On stability of systems of delay differential equations

In this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the formx′(t)=αAx(t)+(1−α)Ax(t−τ),where A is an n×n matrix, τ>0 is constant, and 0⩽α⩽1. We reduce this to systems of first- and second-order problems. Our stability results are given in terms of the eigenvalues of A. The proof of our results are carried out by an application of Pontryaginʹs criterion for quasi-polynomials to the characteristic functions of subsystems of the delay differential equations. We also provide four algorithmic stability tests and include several examples.