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Many dynamical systems are modeled as vector second-order differential equations. This paper presents analysis and synthesis conditions in terms of LMI with explicit dependence in the coefficient matrices of vector second-order systems. These conditions benefit from the separation between the Lyapunov matrix and the system matrices by introducing matrix multipliers, which potentially reduce conservativeness in hard control problems. Multipliers facilitate the usage of parameter-dependent Lyapunov functions as certificates of stability of uncertain and time-varying vector second-order systems. The conditions introduced in this work have the potential to increase the practice of analyzing and controlling systems directly in vector second-order form.

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one-parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non-singular and singular; (c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler-Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double-walled carbon nanotube. Problems have been discussed in the first paper of the aforementioned set. Problems are discussed in the present paper.

SUMMARY The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill-conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

A beam design method based on signal-to-leakage-plusnoise ratio (SLNR) has been recently proposed as an effective scheme for multiuser multiple-input multipleoutput downlink channels. It is shown that its solution, which maximizes the SLNR at a transmitter, can be simply obtained by the generalized eigenvectors corresponding to the dominant generalized eigenvalues of a pair of covariance matrices of a desired signal and interference leakage plus noise. Under time-varying channels, however, generalized eigendecomposition is required at each time step to design the optimal beam, and its level of complexity is too high to implement in practical systems. To overcome this problem, a predictive beam design method updating the beams according to channel variation is proposed. To this end, the perturbed generalized eigenvectors, which can be obtained by a perturbation theory without any iteration, are used. The performance of the method in terms of SLNR is analyzed and verified using numerical results.

In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S. Naboko and G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not. 4 (2009), 736–764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P−1 = PT. An n × n real matrix Y is called a generalized centro-symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro-symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro-symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro-symmetric matrix, a generalized centro-symmetric solution (least squares generalized centro-symmetric solution) can be obtained within a finite number of iterations in the absence of round-off errors, and the least Frobenius norm generalized centro-symmetric solution (the minimal Frobenius norm least squares generalized centro-symmetric solution) can be derived by choosing a special kind of initial generalized centro-symmetric matrices. We also obtain the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix Y0 in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.

The new block and the block diagonal Lagrange iterative methods together with the block generalizations of the Newton–Rayleigh type methods are proposed. It is also shown that the Jacobi–Davidson correction vector is a Newton–Raphson correction vector for the Lagrange functional of the generalized eigenvalue problem. For a simplification of a solution of the Newton–Raphson equation for calculations of correction vectors, a skeleton matrix approximation was introduced and used in the new methods as well in a few known ones. The numerical algorithms of the new methods are described in details and their performances are compared in several numerical test calculations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

In this paper, we describe a linear boundary non-collocated feedback controller to stabilize a string system with variable coefficients and investigate the exponential stability of the closed-loop system. We show that the system associates with a C0 semigroup. Through the asymptotic analysis technique, we get the asymptotic values of all eigenvalues of the system operator A. Furthermore, we prove that there is a sequence of generalized eigenvectors of A that forms a Riesz basis with parentheses for the energy state space. Hence, the system satisfies the spectrum determined growth assumption. Finally, we show that the system is exponentially stable with suitable choice of the feedback gains.

For conservative linear chain-structured mass-spring vibration systems with n degrees of freedom, the conjecture of Mikota [1] on the natural frequencies is verified. In addition, the eigenvalue/eigenvector problem of a related square root matrix is solved.

The article is devoted to the solution of the infinite-dimensional variant of the complex moment problem, and to the uniqueness of the solution. The main approach is illustrated for the best explanation on the one-dimensional case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available., Comment: 19 pages, 3 figures

Four classes of nonlinear wave equations are joined in one generalized nonlinear wave equation. A theorem is proved that the whole series of matrix functions satisfy the generalized wave equation. A justification of rotational properties of matrix solutions is given and a mathematical model of the ring vortex around the acute edge is proposed using of matrix solutions.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$ . If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$ , then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.

Insensitive and robust control design using output-feedback eigenstructure assignment for linear multivariable systems is considered in this paper. A parametric expression of closed-loop eigenvectors and generalized eigenvectors is developed. It can cope with the case where the closed-loop eigenvalues are multiple and/or the same as the open-loop ones so that the system to be designed can be uncontrollable and/or unobservable. The controller designed via output-feedback eigenstructure assignment is expressed by proposed parameter vectors. The freedom provided by output-feedback eigenstructure assignment is used to optimize some performance functions which are used to measure the sensitivity of the closed-loop matrix and the robustness of the closed-loop system. Copyright © 2000 John Wiley & Sons, Ltd.

The purpose of this paper is to provide explicit transient solutions for the multiserver queueing system Geom n /Geom n /c/N+c . The method proposed here can also be used for obtaining transient solutions of Markov chains having the transition matrix of Hesselberg type. To support this, we also consider a more complex model such as GI/M/c/N+c. In our analysis, we use eigenvalues and generalized eigenvectors of transition probability matrices. Since we use the Jordan canonical form from linear algebra, the method is good even if the eigenvalues are repeated. Numerical procedures for computations involved in various examples are also provided.

The Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. Parlett and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problem Ax = λBx is considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three-dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjecture.

For the computation of the singular behavior of an elastic field near a three-dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one-dimensional integral equation on a piecewise circular curve γ in IR3 depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline-Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.

Three techniques for robust eigenvalue assignment are presented. The first is well known and is based on iteratively assigning the closed-loop eigenvectors so as to be maximally orthogonal to one another. The second has been recently presented by the authors and is an improvement of the first which gives better results for problems where complex-conjugate eigenvalue pairs are to be assigned. The final method is new and is founded on the iterative replacement of the current closed-loop eigenvector matrix with a new matrix which is the projection of the columns of the nearest orthogonal matrix into the allowable eigenvector subspaces. Some numerical examples are given which are used to illustrate the improved results obtained using the second technique in place of the first and to compare these with the performance of the last algorithm which is based on an alternative approach.

This paper is concerned with the marginal models associated with a given multivariate first-order autoregressive model. A general theory is developed to determine when reductions in the known orders of the marginal models will occur. When the auto-regressive coefficient matrix has repeated eigenvalues, there may be global reductions in the marginal models. Zeros in the eigenvectors and generalized eigenvectors of the auto-regressive coefficient matrix lead to local reductions in the marginal models. The case when the autoregressive parameter matrix has systematic zeros is also investigated.

This paper gives a new characterization of the smallest eigenvalue for second order linear elliptic partial differential equations, not necessarily self-adjoint, with both natural and Dirichlet boundary conditions, and also give a new alternative numerical method for calculating both the smallest eigenvalue and corresponding eigenvector in the case of natural boundary conditions. The smallest eigenvalue, if appropriate sign changes are made, determines the stability of equilibrium solutions to certain second order nonlinear partial differential equations. The corresponding eigenvector enables one to determine the first approximation of the solution of the nonlinear equation to variations of the initial conditions from the equilibrium solution. These nonlinear equations are important in the applications. For these reasons it is important to have these characterizations of the smallest eigenvalue and eigenvector. Our method converts the determination of the eigenvalue and eigenvector to determining the solution of a stationary stochastic control problem. This latter problem is solved and from it a numerical scheme arises naturally. This method appears to have applications in solving other problems.