The Problem of the Absolute Continuity for Lyapunov-Krasovskii Functionals

The condition of nonpositivity, almost everywhere, of the upper right-hand Dini derivative of a (simply) continuous function is not a sufficient condition for such function to be nonincreasing. That condition is sufficient for the nonincreasing property if the function is locally absolutely continuous. Therefore, if the time function obtained by the evaluation of a Lyapunov–Krasovskii functional at 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 nonpositive, then the conclusion that such function is nonincreasing cannot be drawn. As a consequence, related stability conclusions cannot be drawn. In this note, 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 Lyapunov–Krasovskii functional is locally Lipschitz with respect to the norm of the uniform topology, then the problem of the absolute continuity is overcome.