Stability implications of delay distribution for first-order and second-order systems

In application areas, such as biology, physics and engineering, delays arise naturally because of the time it takes for the system to react to internal or external events. Often a delay is not fixed but varies according to some distribution function. This paper considers the effect of delay distribution on the asymptotic stability of the zero solution of functional differential equations --- the corresponding mathematical models. We first show that the asymptotic stability of the zero solution of a first-order scalar equation with symmetrically distributed delay follows from the stability of the corresponding equation where the delay is fixed and given by the mean of the distribution. This result completes a proof of a stability condition in [Bernard, S., Belair, J. and Mackey, M. C. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B, 1(2):233--256, 2001], which was motivated in turn by an application from biology. We also discuss the corresponding case of second-order scalar delay differential equations, because they arise in physical systems that involve oscillating components. An example shows that it is not possible to give a general result for the second-order case. Namely, the boundaries of the stability regions of the distributed-delay equation and of the mean-delay equation may intersect, even if the distribution is symmetric.