Conditions for convergence of the fundamental matrix of linear time‐invariant time‐delayed singular systems

Abstract

In the introductory part of this paper the so-called general Bromwich-Wagner's theorem has been generalized. Throughout the main part of the paper, an attempt has been made to derive conditions for convergence of the fundamental matrix of linear time-invariant timedelayed singular systems. The paper ends with a counterexample pointing out the fact that certain results of the theory of linear time-invariant singular systems are incorrect.2000 Mathematics Subject Classification: 93C05, 34A30, 34D05.1. Introduction. The introductory part is provided to give the generalization of the so-called general Bromwich-Wagner's theorem from Cauchy's calculus of residues, which has proven to be a powerful tool in the Laplace transforms theory. The main part is concerned with the theory of singular systems. On the basis of the result worded in the form of the theorem at the very beginning, throughout the main part of the paper, an attempt has been made to derive conditions for convergence of the fundamental matrix of linear time-invariant time-delayed singular systems. The paper ends with a counterexample pointing out the fact that certain results of the theory of linear timeinvariant singular systems, as for example a condition for the existence of infinite frequency dynamic modes (of impulsive modes) in a solution formulation of linear time-invariant singular systems, are incorrect.In the complex functions theory, more precisely in the theory of Cauchy's calculus of residues, the following result, which is well known as the general Bromwich-Wagner's theorem (see [4, Theorem 1, page 215]), is a fundamental result from the viewpoint of determining the function t f (t) whose Laplace transform is z F(z).

Stability of discrete descriptive time delayed system in the sense of Lyapunov

“…To the best of our knowledge, some attempts in stability investigation of (LCSTDS) was due to [14,15,16] where sufficient conditions for convergence of appropriate fundamental matrix were established.…”

Lyapunov stability theory: a quite new approach - discrete descriptive time delayed system

“…To the best of our knowledge, some attempts in stability investigation of (LCSTDS) was due to Saric (2001Saric ( , 2002aSaric ( , 2002b where sufficient conditions for convergence of appropriate fundamental matrix were established.…”

Further Results on Descriptor Time Delayed System Stability Theory in the sense of Lyapunov: Pandolfi Based Approach

“…To be familiar with this matter see the list of attached references. To the best of our knowledge, some attempts in stability investigation of (LCSTDS) was due to [12], [13] and [14] where sufficient conditions for convergence of appropriate fundamental matrix were established. Recently, in the paper of [20] the problem of robust stability and established for uncertain (LCSTDS) was addressed and necessary and sufficient conditions were obtained in terms of strict LMI.…”