等:ODE–热方程级联系统的事件触发控制1355明事件触发控制器具有优越的性能,能在很短的时间内让蓝宝石加热炉达到一致的工作状态.图8多智能体系统PDE部分状态5Fig.8Multi-agent system PDE partial State5图9多智能体系统事件触发条件1Fig.9Multi-agent system event trigger Condition1图10多智能体系统事件触发条件2Fig.10Multi-agent system event trigger Condition25结论与展望本文研究了基于事件触发的ODE–热方程级联系统多智能体一致性控制协议的设计,通过控制与PDE部分边界耦合的ODE系统的输入,使PDE部分的热系统最终能够稳定,快速的达成一致,进一步给出了事件触发控制的条件,通过仿真验证可以看出该同步协议具有一定的鲁棒性.相较于之前已有的成果来说,本文的创新在于采用不同的(Neumann)边界条件,控制输入和被控对象分别处于不同的系统,同步采用边界反馈,并证明了事件触发需要满足的条件及其稳定性,但因本文系统存在耦合项,Zeno现象的避免未能给出严格的证明,这将在后续的工作中进行.参考文献:[1]TIAN Yuan.Consensus control for multi-agent systems with impul-sive effects.Chongqing:Southwest University,2020.(田袁.脉冲作用下的多智能体系统一致性控制.重庆:西南大学, 2020.)[2]JI Lianghao,LIAO Xiaofeng.Consensus analysis of multi-agent sys-tem with multiple time delays.Journal of Physics,2012,61(15):8–16.(纪良浩,廖晓峰.具有不同延时的多智能体系统一致性分析.物理学报,2012,61(15):8–16.)[3]FU Q,YU P F,XU G Z,et al.Containment control for partial differ-ential multi-agent systems.Physica A:Statistical Mechanics and its Applications,2019,529:121549.[4]ZHOU Tuo,LIU Quanli,WANG Dong,et al.Leader-following con-sensus for linear multi-agent systems based on integral-type event-triggered strategy.Control and Decision,2022,37(5):1258–1266.(周托,刘全利,王东,等.积分事件触发策略下的线性多智能体系统领导跟随一致性.控制与决策,2022,37(5):1258–1266.)[5]OLFATI-SABER R,MURRAY R.Consensus problems in networksof agents with switching topology and time-delays.IEEE Transac-tions on Automatic Control,2004,49(9):15201533.[6]XIE Jun,CHEN Kaixuan,YUE Dong.Distributed economic dispatchbased on consensus algorithm of multi agent system for power sys-tem.Electric Power Automation Equipment,2016,36(2):112–117.(谢俊,陈凯旋,岳东.基于多智能体系统一致性算法的电力系统分布式经济调度策略.电力自动化设备,2016,36(2):112–117.)[7]LI L C,FENG J S.Current situation and development trend of sap-phire crystal growth technology in China.Journal of Synthetic Crys-tals,2012,41(S1):221–226.[8]MENG X Y,MENG Z Y,CHEN T W,et al.Pulse width modulationfor multi-agent systems.Automatica,2013,49(7):2125–2132.[9]XI J X,WANG C,YANG X J,et al.Limited-budget output consen-sus for descriptor multiagent systems with energy constraints.IEEE Transactions on Cybernetics,2020,50(11):4585–4598.[10]OLFATI-SABER R,ALEX F,MURRAY R M.Consensus and coop-eration in networked multi-agent systems.Proceedings of the IEEE, 2007,95(1):215–233.[11]CHRISTOFIDES P D.Nonlinear and robust control of PDE systems:Methods and applications to transport-reaction processes.Applied Mechanics Reviews,2002,55(2):29–30.[12]WANG J,KRSTIC M.Delay-compensated control of sandwichedode-pde-ode hyperbolic systems for oil drilling and disaster relief.Automatica,2020,120:109131.