Stability of delay differential systems by several Lyapunov functions (Q2720032)

For the functional-differential equations with finite delay of the form NEWLINE\[NEWLINEx'(t)= f(t, x_t),\quad t\geq \tau,\tag{1}NEWLINE\]NEWLINE where \(x_t(\vartheta)= x(t+ \vartheta)\) for \(-r\leq\vartheta\leq 0\) with some \(r\geq 0\) and \(f: [\tau,\infty)\times C\to\mathbb{R}^l\), where \(C= C([-r, 0],\mathbb{R}^l)\) is the space of continuous functions, mapping \([-r,0]\) into the \(l\)-dimensional space \(\mathbb{R}^l\), the stability of the zero solution is considered.