Exponential stability of linear periodic difference-delay equations

Abstract

This paper deals with the stability of linear periodic difference delay systems, where the value at time $t$ of a solution is a linear combination with periodic coefficients of its values at finitely many delayed instants $t-\tau_1,\ldots,t-\tau_N$. We establish a necessary and sufficient condition for exponential stability of such systems when the coefficients have H\"older-continuous derivative, that generalizes the one obtained for difference delay systems with constant coefficients by Henry and Hale in the 1970s. This condition may be construed as analyticity, in a half plane, of the (operator valued) harmonic transfer function of an associated linear control system.