[13]JIN F F,GUO B Z.Performance boundary output tracking for one-dimensional heat equation with boundary unmatched disturbance.Automatica,2018,96:1–10.[14]ALAMT T M,PAQUET L.The coupled heat maxwell equations withtemperature-dependent permittivity.Journal of Mathematical Analy-sis and Applications,2022,505(1):125472.1356控制理论与应用第40卷[15]AY ADI H.Exponential stabilization of cascade ode-linearized KDVsystem by boundary dirichlet actuation.European Journal of Control, 2018,43:33–38.[16]XU Zenghe,ZHAI Yuchun,JI Zhiling.An analytical solution for cou-pled convection-diffusion and chemical reaction infixed-beds.The Chinese Journal of Nonferrous Metals,2004,14(11):1918–1925.(徐曾和,翟玉春,纪智玲.固定床中对流扩散与反应耦合问题的一个解析解.中国有色金属学报,2004,14(11):1918–1925.)[17]DIMAROGONAS D V,FRAZZOLI E,JOHANSSON K H.Dis-tributed event-triggered control for multi-agent systems.IEEE Trans-actions on Automatic Control,2012,57(5):1291–1297.[18]LIU T F,JIANG Z P.Distributed control of multi-agent systems withpulse-width-modulated controllers.Automatica,2020,119:109020.[19]LU Renwei.Consistency of nonlinear second-order multiagent withdirected topology and delay.Yangzhou:Yangzhou University,2012.(鲁仁伟.具有有向拓扑和时滞的非线性二阶多智能体的一致性.扬州:扬州大学,2012.)[20]SI Qin,LUO Cheng.Global attractors of operator semigroups ontopological space.Acta Analysis Functionalis Applicata,2017,19(4): 412–418.(斯琴,罗成.拓扑空间上算子半群的全局吸引子.应用泛函分析学报,2017,19(4):412–418.)[21]LIU Rui,ZHAO Huaxin,MA Qiangqiang.Generalized-semigroupsand dissipative operator.Journal of Shenyang Normal University (Natural Science Edition),2011,29(3):362–364.(刘瑞,赵华新,马强强.广义C0半群与耗散算子.沈阳师范大学学报(自然科学版),2011,29(3):362–364.)作者简介:杨辉硕士研究生,目前研究方向为分布参数控制系统理论与应用,E-mail:1104660379@;宗西举博士,教授,英国奥斯特大学兼职博士生导师,目前研究方向为分布参数控制系统理论、复杂系统控制理论、控制理论在电力系统中的应用,E-mail:cse**************.cn;郑泽阳硕士研究生,目前研究方向为分布参数控制系统理论与应用,E-mail:*****************;徐秀妮讲师,目前研究方向为分布参数控制系统理论与应用, E-mail:***************.。
结构动力学多自由度
▪ 振型方程：
(K i2M)ji 0 (i 1, 2, 3, n)
▪∵
K 2i M 0
▪ ∴ 第i 个振型方程中的n 个方程中只有n-1个是独立的！ ▪ ——无法得到j1i、 j2i、 … 、 jni 的确定值， ▪ 但可以确定各质点振幅之间的相对比值： ▪ —— 振型的幅值是任意的，但形状是惟一的。
一致质量矩阵：
L
pava m13v1 0 fI ( x)v( x)dx
L
0
m( x) 3( x)v3
L
1( x)v1dx
mij 0 EI ( x)i ( x) j ( x)dx
L
cij 0 c( x) i ( x) j ( x)dx
其中，c(x)表示分布的粘滞阻尼特性。
一致节点荷载
L
vˆ 表示体系的形状，不随时间变化。
v 2vˆ sin(t ) 2v 2mvˆ sin(t ) kvˆ sin(t ) 0
k 2m vˆ 0
k 2m vˆ 0
即： k 2m 0
上式的N个根，表述体系可能存在的N个振型的频率。
1
2
3
N
2)
2
)
y32
(t
)
jˆ
32
s
in
(
2
t
2
)
1
jˆ
2i
yi
(t
)
jˆ3i
s
in(i
t
i
)
jˆ ni
1
jˆ 21
jˆ 31
jˆ 32
1
jˆ 22
将N个振型中的每一振型形式，用F表示N个振型所组成的方阵。
11 12 13 1N
1汉英力学名词(1993)
