Asymptotic stability of linear delay differential equations (Q2761989)

The authors give necessary and sufficient conditions for the asymptotic stability of the zero solution to the system of linear delay differential equations of the form NEWLINE\[NEWLINE x'(t)=\alpha Ax(t)+\beta Ax(t-\tau), NEWLINE\]NEWLINE where \(A\) is an \(n\times n\)-matrix, \(\tau>0\) is constant, and \(\alpha>0\), and \(\beta<0\). They reduce this to systems of 1st- and 2nd-order problems. The stability results are given in terms of the eigenvalues of \(A\). The proof of the results are carried out by an application of Pontryagin's criterion for quasi-polynomials to the characteristic functions of subsystems of the delay differential equations. The authors give some examples.