A stability criterion for a linear Volterra equation of convolution type (Q756079)

The author studies the stability of the equation \(x'(t)=Ax(t)+\int^{t}_{0}B(t-s)x(s)ds,\quad t\geq 0,\) in \({\mathbb{C}}^ n\). A necessary condition for stability (and asymptotic stability) is that the characteristic equation \(\det(zI-A-\hat B(z))=0\) should have no roots with positive real part. A sufficient condition for stability is that in addition there are at most finitely many roots on the imaginary axis and that they are all simple. Here the author assumes that \(\int^{\infty}_{0}s^ 2\| B(s)\| ds<\infty\) and then shows that in this case the zero solution is stable if and only if the resolvent is bounded. In the main result the author gives a condition on the roots of the characteristic equation on the imaginary axis that is necessary and sufficient for the boundedness of the resolvent.