Remarks on Lyapunov-Krasovskii Functionals with Weak Decay Rates

In the existing Lyapunov literature, the decay rate of a Lyapunov-Krasovskii functional along trajectories is commonly required to be comparable to the norm of the state variables (or output variables) over the delay interval. In this work we consider a class of Lyapunov-Krasovskii functionals whose decay rates along trajectories depend only on the current values of the state or output variables. We show that for one type of input-to-output stability property (which leads to input-to-state stability when the output and the state variables are the same), a Lyapunov-Krasovskii functional with the weaker decay rate will still be sufficient for the corresponding stability property