On the stability of a periodic system of delay differential equations (Q2703890)

The authors consider the system of delay differential equations NEWLINE\[NEWLINEJ_{2k}{dx(t)\over dt}= H_1(t) x(t)+ H_2(t) x(t-\omega),NEWLINE\]NEWLINE with \(x: \mathbb{R}\to \mathbb{R}^{2k}\), \(\omega> 0\), \(H_1\) and \(H_2\) are \(\omega\)-periodic real symmetric matrix functions, Lebesgue-integrable on the interval \([0,\omega]\), the matrices \(H_i(t)\) are positive definite a.e. on \(\mathbb{R}\), and \(J_{2k}\) is a constant skew-symmetric nonsingular matrix. The authors give necessary and sufficient conditions for this system to be asymptotically stable. The conditions given concern the canonical equations \(J_{2k}{dx\over dt}= H(t)x\), with \(H= H_1+ H_2\) and \(H= H_1- H_2\).