Stability radii of some time-varying linear stochastic differential systems

Characterizations of a stability radius of a time-varying linear stochastic Ito system with respect to structured stochastic multiperturbations are given.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Quasi-integrals of even-dimensional linear differential systems with skew-symmetric coefficient matrices

For a linear nonstationary system with skew-symmetric continuously differentiable coefficient matrix of arbitrary even dimension, we construct its quasi-integrals and obtain effective coefficient estimates for their deviations from the corresponding integrals in the stationary case. For each trajectory of motion described by such a system and lying on the sphere of the corresponding radius, these estimates permit precisely indicating a domain on the sphere containing the trajectory on a nonsmall time interval. The estimates can also be used for expanding the multidimensional motion of a mechanical object into multicomponent elements of lower dimension.     

Convergence of discrete approximation for differential linear stochastic complementarity systems

Numerical Algorithms - We provide a new algorithm to generate images of the generalized fuzzy fractal attractors described recently by Oliveira and Strobin. We also provide some new results on the...     

Stabilization algorithm for linear time-varying systems

We consider the state feedback stabilization problem for a linear time-varying system. Attention is mainly paid to the reduction of the system to canonical form; to this end, we suggest an algorithm for constructing the transformation matrix. This algorithm is based on the solution of a hybrid system and, in contrast to the classical approach, does not require the multiple differentiability of the system parameters.     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

Spline monotone approximation with linear differential operators

Let f∈C³[a,b] and L be a linear differential operator such that L(f)≥0. Then there exists a sequence Q_n, n≥1, of polynomial splines with equally spaced knots, such that Q^(r), approximates f^(r), 0≤r≤s, simultaneously in the uniform norm. This approximation is given through inequalities with rates, involving a measure of smoothness to f^(s); so that L (Q_n)≥0. The encountered cases are the continuous, periodic and discrete.     

Polynomial matrix-fraction representations for linear time-varying systems

This paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time-varying discrete-time systems. It is shown that the time-varying analog of the polynomial model always has a free state-space. Necessary and sufficient conditions for the existence of Bezout factorizations are obtained in terms system theoretic properties of state-space realizatios.     

The reducibility of second-order linear time-varying homogeneous systems

Problems of the theory of the reduction of first- and second-order homogeneous time-varying systems are briefly described. Problems of the motion of a gyrohorizon compass and of the periodic motion of a rotor, attached to a flexible shaft are considered as interesting examples.     

Sensitivity of time-varying,linear optimal control systems

This paper analyzes the following fundamental question: In what precise sense, if any, is the closed-loop realization of an optimal control system less sensitive to parameter variations than the nominally equivalent open-loop system? This question is answered for time-varying, linear optimal control systems. It is shown that there is a closed-loop sensitivity reduction in terms of an inequality involving a particular integral-square of the first-order variation of the state due to first-order variations of plant parameters.     

On the spectrum of discrete time-varying linear systems

In this paper, we investigate the influence of small perturbations of the coefficients of discrete time-varying linear systems on the Lyapunov exponents. For that purpose we introduce the concepts of central exponents of the system and we show that these exponents describe the possible changes in the Lyapunov exponents under small perturbations. Finally, we present several formulas for the central exponents in terms of the transition matrix of the system and the so-called upper sequences. The results are illustrated by numerical examples.     

Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices

This paper investigates the relative controllability of delay differential systems with linear impulses and linear parts defined by permutable matrices. We use the impulsive delay Grammian matrix to discuss the relatively controllability of impulsive linear delay controlled systems and we use the Krasnoselskii's fixed point theorem to discuss the relatively controllability of impulsive semilinear delay controlled systems. Finally, two examples are presented to illustrate our theoretical results.     

Weak approximation for linear systems of quadrics

We give local conditions at ∞ ensuring that the intersection of n quadrics in PN, N?n, satisfies weak approximation.     

An algebraic analysis approach to linear time-varying systems

^** Email: zerz{at}mathematik.uni-kl.de This paper introduces an algebraic analysis approach to lineartime-varying systems. The analysis is carried out in an ‘almosteverywhere’ setting, i.e. the considered signals are smoothexcept for a set of measure zero, and the coefficients of thelinear ordinary differential equations are supposed to be rationalor meromorphic functions. The methodology is based on a normalform for matrices over the resulting ring of differential operators,which is a non-commutative analogue of the Smith form. Thisis used to establish a duality between linear time-varying systemson the one hand and modules over the ring of differential operatorson the other. This correspondence is based on the fact thatthe signal space is an injective cogenerator when consideredas a module over this ring of differential operators.     

Accurate solutions of diagonally dominant tridiagonal linear systems

In this paper, we settle Higham’s conjecture for the LU factorization of diagonally dominant tridiagonal matrices. We establish a strong componentwise perturbation bound for the solution of a diagonally dominant tridiagonal linear system, independent of the traditional condition number of the coefficient matrix. We then accurately and efficiently solve the linear system by the GTH-like algorithm without pivoting, as suggested by the perturbation result.