The problem of the absolute continuity for Liapunov-Krasovskii functionals

The condition of non-positivity, almost everywhere, of the upper right-hand Dini derivative of a (simply) continuous function is not a sufficient condition for such function to be non-increasing. That condition is sufficient for the non-increasing property if the function is locally absolutely continuous. Therefore, if the time function obtained by the evaluation of a Liapunov-Krasovskii functional on the solution of a time-delay system is not locally absolutely continuous, but simply continuous, and its upper right-hand Dini derivative is almost everywhere non-positive, then the conclusion that such function is non-increasing cannot be drawn. And, as a consequence, related stability conclusions cannot be drawn. In this paper such problem is investigated for input-to-state stability concerns of time invariant time-delay systems forced by measurable locally essentially bounded inputs. It is shown that, if the Liapunov-Krasovskii functional is locally Lipschitz with respect to the norm of the uniform topology, then the problem of the absolute continuity is overcome.