矩阵方程X＋A^TX^—1A=Q存在可稳定化解的充分必要条件

１引言一般的时离散代数Ｒｉｃｃａｔｉ方程具有下面的形式：这里如果方程（１）中的系数矩阵满足：（ｎ＝ｍ）则方程（１）变为当Ｑ＝ＱＴ＞０时,Ｅｎｇｗｅｒｄａ,詹兴致等人研究了方程（２）存在正定解的充分必要条件［１］［２］［３］．本章利用方程（２）与（１）的关系,从另一角度讨论了Ｑ为对称矩阵时,方程（２）存在可稳定化解的充分必要条件．２基本概念与记号首先我们简单回顾一下以前的概念与记号．矩阵束Ｍ—Ｎ,Ｍ,Ｎ为正则的,也就是说ｄｅｔ（λＭ－Ｎ）＝０;如果λ０为ｄｅｔ（λＭ－Ｎ）的ｋ重根,则称λ０为它的ｋ…     

一些矩阵分解的敏度分析

１．引言 代数Ｒｉｃｃａｔｉ方程是线性系统理论与设计的核心课题之一．矩阵的Ｈｅｓｓｅｎｂｅｒｇ分解、Ｈａｍｉｌｔｏｎ矩阵的平方约化分解、辛矩阵的ＱＴ分解是数值求解代数Ｒｉｃｃａｔｉ 方程的基本工具．关于 Ｈｅｓｓｅｎｂｅｒｇ分解的研究工作有很多（参阅 ［４］及其参考文献）．最近, Ｓｕｎ［４］利用矩阵分裂算子研究了Ｈｅｓｓｅｎｂｅｒｇ分解因子的扰动分析,并根据所得的扰动上界定义了分解因子的条件数．本文第 ２节将运用局部展开方法引入 Ｈｅｓｓｅｎｂｅｒｇ分解因子的条件数．有趣的是所定义的条件数与Ｓｕｎ引…     

无限维离散时间代数Riccati方程的非负解

本文研究了无限维离散时间代数Ｒｉｃｃａｔｉ方程（ＤＡＲＥ）的非负自伴解，给出了（ＤＡＲＥ）有非负 自伴解的充要条件．对幂可稳定化的离散时间系统∑ｄ（Ａ，Ｂ，－），若Ａ是可逆的，Ｂ是紧的，给出 了（ＤＡＲＥ）的非负解集的参数化刻画，并以Ａ的有限维的含于反稳定的不可观察子空间中的不变子 空间为参数．该结果把［５］中关于有限维系统∑ｄ（Ａ，Ｂ，－）的结果推广到了一般的系统∑ｄ（Ａ，Ｂ，－） 中．最后，还给出了∑ｄ（Ａ，Ｂ，－）具有非负稳定化解的充要条件．     

无限维线性-非二次最优控制问题

本文研究一类较文［１６］提出的问题更为一般的线性－非二次最优控制问题．引进了所谓积分拟－Ｒｉｃｃａｔｉ方程并揭示了它与一非线性积分方程族之间的双向联系．凭此建立起积分拟－Ｒｉｃｃａｔｉ方程之解的存在唯一性．随后，利用积分拟－Ｒｉｃｃａｔｉ方程的解完成了最优控制问题的闭环综合．最后还导出了积分拟－Ｒｉｃｃａｔｉ方程之解关于其参数的一个连续依赖性定理，据之可以用适当的有限维最优控制问题的闭环解来逼近本文所考虑的无限维最优控制问题的闭环解．     

