随机系统均方稳定Lyapunov型方程

研究随机系统的均方稳定性．得到了一个Lyapunov型方程，并证明随机系统是均方稳定的充分必要条件是该Lyapunov型方程有正定解.     

全导数为半负定时Lyapunov函数的存在性

在文[1]中Lehnigk证明了对于实方阵A存在严格半正定阵C使系统x=Ax之零解渐近稳定的充分必要条件是Lyapunov矩阵方程A'B+BA=-C有正定矩阵解B.但由于C的构造复杂且对于给定的A只能得到一个C,这个结果事实上是难于应用的. 本文得到:若A没有这样的特征向量,其第l_1,…,l_m位分量为零,对于任意第l_1,…,1_m位的部分正定阵C,系统x=Ax之零解渐近稳定的充要条件是上述矩阵方程有正定矩阵解B.由于不须求解矩阵A的特征问题,且部分正定阵C的类型广泛,易于构造,从而圆满地解决了这一问题。     

一类混合型Lyapunov矩阵方程的对称正定解

本文研究了一类混合型Lyapunov矩阵方程的对称正定解问题。首先将此方程转化为等价的含参矩阵方程,然后运用矩阵分解和紧凸集上不动点定理,给出了方程具有对称正定解的一些必要和充分条件:其次建立两种求方程对称正定解的参数迭代算法,分析了迭代的收敛性及参数的选取方法,并指出这两种算法的适应性和特点;数值算例表明上述算法的可行性和有效性,并对比出两种迭代的敛速。     

解非线性方程的三个Steffensen型双侧逼近迭代算法

本文讨论了三类具有不同单调性质单向广义均差的非线性方程,在一定初值条件下极大、极小解的存在性,以及相应的Steffensen型双侧逼近迭代算法的收敛性,并给出了若干数值例子。     

矩阵方程X-m∑i-1A*iXδAi=Q的正定解

基于正规锥上单调算子的不动点定理,本文研究非线性矩阵方程X-m∑i-1A*iXδAi=Q的正定解.给出了正定解的存在性定理,并且构造了求解的m步定常迭代方法,最后证明了该迭代法的收敛性.     

子空间上矩阵方程AX=B的正定与半正定解

研究了矩阵方程ＡＸ＝Ｂ在子空间上的正定及半正定解并出了解的存在的充要条件及解的表达式。     

一类具有脉冲的向量型时滞微分系统的振动性

研究了一类向量型时滞脉冲微分方程系统的振动性问题,通过作变最变换,将常系数脉冲微分方程系统变为变系数非脉冲微分方程系统,得到了模型非振动解的渐近性态和方程任意解振动的充分条件,利用留数理论,得到了方程广义振动和广义非振动的充分条件。     

广义Degasperis—Proces方程的非解析行波解

本文利用动力系统分岔理论和广义函数理论,并结合相图分析的方法对广义Degasperis-Proces方程的非解析波解进行了研究,给出了不同非解析波存在的分岔条件,并给出了这些非解析行波解的精确参数表达式,此外本文对不同非解析行波解进行了分类,证明了Peakon解和Vallyon解是广义解而非弱解,而Compacton解是弱解。本文提出的方法为非线性波方程的非解析行波解的研究提供了一条可行的思路,给出的结论丰富了非解析行波解的研究结果。     

一类KdV-Burgers方程的概周期解

KdV-Burgers方程出现在许多物理模型中,是非线性科学领域中的重要模型之一.本文讨论一类具有阻尼和非齐次项的KdV-Burgers方程的概周期解存在性问题.首先利用Galerkin方法构造出方程的有界解,并利用一些数学不等式给出这个解的先验估计;然后利用所得的先验估计和标准的紧致性方法证明方程广义解的存在性;最后证明当方程的非齐次项函数是关于时间变量的概周期函数时,该广义解就是方程的概周期解.     

广义二阶不变凸向量变分类不等式

研究了广义二阶不变凸向量变分类不等式问题解的存在性，并讨论其与多目标优化问题解之间的关系。引入了两种广义二阶不变凸函数和一种广义二阶单调函数，并给出具体实例说明了它们的存在性。在广义二阶不变凸性假设下，利用分析的方法给出了向量变分类不等式与多目标优化问题解之间的关系。利用KKM定理，在广义二阶单调性假设下得到了向量变分类不等式问题解的存在性定理。研究表明在广义二阶单调性假设下，向量变分类不等式存在解，并且在适当的广义二阶凸性条件下，其解与多目标优化问题解之间相互等价。     

随机系统均方稳定性的Lyapunov型方程(英文)

研究随机系统的均方稳定性，得到了一个Ｌｙａｐｕｎｏｖ型方程，并证明随机系统是均方稳定的充分必要条件是该Ｌｙａｐｕｎｏｖ型方程有正定解。     

Numerical treatment of hemivariational inequalities in mechanics: two methods based on the solution of convex subproblems

Hemivariational inequality problems describe equilibrium points (solutions) for structural systems in mechanics where nonmonotone, possibly multivalued laws or boundary conditions are involved. In the case of problems which admit a potential function this is a nonconvex, nondifferentiable one. In order to avoid the difficulties that arise during the calculation of equilibria for such mechanical systems, methods based on sequential convex approximations have recently been proposed and tested by the authors. The first method is based on ideas developed in the fields of quasidifferential and difference convex (d.c.) optimization and transforms the hemivariational inequality problem into a system of convex variational inequalities, which in turn leads to a multilevel (two-field) approximation technique for the numerical solution. The second method transforms the problem into a sequence of variational inequalities which approximates the nonmonotone problem by an iteratively defined sequence of monotone ones. Both methods lead to convex analysis subproblems and allow for treatment of large-scale nonconvex structural analysis applications.The two methods are compared in this paper with respect to both their theoretical assumptions and implications and their numerical implementation. The comparison is extended to a number of numerical examples which have been solved by both methods.     

