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The stability of interval matrices has been studied by many authors. In this paper, the stability of interval parameter matrices is introduced and some criteria for this stability are given. Our results are effective for testing the stability of interval matrices and are more general than previous results. Furthermore, criteria for the stability of time-varying matrices are obtained.

In this paper new sufficient conditions for stability and complete instability of an interval matrix JV(P, Q) are presented. They include the corresponding results derived by Heinen (1984) and Xu Daoyi (1985) as particular cases. Secondly, we present the concept of composite type for stability, and simple criteria for this type.

Nonnegative solutions of an interval linear system A I x = b I ( A I being an interval matrix and b I an interval vector) with additional column sum restrictions of the type σ i α ij a ij ∈[ c -j , c j ] ∀j are described by a system of linear inequalities with auxiliary variables.

In a recent paper by G. Mayer [8] the convergence of the sequence {[A]k} of the powers of an interval matrix [A] to the nullmatrix was investigated. In this note we give some conditions for the boundedness of the sequence {k−α[A]k}, where α is a nonnegative number. The connection to the α-stability of the set [A] is discussed.

Mittels des Satzes von Brouwer wird ein Satz bewiesen, nach dem man entscheiden kann, ob eine gegebene Intervallmatrix [S] die Einschlieβungseigenschaft bezuglich der unbekannten InversenA?1 einer bekannten MatrixA hat. Es wird vorgefuhrt, in welchen Fallen der Satz anwendbar ist und wie [S] in diesen Fallen gewahlt werden kann. A theorem is proved by means of Brouwer's theorem which allows to decide whether a given interval matrix [S] has the inclusion property relative to the unknown inverseA?1 of a known matrixA. It is demonstrated in which cases the theorem may be used and how to choose [S] in these cases.

We give a necessary and sufficient condition for the sequence { A k}of the powers of an interval matrix A to converge to the null matrix O . We construct a point matrix B which has spectral radius ϱ ( B ) less than one if { A k}converges.

New sufficient conditions for the stability of interval matrices are presented, based on the Lyapunov stability approach as well as on the frequency-domain approach. The conditions help to overcome the conservatism of the existing criteria for the stability of interval matrices based on a matrix approach. Several numerical examples are given to demonstrate the new conditions and to compare them with results previously reported.

In this paper some necessary and sufficient conditions for stability, complete instability and composite stability type of a class of interval matrices are given.

In association with robust control-system design and analysis, the Hurwitz property of interval matrices and interval polynomials has recently been actively investigated. However, its discrete counterpart, the convergence property, has seemingly not been much discussed. In this paper, this property is studied in comparison with the Hurwitz counterpart. Some conditions under which interval matrices or interval polynomials are convergent are derived.

We introduce interval multisplittings to enclose the setS={A?1b|A?[A], b?[b]}, where [A] denotes an interval matrix and [b] an interval vector. The resulting iterative multisplitting methods have a natural parallelism. We investigate these methods with respect to convergence, speed of convergence and quality of the resulting enclosure forS.

A counter-example is given to a recent result on the stability analysis of interval matrices by Jiang (1988). A sufficient condition is then presented for the asymptotic stability of the discrete-time interval system X(k + 1) = A I X(k), by testing the norms of extreme matrices of the interval matrix A I such that they are all less than one.

A new sufficient condition is presented for the stability of interval matrices based on Kharitonov's theorem (1978).

In a recent paper by Jiang (1988), a necessary and sufficient condition for asymptotic stability of the interval matrix of discrete-time systems is presented. In this contribution it is shown that this condition is not true in general. A counter-example is given and additional remarks are made.

It is shown that only two matrix sequences are to be constructed to solve a system of linear interval equations with an inverse stable, strongly regular interval matrix.

This paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right-hand sides vary in some real intervals. Finite and iterative methods are given, based on results from linear complementarity theory. Also regularity conditions, regularity testing, and computing the exact inverse of an interval matrix are dealt with.

Let A I = { A ∈ R n × n | A - ⩽ A ⩽ A + } be an interval matrix and 1 ⩽ p ⩽ ∞ . We introduce the concept of Schur and Hurwitz diagonal stability, relative to the Holder p-norm, of A I , abbreviated as SDS p and HDS p , respectively. This concept is formulated in terms of a matrix inequality using the p-norm, which must be satisfied by the same positive definite diagonal matrix for all A ∈ A I . The inequality form is different for SDS p and HDS p . The particular case of p = 2 is equivalent to the condition of quadratic stability of A I . The SDS 2 inequality is equivalent to the Stein inequality ∀ A ∈ A I : A T PA - P ≺ 0 , and the HDS 2 inequality is equivalent to the Lyapunov inequality ∀ A ∈ A I : A T P + PA ≺ 0 ; in both cases P is a positive definite diagonal matrix and the notation “ ≺ 0 ” means negative definite. The first part of the paper • provides SDS p and HDS p criteria, • presents methods for finding the positive definite diagonal matrix requested by the definition of SDS p and HDS p , • analyzes the robustness of SDS p and HDS p and • explores the connection with the Schur and Hurwitz stability of A I . The second part shows that the SDS p or HDS p of A I is equivalent to the following properties of a discrete- or continuous-time dynamical interval system whose motion is described by A I : • the existence of a strong Lyapunov function defined by the p-norm and • the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system’s trajectories.

