On the stability of periodic differential equation with delays (Q2387075)

The paper deals with the exponential stability of the zero-solution of the delayed equation \[ \dot x(t)=\sum_{i=1}^m A_i(t) x(t-g_i(t)),\quad t\in [0,\infty), \quad x(t)=\varphi(t),\quad t\in [a,0], \] where the \(\omega\)-periodic (\(\omega>0\)) \(n\times n\)-matrices \(A_i\) and the \(\omega\)-periodic delays \(g_i\) are measurable and essentially bounded, and \(t-g_i(t)\geq s_1\) for some \(s_1\). The solution of this problem is understood as a locally absolutely continuous on \([a,\infty)\) \(n\)-dimensional vector-function satisfying the above initial condition and the above equation almost everywhere on this ray. Considering the equivalent integral equation and applying iterated kernels, the authors get a Cauchy-type formula for the solutions. The obtained formula is used to get several sufficient conditions for the exponential stability of the zero solution of the considered equation. It is worth to note that usually (compare with Theorem 4 in the paper) the exponential stability is proved by assuming that some data functions are bounded and analytic.