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This document is an editorial for a special issue of the International Journal of Robust & Nonlinear Control on the topic of time-delay systems. It provides a historical overview of functional differential equations and highlights the contributions of Nikolay Nikolayevich Krasovskii and Aristide Halanay to the field. The special issue includes 21 research papers covering stability, robustness, observer theory, predictors, and delay-based approaches to partial differential equation models. Each paper is briefly described, covering topics such as stabilization of uncertain systems, sampled-data control for nonlinear systems, observer theory for differential delay systems, sensor fault estimation, and control approaches for vehicular platoons. The summaries aim to provide library patrons with a quick understanding of the papers' contents and their relevance to their research interests. [Extracted from the article]

This paper presents the analysis of the stability properties of proportional-integralderivative (PID) controllers for dynamical systems with multiple state delays, focusing on the mathematical characterization of the potential sensitivity of stability with respect to infinitesimal parametric perturbations. These perturbations originate, for instance, from neglecting feedback delay, a finite-difference approximation of the derivative action, or neglecting fast dynamics. It is also shown that adding a low-pass filter, which might be necessary to reduce sensitivity to high-frequency noise, under certain conditions on the derivative feedback gain may destabilize the closed loop even for arbitrarily large cutoff frequencies. The analysis of these potential sensitivity problems leads us to the introduction of a "robustified" notion of stability called strong stability, inspired by the corresponding notion for neutral functional differential equations. We prove that strong stability can be achieved by adding a low-pass filter with a sufficiently large cutoff frequency to the control loop on the condition that the filter itself does not destabilize the nominal closed loop system. Throughout the paper, the theoretical results are illustrated by examples that can be analyzed analytically, including, among others, a third-order unstable system for which both proportional and derivative control action are necessary for achieving stability, while the regions in the gain parameter-space for stability and strong stability are not identical. Besides the analysis of strong stability, a computational procedure is provided for designing strongly stabilizing PID controllers. Computational case studies illustrating this design procedure complete the presentation. [ABSTRACT FROM AUTHOR]

This article focuses on the characterization of a particular spectral property called Multiplicity-induced-dominancy applying for linear dynamical systems described by delay-differential equations. More precisely, we characterize the property in the scalar neutral case with respect to the system parameters. Particular attention is paid to the so-called over-order multiplicities corresponding to real double and triple characteristic roots. [ABSTRACT FROM AUTHOR]

The problem of the asymptotic stability independent of delays for a class of linear systems including multiple delays is addressed. Both cases where the delays are allowed to vary independently of each other and where they are restricted to a one-dimensional subspace of the delay-parameter space are considered. It the latter case it turns out that the resulting dependency between the delays (rationally independent, rationally dependent, commensurate) plays an important role. The stability conditions are expressed in terms of the spectral properties of some appropriate complex matrices. As a consequence of the stability study, a complete characterization of the delay interference phenomenon is given. Furthermore, a connection is established with the stability theory for continuous-time delay-difference equations, subjected to delay perturbations. Various illustrative examples complete the paper. [ABSTRACT FROM AUTHOR]

In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions. [ABSTRACT FROM AUTHOR]

Abstract: Definitions for pseudospectra and stability radii of an analytic matrix function are given, where the structure of the function is exploited. Various perturbation measures are considered and computationally tractable formulae are derived. The results are applied to a class of retarded delay differential equations. Special properties of the pseudospectra of such equations are determined and illustrated. [Copyright &y& Elsevier]