Abstract: Continuous-time difference equations are dynamical systems in which the variable of interest x(t) is expressed as a function of x itself evaluated at previous times. These equations define a class of time-delay systems, which can be seen as the simplest class of neutral time-delay systems. In addition to their relations to other classes of time-delay systems, the study of continuous-time difference equations is also motivated by their links to hyperbolic partial differential equations in one space dimension. After presenting continuous-time difference equations and the motivations for their study, this talk will present some classic and more recent results on their stability analysis. We will focus on the linear case, which is already rich enough and allows one to observe many interesting phenomena. We begin by presenting some classic results on the stability of time-invariant linear difference equations with finitely many pointwise delays, notably the Hale–Silkowski strong stability criterion, originally obtained by spectral methods. We then present the case of systems with a time-varying matrix, showing that similar results hold, but they require different techniques as spectral methods are not available for general time-varying matrices. Finally, we present a simple equation with a single time-varying delay, showing that many subtleties appear due to the fact that the delay is time-varying. The recent results presented in this talk are based on a series of joint works with Yacine Chitour, Jaqueline Godoy Mesquita, and Mario Sigalotti.
