Well-posedness and stability of boundary delay equations

Abstract

In this paper, we introduce the notion of boundary delay equations, establishing a unified framework for analyzing linear time-invariant systems with pure time-delayed boundary conditions. We establish mild sufficient conditions for the existence, uniqueness, and positivity of solutions. Furthermore, we derive spectral criteria for exponential stability. The conditions on the perturbation generalize well-known criteria for the generation of domain perturbations of positive semigroup generators. As an application, we present necessary and sufficient conditions for the exponential stability of positive hyperbolic systems with time-delayed boundary conditions.