Predictor Control for Non Forward Complete Nonlinear System With Time-Varying Input Delay

We consider
$\dot {\textrm {X}}(\textrm {t})=\textrm {X(t)}^{2}+\textrm {U}(\textrm {t}-\textrm {D}(\textrm {t}))$
, where D(t) is a long time-varying delay. If D(t) = 0,
$\textrm {U(t)}=-\textrm {X(t)}^{2}-\textrm {cX(t)},\,\,\textrm {c}>0$
is a simply control, but it just delays finite time escape for this system. We design a predictor control and prove that the attraction region is
$\textrm {X}(0) + \sup _{\theta \in [\varphi (0),\,0]} \int _{\varphi (0)}^{\theta } {\frac {\textrm {U}(\theta)}{\varphi ^{\prime }(\varphi ^{-1}(\theta))}\textrm {d}\theta } < \frac {1}{\sigma (0)}$
, with
$\varphi (\theta)=\theta -\textrm {D}(\theta)$
, and
$\sigma (\theta)=\varphi ^{-1}(\theta)$
. Further, the predictor control locally exponentially stabilizes this system.