Uniform asymptotic stability in functional differential equations

Abstract:

The classical uniform asymptotic stability result for a system of functional differential equations \begin{equation}\tag {$1$} x’ = F(t,{x_t})\end{equation} calls for a Liapunov functional $V(t,\phi )$ satisfying $W(|\phi (0)|) \leqslant V(t,\phi ) \leqslant {W_1}(|\phi (0)|) + {W_2}(|||\phi |||),{V’_{(1)}} \leqslant - {W_3}(|\phi (0)|)$, and $|f(t,{x_t})|$ bounded for $|||{x_t}|||$ bounded. We show that it is not necessary to require $|f(t,{x_t})|$ bounded. Here, $||| \cdot |||$ is the ${L^2}$-norm.