Sensitivity of Schur stability of monodromy matrix

For Schur stable linear difference equation system with periodic coefficients, we prove continuity theorems on monodromy matrix which show how much change is permissible without disturbing the Schur stability, and some examples illustrating the efficiency of the theorems are given.     

On matrix types of prime varieties of associative algebras

Some answer is obtained to the problem by A. R. Kemer: We show that for a given number k and all sufficiently large p every prime variety of associative algebras of matrix type k over a field of characteristic p > 0 is regular.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

A perturbation method for optimizing matrix stability

We present the first practical perturbation method for optimizing matrix stability using spectral abscissa minimization. Using perturbation theory for a matrix with simple eigenvalues and coupling this with linear programming, we successively reduce the spectral abscissa of a matrix until it reaches a local minimum. Optimality conditions for a local minimizer of the spectral abscissa are provided and proved for both the affine matrix problem and the output feedback control problem. Experiments show that this novel perturbation method is efficient, especially for a matrix with the majority of whose eigenvalues are already located in the left half of the complex plane. Moreover, unlike most available methods, the method does not require the introduction of Lyapunov variables. The method is illustrated for a small size matrix from an affine matrix problem and is then applied to large matrices actually arising from more sophisticated control problems used in the design of the Boeing 767 jet and a nuclear powered turbo-generator.     

Pseudospectra for matrix pencils and stability of equilibria

The concept of ε-pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, ε-pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of ε-pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to time-dependent ordinary or partial differential equations; two cases, based on Burgers' equation, will be shown. This research has been supported by the Netherlands Organization for Scientific Research (N.W.O.)     

On different types of stability of linear polytopic systems

Different types of stability of linear uncertain systems are considered. Stability conditions in terms of nonsingularity are obtained. Numerical examples of using the Bernstein expansion of the determinant function are given.     

On solvability of some types of matrix equations over lattices

We present a formula for calculating the greatest solution to matrix equations over Boolean lattices and establish the smallest (greatest) solution to a matrix equation over a distributive lattice whose matrix enjoys certain properties.     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

On the theory of stability of matrix differential equations

We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.     

Fibre stability in anisotropic matrix at small strains

As acknowledged in almost all monographs on the fracture of composite materials, one of the major fracture mechanism in unidirectional fibrous composites under uniaxial compression along the reinforcing elements is the stability loss of the material structure (the structural instability). According to this mechanism, theoretical investigations of the fracture along the fibres are reduced to those of the stability loss in the material structure, and the value of external critical forces is accepted as the value of failure forces. At present, numerous theoretical investigations have been carried out in this field with the use of the three-dimensional linearized theory of stability in the framework of the piecewise-homogeneous body model. However, in all the investigations it is assumed that the matrix and the fibre material are isotropic. It is evident that in many cases it is necessary to take into account the anisotropy of the matrix material when investigating the stability loss of fibres. In view of the above, in the framework of the piecewise-homogeneous body model using the three-dimensional linearized theory of stability, the present paper considers the stability loss of the fibre in the anisotropic (transversally isotropic) matrix. The effect of the properties of the matrix material on the critical values of the external loading is examined.Submitted to the 10th International Conference of Mechanics of Composite Materials (Riga, April 20–23, 1998).Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 603–611. September–October, 1997.     

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions

Classical interpolation problems are concerned with the problem of finding an analytic function on the unit disk bounded by one which takes on prescribed values at certain prescribed points inside the disk (Pick-Nevanlinna) or on the boundary of the disk (Loewner). We consider the problem for matrix-valued functions having as many as poles ( a nonnegative integer) inside the disk but still uniformly bounded by one on the boundary of the disk. Our technique is an adaptation of that of Sz.-Nagy and Koranyi to spaces with an indefinite inner product. The problem arises in the broadband matching problem for electrical circuits and certain multichannel scattering problems in physics.Research supported by US National Science Foundation Grant MCS-8101678.     

A stability analysis of the jacobi matrix inverse eigenvalue problem

In this paper, we give a perturbation bound for the solution of the Jacobi matrix inverse eigenvalue problem.China State Major Key Project for Basic Researches.     

Wavelet frames with irregular matrix dilations and their stability

Let , B and A_j () be real nonsingular n×n matrices, λ_k () be real numbers. In this paper we present a sufficient condition for the system to be a frame for . This sufficient condition also shows the stability of the system with respect to the perturbation of matrix dilation parameters and the perturbation of translation parameters .     

Rate of stability of solutions of matrix polynomial and quadratic equations

In this paper the rate of stability of solutions of matrix polynomial equations of the typeA ₀+A ₁ X+A ₂ X ²+...+A _m X ^m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A ₀+A ₁ λ+A ₂ λ ²+...+A _m λ ^m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B ^* is positive semidefinite andC=C ^*,A=?D ^*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings.     

Monitoring the stability of the triangular factorization of a sparse matrix

We propose a method for bounding the numerical instability in Gaussian elimination which may result from the use of interchanges calculated from the sparsity pattern alone or calculated for another matrix having the same sparsity pattern. The computational cost of obtaining the bound is small compared with that of the elimination. Also we propose the use of a variant of iterative refinement to estimate the accuracy of the solution obtained.     

Finite time stability analysis of systems based on delayed exponential matrix

In this paper, we analyze finite time stability for a class of differential equations with finite delay. Some sufficient conditions for the finite time stability results are derived based on delayed matrix exponential approach and Jensen’s and Coppel’s inequalities. Finally, we demonstrate the validity of designed method and make some discussions by using a numerical example.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.