In this paper we construct rigorous and realistic enclosures for the solutions of parameter-dependent systems of equations F( x , t ) = 0 . The enclosure of the solution x = x( t ) is given in the form x 0 + S( t − t 0 ) with interval vectors x0, t0 and an interval matrix S. This allows a detailed analysis of the sensitivity of the solution x( t ) to small changes of t and the influence of specific changes in selected parameters on the solution. The linear case is treated separately.

Transition states (index-1 saddle points) play a crucial role in determining the rates of chemical transformations but their reliable identification remains challenging in many applications. Deterministic global optimization methods have previously been employed for the location of transition states (TSs) by initially finding all stationary points and then identifying the TSs among the set of solutions. We propose several regional tests, applicable to general nonlinear, twice continuously differentiable functions, to accelerate the convergence of such approaches by identifying areas that do not contain any TS or that may contain a unique TS. The tests are based on the application of the interval extension of theorems from linear algebra to an interval Hessian matrix. They can be used within the framework of global optimization methods with the potential of reducing the computational time for TS location. We present the theory behind the tests, discuss their algorithmic complexity and show via a few examples that significant gains in computational time can be achieved by using these tests.

This paper gives a new proof of Mayer's theorem concerning the convergence of powers of an interval matrix.

In this note, we simplify the main theorem on power-boundedness of interval matrices of Garloff (1986). In addition, another new result about powers of interval matrices is given.

The stability of discrete-time system whose state matrix is an interval matrix is discussed. In terms of Lyapunov stability theory, some sufficient conditions are obtained which guarantee the stability of gray discrete-time systems (or interval discrete-time systems). Illustrative examples are given. >

In this paper, a sufficient and necessary condition for the asymptotic stability of the discrete linear interval system X(k + 1) = A1X(k) is presented; i.e. all the eigenvalues of each matrix AϵA1, (an interval matrix) have magnitudes less than 1 if and only if all the eigenvalues of every ‘extreme matrix’ of the interval matrix A1, have magnitudes less than 1.

Using interval matrices properties, we derive tests for weak stability, Schur stability, positive definiteness of a given real matrix as well as a test for a given Z-matrix to be a nonsingular M-matrix. Moreover, we also provide some necessary conditions for an interval matrix to be nonsingular.

Three types of errors are encountered in numerical analysis, namely: 1. Round-off errors, arising when numbers are rounded to fit a certain precision arithmetic; e.g. the case where 1/6 = 0.1666 ... is approximated to 0.167 on a three-digit machine. 2. Truncation errors, resulting when convergent series are truncated down to a number of terms, e.g. the case where π = 3.141 592 65 ... is approximated by π= 3.14. 3. Data errors, associated with the specific physical model under study. They represent a parameter’s uncertainties when it is determined through experimental measurements.

We show a possibility to determine the asymptotic convergence factor α T of the total step method x m+1 = Ax m +b in interval computations. Especially we set up a necessary and sufficient condition for α T to be less than the spectral radius of the absolute value of A if this interval matrix fulfils some assumptions.

Given an interval matrix system with discrete- or continuous-time dynamics, the flow-invariance of an arbitrarily time-dependent rectangular set with respect to this system is introduced as a concept of geometric nature. The case of rectangular sets with exponential time-dependence is separately explored. If there exist flow-invariant rectangular sets approaching the state space origin for an infinite time horizon, then the interval matrix system exhibits two special types of asymptotic stability, which we have called componentwise asymptotic stability and componentwise exponential asymptotic stability. An interval matrix system is shown to be componentwise exponential asymptotically stable if and only if it is componentwise asymptotically stable. A necessary and sufficient condition for the componentwise asymptotic stability of an interval matrix system is derived in a matrix form. Brief comments focus on three classes of interval matrix systems for which the componentwise asymptotic stability is equivalent to the standard asymptotic stability (defined in terms of any consistent finite-dimensional norm).

A Self-Validating Method for Solving Linear Programming Problems with Interval input Data. Linear programming problems are very important in many practical applications. They are usually solved by the simplex method. The computational results are, in general, good approximations to the solution of the problem. However, in some cases the computed approximation may be wrong due to round-off and cancellation errors. In practice it occurs frequently that the input data of a linear programming problem are not known exactly but are afflicted with tolerances. In this case it has to be precisely defined what a “solution” to such a problem is. A sensitivity or postoptimality analysis is necessary. In the following a method for linear programming problems with interval input data is described which computes guaranteed lower and upper bounds for all optimal vertices and the optimal value. The method controls rigorously all round-off errors and gives an automatic sensitivity analysis. As an example a diet problem is treated to demonstrate how the method in principle works. © Springer-Verlag/Wien 1988

This paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots, and applies them to the problem of robust pole-placement. We formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed-loop poles within desired intervals for plants with unknown-but-bounded parameter uncertainties. The algorithms are computationally reasonable and provide a useful addition to currently existing control CAD tools.

Jiang (1987) gave a sufficient condition for the asymptotic stability of interval matrices. We shall show here that his results can be derived in a trivial way using the Lyapunov equation. An extension of his results and a similar result for discrete systems is stated.

It will be shown that the sufficient conditions for the asymptotic stability of interval matrices proved in Mansour (1988) can be further simplified by considering only a part of the extreme matrices.

Jiang (1988) stated that the Schur stability of an interval matrix A I is equivalent to that of the vertices of A I . We point out, via a counter-example, that this result is incorrect.

The main results obtained recently by Shi and Gao are already known and can be strengthened.