Editorial for the special issue ‘Time‐delay systems: Recent trends and applications’

Abstract

Differential-difference equations, differential integral equations, and functional differential equations have been subjects of recurrent study with origins in geometry and number theory. Picard, in 1908, emphasized the importance of considering hereditary effects in physical systems, suggesting the need for understanding the behaviour of complex functional equations that account for past influences * . Volterra's works in the early 20th century further delved into integro-differential equations, particularly in modeling viscoelasticity and species interaction.Driven by the necessity of modeling engineering systems and control, the field gained in popularity after the Second World War, specially in Soviet Union and United States. Despite engineers' awareness of hereditary effects, the lack of theoretical groundwork limited detailed discussions. Over the past five decades, extensive development in the theory of functional differential equations has occurred, becoming integral in various applications like viscoelasticity, mechanics, and biology. The 1950s witnessed notable activity in the field, with key publications from researchers such as Myshkis, Krasovskii, Razumikhin, Bellman, Cooke, Halanay, Yoshizawa and Hale shedding light on the subject until the early 1960s. Most research primarily focused on linear equations, especially concerning stability analysis under nonlinear perturbations. The use of Laplace transforms and Lyapunov's second method aided in understanding stability properties and understanding the ways the parameters affect the qualitative behavior of the dynamics. Presently, functional differential equations are considered as well developed as ordinary differential equations. However, the journey from ODE ideas to FDE expressions was lengthy, with the need for a new approach to tackle complex problems that were challenging through traditional means. Despite initial resistance, the modern analytic and geometric theory of FDEs has flourished, offering insights into various scientific disciplines.Year 2024 marks the 100th anniversary of the birth of two prominent figures in the field of time-delay systems: Nikolay Nikolayevich Krasovskii and Aristide Halanay, whose monographs, 1,2 and most notably their subsequent English translations 3,4 served as first and foremost references on the field-along with other major books dating from the 1960s. More than 60 years after the seminal monograph by Krasovskii (1959) and the one by Halanay (1963), time-delay systems can be considered as an established area, with research flourishing in many heterogeneous directions.The special issue collects 21 works by some of the leading researchers on the topic, focusing on fundamental problems, recent trends, and applications. We divided the contributions in five main sections, each of them devoted to a major active area of research in the broad field of time-delay systems. We start with four contributions on problems concerning stability and stabilization of time-delay systems. Then, we move to results focusing on uncer...