BZ反应||Belousov-Zhabotinski reaction, BZ reactionFPU问题||Fermi-Pasta-Ulam problem, FPU problemKBM方法||KBM method, Krylov-Bogoliubov-Mitropolskii method KS[动态]熵||Kolmogorov-Sinai entropy, KS entropyKdV 方程||KdV equationU形管||U-tubeWKB方法||WKB method, Wentzel-Kramers-Brillouin method[彻]体力||body force[单]元||element[第二类]拉格朗日方程||Lagrange equation [of the second kind] [叠栅]云纹||moiré fringe; 物理学称“叠栅条纹”。
[叠栅]云纹法||moiré method[抗]剪切角||angle of shear resistance[可]变形体||deformable body[钱]币状裂纹||penny-shape crack[映]象||image[圆]筒||cylinder[圆]柱壳||cylindrical shell[转]轴||shaft[转动]瞬心||instantaneous center [of rotation][转动]瞬轴||instantaneous axis [of rotation][状]态变量||state variable[状]态空间||state space[自]适应网格||[self-]adaptive meshＣ0连续问题||C0-continuous problemＣ1连续问题||C1-continuous problemＣＦＬ条件||Courant-Friedrichs-Lewy condition, CFL condition ＨＲＲ场||Hutchinson-Rice-Rosengren fieldＪ积分||J-integralＪ阻力曲线||J-resistance curveＫＡＭ定理||Kolgomorov-Arnol'd-Moser theorem, KAM theoremＫＡＭ环面||KAM torusｈ收敛||h-convergenceｐ收敛||p-convergenceπ定理||Buckingham theorem, pi theorem阿尔曼西应变||Almansis strain阿尔文波||Alfven wave阿基米德原理||Archimedes principle阿诺德舌[头]||Arnol'd tongue阿佩尔方程||Appel equation阿特伍德机||Atwood machine埃克曼边界层||Ekman boundary layer埃克曼流||Ekman flow埃克曼数||Ekman number埃克特数||Eckert number埃农吸引子||Henon attractor艾里应力函数||Airy stress function鞍点||saddle [point]鞍结分岔||saddle-node bifurcation安定[性]理论||shake-down theory安全寿命||safe life安全系数||safety factor安全裕度||safety margin暗条纹||dark fringe奥尔-索末菲方程||Orr-Sommerfeld equation奥辛流||Oseen flow奥伊洛特模型||Oldroyd model八面体剪应变||octohedral shear strain八面体剪应力||octohedral shear stress八面体剪应力理论||octohedral shear stress theory巴塞特力||Basset force白光散斑法||white-light speckle method摆||pendulum摆振||shimmy板||plate板块法||panel method板元||plate element半导体应变计||semiconductor strain gage半峰宽度||half-peak width半解析法||semi-analytical method半逆解法||semi-inverse method半频进动||half frequency precession半向同性张量||hemitropic tensor半隐格式||semi-implicit scheme薄壁杆||thin-walled bar薄壁梁||thin-walled beam薄壁筒||thin-walled cylinder薄膜比拟||membrane analogy薄翼理论||thin-airfoil theory保单调差分格式||monotonicity preserving difference scheme 保守力||conservative force保守系||conservative system爆发||blow up爆高||height of burst爆轰||detonation; 又称“爆震”。
数学物理方程(很好的学习教材)
数学物理方程(很好的学习教材)
二、数学物理方程的一般分类
一般分类 按自变量的个数，分为二元和多元方程； 按未知函数及其导数的幂次，分为线性微分方程和 非线性微分方程； 按方程中未知函数导数的最高阶数，分为一阶、二 阶和高阶微分方程。
由能量守恒定律 c ρdx du=dQ =[q(x,t)-q(x+dx,t)]dt =-qx(x,t)dxdt
于是有 c ρut = -qx 由热传导定律 q(x,t) = -k ux(x,t) 代入前面的式子，得到 c ρut = k uxx ut = a2 uxx
a2 = k/(cρ)
数学物理方程(很好的学习教材)
四、常见数学物理方程的定解条件
波动方程
方程形u式 tt ： a2u f 定解条初 件边始界条条件件：：包第含一 位初 第类 移始 二或 ”“ 类者 和或初者始第“三速
输运方程
方程形u式 t a： 2uf 定解条边 件初 界始 条条 件件 ：： 第物 一 第理 类 始 二量 或 时 类在 者 刻 或初 的 者值 第
三类线性边界条件
第一类边界条 u(x件 ,y,： z,t)边界x0,y0,z0 f(x0,y0,z0,t)
第二类边界条件： u n边界x0,y0,z0
f(x0,y0,z0,t)
第三类边界条 u件 H： u
n边界x0,y0,z0
f(x0,y0,z0,t)
初始条件
定解条件
边界条件
数学物理方程(很好的学习教材)
