Synchronization of complex switched delay dynamical networks with simultaneously diagonalizable coupling matrices

This paper studies local exponential synchronization of complex delayed networks with switching topology via switched system stability theory. First, by a common unitary matrix, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when all subnetworks are synchronizable, a delay-dependent sufficient condition is given in terms of linear matrix inequalities （LMIs） which guarantees the solvability of the synchronization problem under an average dwell time scheme. We extend this result to the case that not all subnetworks are synchronizable. It is shown that in addition to average dwell time, if the ratio of the total activation time of synchronizable and non-synchronizable subnetworks satisfy an extra condition, then the problem is also solvable. Two numerical examples of delayed dynamical networks with switching topology are given, which demonstrate the effectiveness of obtained results.

The Explicit Expressions of Level-k Circulant Matrices of Type(n_1,n_2.…,n_k) and Their Some Properties

In this paper, we give the explicit expressions of level-k circulant matrices of type (n1,n2,…nk) and of order n1n2…nk,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level-k circulant matrices are derived,and it is also proved that the sort matrices are unitarily diagonalizable.

Weakly Approximate Diagonalization of Homomorphisms into Finite von Neumann Algebras

Let A be a unital C^(∗)-algebra and B a unital C^(∗)-algebra with a faithful traceτ.Let n be a positive integer.We give the definition of weakly approximate diagonalization(with respect toτ)of a unital homomorphismφ:A→Mn(B).We give an equivalent characterization of McDuff Ⅱ_(1) factors.We show that,if A is a unital nuclear C^(∗)-algebra and B is a type Ⅱ_(1) factor with faithful traceτ,then every unital^(∗)-homomorphism φ:A→M_(n)(B)is weakly approximately diagonalizable.If B is a unital simple infinite dimensional separable nuclear C^(∗)-algebra,then any finitely many elements in Mn(B)can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same.

ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRICMULTIPLE EIGENVALUE PROBLEMS

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for A symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

A Note on the Hoffman-Wielandt Theorem for Generalized Eigenvalue Problems

Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where∧ = diag（α1, α2, …, αn）, Ω= diag（βl, β2, …βn）,Г=diag（γ1,γ2,…,γn）, △=diag（δl,δ2,…,δn）.Let ρ（（α,β）, （γ,δ））=｜αδ-βγ｜/√｜α｜^2＋｜β｜^2√｜γ｜^2＋｜δ｜^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such thatn∑i=1[ρ（（αi,βi）,（γτ（i）,δτ（i）））]^2≤n[1-1/κ^2（Y）κ^2（Q）（1-d2F（Z,W）/n）],where κ（Y） = ｜｜Y｜｜2｜｜Y^-1｜｜2,Z= （A,B）,W= （C, D） and dF（Z,W） = 1/√2｜｜Pz＊ -Pw＊｜｜F.

Sufficiency of Kalman's Rank Condition for the Approximate Boundary Controllability on an Annular Domain

In this paper the author establishes the sufficiency of Kalman’s rank condition on the approximate boundary controllability at a finite time for diagonalizable systems on an annular domain in higher dimensional case.

Sign Patterns of Orders up to 4 that Allow Diagonalizability

Finding the necessary and sufficient conditions for a sign pattern to allow diagonalizability is an open problem. In this paper,we identify sign patterns of up to four orders that allow diagonalizability.