Diagonal Riccati stability and positive time-delay systems

We consider a class of algebraic Riccati equations arising in the study of positive linear time-delay systems. We show that this class admits diagonal positive definite solutions. This implies that exponentially stable positive linear time-delay systems possess Lyapunpov–Krasovskii functionals of a simple quadratic form. We also show that for this class of equations, the existence of positive-definite solutions is equivalent to a simple spectral condition on the coefficient matrices.