The asymptotic stability of systems with delay (Q1109209)

Consider the differential system \[ (1)\quad \dot x=f_ i(x(t))+ \sum^{m}_{j=1} F_{ij}(x(t))u_ j(x(t-\tau)),\quad \tau =const \geq 0, \] \[ x\in R^ n,\quad u\in R^ m,f_ i,F_{ij},u_ j\in C^ 1(\Omega),\quad \Omega \subset R^ n,\quad m\leq n. \] It is assumed that the zero solution \((x=0)\) of (1) in the case of no delay \((\tau =0)\) is asymptotically stable and that there exists in \(\Omega_ 0\subset \Omega\) a Lyapunov function. The author finds sufficient conditions for the existence of a finite domain of attraction of the solution \(x=0\) of (1).