On stability of systems of delay differential equations (Q1570987)

The authors study a special class of the linear autonomous systems with constant delay of the form \[ \dot x(t)=\alpha Ax(t)+(1-\alpha)Ax(t-\tau), \] where \(A\) is an \(n\times n\)-matrix, \(\tau>0\) is constant, \(0\leq\alpha\leq 1\). Necessary and sufficient conditions for the asymptotic stability of the system are given in terms of the eigenvalues of \(A\). Due to the special form of the system the problem is reduced to systems of first- and second-order problems. The proof of the results are carried out by an application of the well-known Pontryagin criterion on the quasi-polynomial zeros [\textit{L. S.Pontryagin}, Am. Math. Soc., Translat., II. Ser. 1, 95-110 (1955); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 6, 115-134 (1942; Zbl 0068.05803)]. To determine the asymptotic stability four algorithmic tests are given. The paper is provided with two illustrative examples.