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

n 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.