Estimation of magnitude of retardation in linear differential systems with deviated argument (Q789592)

The authors consider the following systems: \[ (1)\quad \dot x(t)=Ax(t)+Bx(t-\tau), \] \[ (2)\quad \dot x(t)=Ax(t)+Bx(t- \tau)+Q(x(t),x(t-\tau)), \] \[ (3)\quad \dot x(t)=(A+B)x(t), \] where \(x\in {\mathbb{R}}^ n\), \(t\geq t_ 0\), \(\tau>0\), A and B are constant matrices, Q is a function such that there exist solutions of (2) on \([t_ 0,\infty).\) They calculate the estimate \(\tau_ 0\) of the magnitude of \(\tau\) such that if the solution \(x_ 0(t)\equiv 0\) of (3) is asymptotically stable, then for \(\tau<\tau_ 0\) (i) the solution x(t)\(\equiv 0\) of (1) is asymptotically stable, (ii) the solution x(t)\(\equiv 0\) of (1) is stable under constantly acting perturbations.