离散时间代数Riccati方程的解矩阵的下界与上界

本文讨论离散时间代数Ｒｉｃｃａｔｉ方程ＡＴＸＡ－Ｘ－（ＡＴＸＢ＋Ｌ）（Ｒ＋ＢＴＸＢ）＾－１（ＬＴ＋ＢＴＸＡ）＋Ｑ＝０的唯一对称正定解的上界和下界。     

数值求解延时微分方程的步长准则

１．引言 用一个数值方法求解下列延时微分方程：其中, ｆ： Ｒ × Ｃｄ × Ｃｄ → Ｃｄ为给定函数, Ｕ（ｔ）当上＞ ０时为未知函数,τ＞ ０为常数延时量,ф（ｔ）∈Ｃｄ为已知向量值函数．为了检验一个数值方法的数值稳定性,常用如下试验方程：来观察方法的数值稳定性,这里ａ,ｂ∈Ｃ（Ｃ为复数集）为已知常数,ф（ｔ）为给定的连续函数（ｔ≤０）． 定义 １［2］．延时微分方程（简记为ＤＤＥＳ）（３）被称为是渐近稳定的,如果（３）的每一个解Ｕ（ｔ）满足 方程(３)的特征方程为： 定义 ２［２］．一数值方法求解ＤＤＥＳ称为…     

解高维广义对称正则长波方程的Fourier谱方法

１引言对称正则长波方程（ＳＲＬＷＥ）是正则化长波方程（ＲＬＷＥ）的一种对称叙述［１］用于描述弱非线性作用下空间变换的离子声波传播．［１］得到了方程组（１．１）的双曲正割平方孤立波解、四个不变量和数值结果、明显地,从（１．１）中消去ρ,得到一类正则长波方程（ＲＬＷＥ）代替（１．２）中第三项、第四项对ｔ的导数为对ｘ的导数,得到Ｂｏｕｓｓｉｎｅｓｑ方程．［２］对一类广义对称正则长波方程组提出了谱方法,证明了古典光滑解的存在性和唯一性,建立了近似解的收敛性和误差估计。［３］研究了高维对称正则长波方程整体…     

Lyapunov不等式的最佳解与一般线性方法的可行性

１引言在刚性常微分方程初值问题数值解法理论中,常要求数值方法中的某个系数矩阵Ｂ具有性质：使Ｌｙａｐｕｎｏｖ不等式有对角正定解Ｄ存在,即存在对角正定矩阵Ｄ,使ＤＢ＋ＢＴＤ为正定矩阵·在许多情形,这种解Ｄ是存在的,例如Ｇ－Ｌ-、ＩＡ-和ⅡＡ－ＲＫ方法［３］．然而也存在一些方法,它是Ａ－稳定、Ｌ－稳定,甚至是代数稳定的,但这种Ｄ却不存在．例如ⅢＣ（ｓ≥３）ＲＫ方法［３］和当θ＝１时的块θ－方法［５］（至少对ｒ＝２,３,４是如此）．前者是Ｌ－稳定的,也是代数稳定的,后者也是Ｌ－稳定的．这意味着,可能有不少…     

三维无界区域上耗散Klein-Gordon-Schrodinger方程的指数吸引子

文 ［１］证明了耗散Ｋｌｅｉｎ－Ｇｏｒｄｏｎ－Ｓｃｈｒｏｄｉｎｇｅｒ方程在Ｘ（Ｒ~３）中具有一个最大吸引子,本文在文 ［１］的基础上得到了该方程在Ｘ（Ｒ~３）中拥有一个指数吸引子．     

奇阶中立型时滞微分方程解的振动性

１．引言考虑奇阶非线性泛函微分方程［ｘ（ｔ）－ｃｘ（ｔ—(）］（ｎ）＋ｐ（ｔ）ｆ（ｘ（ｔ－σ））＝０（１）对方程（１）我们作如下假设（Ｈ）：（Ｈ１）ｎ＞１是奇整数,ｐ∈Ｃ（（ｔ０,∞）,（ｔ０,∞））;（Ｈ２）τ＞０,σ＞０且０≤ｃ≤１;（Ｈ３）ｆ∈Ｃ（Ｒ,Ｒ）是单调增加,ｘｆ（ｘ）＞０,Ｘ≠０且当｜ｘ｜→∞时有｜ｆ（ｔ）｜→∞．设δ＝ｍａｘ｛τ,σ｝,∈Ｃ（［Ｔ－δ,Ｔ］,Ｒ）．方程（１）在［Ｔ,∞）上的解是指函数ｘ∈Ｃ（［Ｔ,∞）,Ｒ）,使得ｘ（ｔ）＝(（ｔ）,Ｔ－δ≤ｔ≤Ｔ,［ｘ（ｔ）－ｃｘ…     

Global exponential stability of static neural networks with delay and impulses: Discrete-time case

In this paper, we investigate the exponential stability of discrete-time static neural networks with impulses and variable time delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with implicit-explicit-θ (IMEX-θ) method. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type— the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using a very excellent ideology introduced recently the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. New analysis techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the ones used in continuous-time case. Several criteria for global exponential stability of the static neural networks in discrete-time case are established in terms of linear matrix inequalities (LMIs) and numerical simulations are given to validate the obtained theoretical results.     

