| Absztrakt: | We consider a system of functional differential equations (1)
$x'(t)=F(t,x\sb t)$ having finite delay h and satisfying $F(t,0)=0$.
The object is to give conditions on Lyapunov functionals to ensure
stability without asking that F(t,$\phi)$ be bounded for $\phi$
bounded. We begin by surveying seven classical examples in which
we note that an $L\sp 2$- norm of the solution x(t) frequently
appears in the derivative of the Lyapunov functional, but that
investigators ignore it in favor of a pointwise norm on x(t).
We show that the $L\sp 2$-norm can be much more useful than the
pointwise norm. \par
A measurable function $\eta$ : $R\to R$
is said to be integrally positive with parameter $\delta >0$
($\eta\in IP(\delta))$ if whenever $\{t\sb i\}$ and $\{\delta\sb
i\}$ satisfy $t\sb i+\delta\sb i<t\sb{i+1}$, $\delta\sb i\ge
\delta$, then $\sum\sp{\infty}\sb{i=1}\int\sp{t\sb i+\delta\sb
i}\sb{t\sb i}\eta (t)dt=\infty.$ Let $W\sb i$ denote a nonnegative
increasing function with $W\sb i(0)=0$ and let $\Vert\vert \phi
\Vert\vert$ denote the $L\sp 2$-norm of $\phi$. The following
is a typical result. Theorem. Let $\eta \in IP(h\sb 1)$ with
$0<h\sb 1<h$ and suppose there is a continuous functional V(t,$\phi)$
satisfying $(i)\quad W\sb 1(\vert \phi (0)\vert)\le V(t,\phi)\le
W\sb 2(\vert \phi (0)\vert +W\sb 3(\Vert\vert \phi \Vert\vert)$
and $(ii)\quad V'\sb{(1)}(t,x\sb t)\le -\eta (t)W\sb 4(\Vert\vert
x\sb t\Vert\vert.$ Then $x=0$ is asymptotically stable. It is
interesting to note that this result fails for ODE's.}
\RV{T.Burton |