线性时滞系统的稳定性和镇定问题

本文讨论了线性时滞系统(?)(t)=Ax(t)+Bx(t-r),给出了系统绝对稳定的充分条件,且直接用系数矩阵描述,并讨论了其受控系统(?)(t)=Ax(t)+Bx(t-r)+Cu(t)的镇定问题。     

不确定随机时滞系统的时滞相关鲁棒镇定

研究一类不确定随机时滞系统的时滞相关鲁棒镇定问题.通过引入参数化的中立型模型变换,构造Lyapunov-krasovskii泛函,运用线性矩阵不等式方法,得到了使得闭环系统为均方指数稳定的保守性较小的时滞相关鲁棒镇定条件.     

一类非线性时滞系统的时滞相关镇定及跟踪控制问题

本文讨论了一类满足Lipschitz条件的非线性时滞系统的镇定与跟踪控制问题.基于非线性状态反馈控制器,利用Lyapunov-Krasovskii泛函和矩阵理论,得到了系统时滞相关全局渐近镇定的新判据,并且保证了输出和状态跟踪控制的误差全局渐近收敛于零.本文推广了文献[9]所得到的结论.因此,本文所研究的模型及所给出的判定条件更具有一般性和实用性.     

一类离散不确定线性时滞系统的鲁棒镇定

研究具有时变不确定参数的离散线性时滞系统的鲁棒控制问题,其中不确定性满足匹配条件,利用Ｌｙａｐｕｎｏｖ确定性理论,提出了鲁棒稳定性控制器一种新的设计方法,得到了这类离散不确定线性时滞系统可鲁棒镇定的充分条件。     

模糊广义系统的时滞依赖稳定与镇定条件

研究T-S模糊广义系统的时滞依赖稳定与镇定问题.利用Lyapunov泛函方法,得到一个线性矩阵不等式(LMIs)形式的时滞依赖稳定条件.本文所提方法考虑以前方法中通常忽略的有用的项,引入松弛变量矩阵和自由权重矩阵,估计Lyapunov泛函导数的上界;在此基础上,设计状态反馈模糊控制器,保证了闭环系统是局部正则、局部无脉冲和渐近稳定的.所得结果无需矩阵分解和利用锥补线性化方法进行迭代,最后通过两个仿真示例表明了本文结果具有较小的保守性.     

非线性离散开关系统的鲁棒镇定问题

利用切换Lypunov函数方法,把非线性离散开关系统的鲁棒镇定问题转化成一个矩阵不等式的最优解问题,给出了在任意切换下具有非线性扰动的线性开关系统的可鲁棒镇定的充分条件,并进一步讨论了同类时滞开关系统的鲁棒镇定问提.最后把以上结论推广到广义开关系统,由于结果均以矩阵不等式形式给出,便于验证和实现.     

一类含参数的分块对称矩阵的正定性及应用

首先给出一种判断分块对称矩阵正定的方法,提供了确定一组尽可能小的参数,使一类含参数的分块对称矩阵正定的简单算法,然后,将其结果用于研究线性定常大系统的分散镇定性,得到了一类可分散镇定的线性大系统,并给出了相应的分散镇定算法,同文献中提供的方法相比,该算法不仅扩大了所考虑的系统范围,而且不会引起过高的反馈增益,同时还简单易算。     

具有时变时滞的不确定离散广义系统的稳定与镇定

本文主要研究具有时变时滞的不确定离散广义系统的稳定性与镇定.利用线性矩阵不等式方法,给出了具有时变时滞的离散广义系统稳定的充分条件,推广了历算广义系统稳定与镇定的相关结果.     

一类不确定多输入模糊双线性系统的输出反馈控制

针对一类具有范数有界参数不确定性的多输入模糊双线性系统,在系统的状态不完全可测的情况下,提出一种模糊镇定控制方法。设计模糊观测器估计系统的状态,基于模糊观测器,设计模糊控制器保证闭环系统的渐近稳定性。模糊控制器可以通过求解线性矩阵不等式(LMI)求得。仿真示例验证了设计方法的有效性。     

非线性随机时滞微分系统的脉冲镇定[英文]

本文研究一类非线性随机时滞微分系统的脉冲镇定.利用Lyapunov函数,Razumikhin和一些分析的技巧得到系统基于线性矩阵不等式形式的均方稳定性判据,该判据表明适当的脉冲可以用来镇定不稳定的随机时滞系统.与此同时,数值例子及仿真证明了本文方法的有效性.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

An eigenvalue problem arising in a patch formation model

A simple numerical scheme has been developed for the solution of the eigenvalue problem arising in a patch formation model given by Del Grosso et al. [1]. The scheme is based on finding bounds which separate the eigenvalues. The exact eigenvalues are obtained by solving an algebraic equation given by the corresponding regular Frobenius series solution. At the same time eigenfunctions may also be obtained from this series solution.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

On the general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations with singularities

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

Evans Function for Lax Operators with Algebraically Decaying Potentials

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.