Stabilizability of Discrete-Time Nonlinear Systems

This work deals with the property of stabilizing a nonlineardiscrete-time control system to a specified equilibrium pointby appropriate state feedback. For the most part, this paperpresents Lyapunov-like sufficient conditions for stabilizabilityof discrete-time systems that are affine in control. Stabilizationresults for general nonlinear discrete-time systems are alsoincluded.     

Exponential stability of discrete-time neural networks with delay and impulses

In this paper, we investigate the exponential stability of discrete-time neural networks with impulses and time-varying delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with different discretization methods. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type - the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using the excellent ideology introduced recently by Chen and Zheng [W.H. Chen, W.X. Zheng, Global exponential stability of impulsive neural networks with variable delay: an LMI approach, IEEE Trans. Circuits Syst. I 56 (6) (2009) 1248-1259], the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. Novel techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the continuous-time case. Several criteria for global exponential stability of the discrete-time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods.     

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.     

Dynamics Aggregation in Stochastic Control Problems

For stochastic control problems with mixed state-control constraints, we develop a dynamics aggregation method which replaces the stochastic differential or difference equation with a simpler constraint. Solutions of such simplified problems are used to construct a sequence of approximations to the original problem. Convergence properties of the method for both discrete-time and continuous-time models are analyzed in detail.     

Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

Existence of optimal controls on hybrid time domains

Hybrid control systems are considered, combining continuous-time dynamics and discrete-time dynamics, and modeled by differential equations or inclusions, by difference equations or inclusions, and by constraints on the resulting dynamics. Solutions are defined on hybrid time domains. Finite-horizon and infinite-horizon optimal control problems for such control systems are considered. Existence of optimal open-loop controls is shown. The assumptions used include, essentially, the existence for the (non-hybrid) continuous-time case; the existence for the (non-hybrid) discrete-time case; mild conditions on the endpoint penalties; and closedness and boundedness, in the finite-horizon case, of the set of admissible hybrid time domains. Examples involving switching systems and hybrid automata are included.     

A class of discrete time generalized Riccati equations

In this paper, a class of discrete-time backward non-linear equations defined on some ordered Hilbert spaces of symmetric matrices is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time non-linear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in the paper provide the sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations as: the maximal solution, the stabilizing solution and the minimal positive semi-definite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities (LMI). One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution, we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by the positive operators on some ordered Hilbert spaces chosen adequately. The results derived in this paper provide useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control both in the deterministic and the stochastic framework. The proofs are deterministic and are accessible to the readers less familiarized with the stochastic reasonings.     

中立型时间延滞抛物方程初边值问题的有限元方法

１引言在生物学、统计学、控制论及航天技术等领域的研究中,经常出现由时间延滞偏微分方程所刻划的数学模型．目前仅有［１］等对这类方程在解的性质方面作过研究．本文考虑最简单的中立型时间延滞抛物方程初边值问题的有限元方法,其中 为常数, 为正常数, 为Ｒ中具有光滑边界 的有界区域． 当 时,（１．１）就是通常的抛物方程初边值问题．讨论（１．１）有限元逼近的难点在于函数 对时间导数一般不存在,且ｔ时刻函数ｕ（ｘ,ｔ）总与ｔ－时刻函数ｕ（ｘ,ｔ－ｒ）有关．为克服这一困难,我们将时间以ｒ为单位进行剖分,在一定条件下…     

Digital stabilizer design for a switched linear system

We consider the problem of designing a digital controller stabilizing a continuoustime switched linear system. Our approach to stabilization includes the construction of a continuous-discrete time closed-loop system, the passage to its discrete-time model, and the subsequent discrete-time controller design based on simultaneous stabilization methods.