Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback

In many situations in physics, engineering and biology time delays arise naturally due to the time needed to transport information from one part of the system to another and/or to react to incoming information. When differential equations are used in the mathematical modelling, then incorporating time delays leads to a description by a delay differential equation. We consider here a class of second-order scalar delay equations without instantaneous feedback, where the delays enter according to a distribution function. This is a natural description whenever there is more than one delay. In this article we show that for this class of systems one can derive stability information about the distributed-delay system by considering the single-delay system where the delay is the mean delay of the distribution function. More specifically, we prove that the asymptotic stability of the zero solution of the second-order delay equation with symmetric delay distribution is implied by the stability of the associated mean-delay equation. Our proof is based on the comparison of stability charts of the two equations.