Singularity conditions for the non-existence of a common quadratic Lyapunov function for pairs of third order linear time invariant dynamic systems

The condition that a finite collection of stable matrices {A₁, … , A_M} has no common quadratic Lyapunov function (CQLF) is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters. These conditions are applied to the case of two stable 3 × 3 matrices, where they are used to derive necessary and sufficient conditions for the non-existence of a CQLF.     

The Lyapunov spectrum of families of time-varying matrices

For -families of time varying matrices centered at an unperturbed matrix, the Lyapunov spectrum contains the Floquet spectrum obtained by considering periodically varying piecewise constant matrices. On the other hand, it is contained in the Morse spectrum of an associated flow on a vector bundle. A closer analysis of the Floquet spectrum based on geometric control theory in projective space and, in particular, on control sets, is performed. Introducing a real parameter , which indicates the size of the -perturbation, we study when the Floquet spectrum, the Morse spectrum, and hence the Lyapunov spectrum all coincide. This holds, if an inner pair condition is satisfied, for all up to at most countably many -values.

     

Estimating the spectral radius of a real matrix by discrete Lyapunov equation

For any real matrix A, this paper is concerned with the estimation of the spectral radius of A. The relationship between the weighted norm and the discrete Lyapunov equation of the matrix A is obtained. On the basis of the relationship, an iterative algorithm is presented to obtain the spectral radius of A and to estimate the solution of the corresponding linear discrete system. Several numerical examples are given to show that the iterative algorithm is effective.     

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T?=?c ₁ T ₁?+?c ₂ T ₂?+?c ₃ T ₃???c ₄(T ₁ T ₂?+?T ₃ T ₁?+?T ₂ T ₃), where T ₁, T ₂, T ₃ are three n?×?n tripotent matrices and c ₁, c ₂, c ₃, c ₄ are complex numbers with c ₁, c ₂, c ₃ nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.     

A Relation Between the Weighted Logarithmic Norm of a Matrix and the Lyapunov Equation

In this note, we define a weighted logarithmic norm for any matrix. In the case when a stable matrix A is considered, we obtain the relationship between the maximal eigenvalue of a symmetric positive definite matrix H which is a solution of the Lyapunov equation and the weight H logarithmic norm of A. It can be seen that the weighted logarithmic norm of A is always a negative value in this case. Several examples illustrate the relationship.     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

On nonsingularity of combinations of two group invertible matrices and two tripotent matrices

Let T ₁ and T ₂ be two n?×?n tripotent matrices and c ₁, c ₂ two nonzero complex numbers. We mainly study the nonsingularity of combinations T?=?c ₁ T ₁?+?c ₂ T ₂???c ₃ T ₁ T ₂ of two tripotent matrices T ₁ and T ₂, and give some formulae for the inverse of c ₁ T ₁?+?c ₂ T ₂???c ₃ T ₁ T ₂ under some conditions. Some of these results are given in terms of group invertible matrices.     

Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations

A unified approach is proposed to solve the estimation problem for the solution of continuous and discrete Lyapunov equations. Upper and lower matrix bounds and corresponding eigenvalue bounds of the solution of the so-called unified algebraic Lyapunov equation are presented in this paper. From the obtained results, the bounds for the solutions of continuous and discrete Lyapunov equations can be obtained as limiting cases. It is shown that the eigenvalue bounds of the unified Lyapunov equation are tighter than some parallel results and that the lower matrix bounds of the continuous Lyapunov equation are more general than the majority of those which have appeared in the literature.     

Minimal residual methods for large scale Lyapunov equations

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.     

Sylvester与Lyapunov方程向后误差分析

利用矩阵Kronecker积的性质,研究Sylvester矩阵方程Ax YB=C与Lyapunov矩阵方程ATX XA=-Q(Q0)的向后误差,获得了这两类矩阵方程向后误差η(X,Y)与η(X)的精确表达式及其更易计算的上下界.这些结果是对有关文献相应结果的改进与补充.     

Solution of Parabolic Equations by Means of Lyapunov Functionals

We propose a new approach to defining the notion of a solution to linear and nonlinear parabolic equations. The main idea consists in studying connections between solutions to dynamic problems in the variational shape and the properties of the corresponding Lyapunov functionals which are strictly decreasing along the trajectories of the above-mentioned dynamic equations except for the equilibrium points. It turns out that the families of Lyapunov functionals constructed by T. I. Zelenyak enable us to propose a new approach to defining solutions to both linear and nonlinear parabolic problems. All results are given in the case of smooth solutions. Note that the Lyapunov functionals can be used for studying solutions with unbounded gradients.     

On the eigenvalue estimation for solution to Lyapunov equation

This paper deals with the problems of eigenvalue estimation for the solution to the perturbed matrix Lyapunov equation. We obtain some eigenvalue inequalities on condition that X is a positive semidefinite solution to the equation A^TXA − X = −Q, which can be used in control theory and linear system stability.     

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.