Strain analysis by a total generalized variation regularized optical flow model

In this paper, we deal with the problem of estimating the local strain tensor from a sequence of micro-structural images realized during deformation tests of engineering materials. Since the strain tensor is defined via the Jacobian of the displacement field, we propose to compute the displacement field by a variational model which takes care of properties of the Jacobian of the displacement. In particular, we are interested in areas of high strain. The data term of our variational model relies on the brightness invariance property of the image sequence. As prior we choose the second order total generalized variation of the displacement field. This splits the Jacobian into a smooth and a non-smooth part. The latter reflects the material cracks. An additional constraint is incorporated to handle physical properties of the non-smooth part for tensile tests. We prove that the resulting convex model has a minimizer and show how a primal-dual method can be applied to find a minimizer. The corresponding algorithm has the advantage that the strain tensor is directly computed within the iteration process. It is further equipped with a coarse-to-fine strategy to cope with larger displacements. Numerical examples with simulated and experimental data demonstrate the very good performance of our algorithm. In comparison to state-of-the-art engineering software, our method can resolve local phenomena much better.     

A note on some properties of the Perron root of nonnegative irreducible matrices

This paper deals with the Perron root of nonnegative irreducible matrices, all of whose entries are continuous bijective functions of some parameter vector. It is known that if the functions are log-convex, then the Perron root is a convex function of the parameter vector. In this paper, we strengthen this result by showing that the log-convexity property is also necessary when the Perron root is required to be convex for any nonnegative irreducible matrix. Furthermore, we show that a less restrictive requirement is sufficient when the matrix is confined to belong to two subsets of irreducible matrices. In particular, in case of positive semidefinite matrices, convexity is sufficient for the Perron root to be convex. This work was supported by the German Ministry for Education and Research.     

Cold Soliton of the General Nonlinear Discontinuity Equation

In this paper we consider generalized nonlinear discontinuity equation and try to obtain second-order approximation solution. In order to obtain these solutions we used modified Lindstedt-Poincare method. Then we extract solitonic solution from special case of generalized nonlinear discontinuity equation. These solution behave as particle like in the cold black holes and low temperature dark matter. Other application of these solution is in superfluid and superconductivity.     

半正定线性系统广义非定常多分裂二阶段迭代方法的收敛性

为了高效求解正定或半正定的大型稀疏线性方程组,在第一阶段采用经典矩阵分裂的基础上,广义非定常多分裂二阶段迭代方法的第二阶段分裂融合了多分裂和矩阵预处理技术,对非定常多分裂二阶段迭代方法进行了推广。为了研究收敛性,将该迭代方法的算法形式和逻辑语言表达形式改写为紧凑的迭代格式。由此得到,广义非定常多分裂二阶段迭代算法在一个充分条件下收敛。最后,具有五对角系数矩阵的大型稀疏线性系统的数值算例验证了广义非定常多分裂二阶段迭代算法的普适性,并且从迭代次数和\,CPU\,时间上体现了算法的高效性。     

一类含两个非线性项的三阶拟线性微分方程解的稳定性

应用Lyapunov函数方法讨论一类含两个非线性项的三阶拟线性微分方程解的稳定性,得到了此类方程解的稳定性的若干新结果,同时推广了作者已有的结果。     

Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure

The direct methods for the solution of systems of linear equations with a symmetric positive‐semidefinite (SPS) matrix A usually comprise the Cholesky decomposition of a nonsingular diagonal block A _???? of A and effective evaluation of the action of a generalized inverse of the corresponding Schur complement. In this note we deal with both problems, paying special attention to the stiffness matrices of floating structures without mechanisms. We present a procedure which first identifies a well‐conditioned positive‐definite diagonal block A _???? of A , then decomposes A _???? by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A _????. The Schur complement S is typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank of A or a lower bound on the nonzero eigenvalues of A are known, then the SVD can be implemented without any ‘epsilon’. Moreover, if the kernel of A is known, then the SVD can be replaced by effective regularization. The results of numerical experiments show that the proposed method is useful for effective implementation of the FETI‐based domain decomposition methods. Copyright © 2011 John Wiley & Sons, Ltd.     

可对称(半)正定化矩阵反问题的解及其最佳逼近

本文给出了矩阵反问题AX=B具有可对称正定化解与可对称半正定化解的必要充分条件,得到了通解的表达式,同时解决了方程的对称半正定化解对己给矩阵的最佳逼近问题。