u u 2u u u 2
2
yy
y数学物理方程(很好y的学y习教材) y
yyu
yy
于是，方程化为：
PhysRevA.87.042115
上海交通大学数学科学学院2020年博士生导师及招生专业和
教授 / Tenure Track Associate
function theory, conformal mappings, potential theory
Professor
4 Victor MOUQUIN / 长聘教轨副教
数学 Mathematics
泊松几何、量子群、表示论 Poisson geometry, quantum
Professor
Mathematics
生物建模）Computatioal Mathematics and Mathematical
Biology(Transport Equations and Reaction Diffusion
Equations, Their Numerical Schemes and
20
林一青 / 长聘教轨副教授
Yiqing LIN / Tenure Track
Associate Professor
统计学 Statistics
金融数学与金融工程 (资产定价与风险管理），随机分析和非 线性概率统计理论 Financial Mathematic and Financial Engineering, Stochastic Analysis, Nonlinear Statistical Theory
6
范金燕 / 教授 Jinyan FAN /
计算数学 Computational 最优化方法（非线性优化理论和方法）Optimization Methods
Professor
Mathematics
(Nonlinear Optimization Theory and Methods)
7 方北香 / 教授 Beixiang Fang / Professor
黑洞的准正模式(quasinormal modes)
Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。
一份关于二维电子气(2DEG)的讲义
where r is the vector in plane of 2DEG. Throughout our considerations we will assume that all the distances are much larger than interatomic distance and thus we will use the effective
Density of States
The density of states g( ) is defined as number of states per the energy interval , + d . It is clear that
g( ) = δ( − α)
α
where α is the set of quantum numbers characterizing the states. In the present case it includes the subband quantum number n, spin quantum number σ, valley quantum number
v (for n-type materials), and in-plane quasimomentum k. If the spectrum is degenerate
with respect to spin and valleys one can define the spin degeneracy νs and valley degeneracy
n-AlGaAs i-GaAs
EC
EF EC
EF
EV
EV
EF
_一类非线性奇性薛定谔方程的精确解
2.1 动力系统.................................................................................................................. 4
一 ？
．
．
＇
心． －． 、
．
— ．
．
，
如 Ｖ ’
’
． 、
． ．
Ｊ
＇
从皆驴 妃 皆 Ｃ
． ．
＾
：
；
、
、
｜
，
，
，
去
Ｉ
，
；
，
：
：
，
．
，
汽 心 冬 …
，
：
：：； 、
＇
Ｖ 峨攤雜 ．
诗 和 、 寺 治 骑 於 乂 等 ： ．
设
塔 琪 少产 ＾
＾
：＾
： ．
ｖ ，，—
界苗装 茲 誦 議＾咖 闺。 ． ＾
爲 ’
，＇
，
＇
．
成 个 ＇、
．
‘
眠’ ＇ 凡以
娘 １
＇
＇
＾
。
、 ．
活、巧
：
；
．．
： ＾
＇ ；
。、
．
＾
， ；
勞 心 ．
’
幕
和 ；
除
学
、
（
部
、 所
；
）
一阶微分方程解的存在唯一性定理
x0 x x0 h ………(3.1.8)
微分之，得到: d(x) f (x,(x))
dx
又把 x x0 代入(3.1.8)，得到: (x0 ) y0
因此，y (x)是方程(3.1.1)定义于 x0 x x0 h
上，且满足初始条件(3.1.2)的解。
同理，可证在 x0 h x x0 也成立。
x
k1(x) y0 x0 f (,k ( )) d M (x x0 ) Mh b
k 1 (x) 在 x0 x x0 h 上有定义，连续。
解：
0
(x)
1, 1 ( x)
1
x
x2 2
,
2 ( x)
1
x
x2
x3 6
,
3 ( x)
1
x
x2
x3 3
x4 24
,
4 ( x)
1
x
x2
x3 3
x4 12
x5 200
§ 3.1 Existence & Uniqueness Theorem & Progressive Method
等价
x
y y0 f (x, y)dx
x0
(x)即唯一解
(x) (x)
(x)另解
x
(x) y0 f (x,(x))dx
x0
n次近似解
x
n (x) y0 f (x,n1(x))dx
x0
证明极限存在
(x)即是解
lim
n
n
(