Growth conditions to ensure input-to-state stability of time-delay systems under point-wise dissipation

We propose relaxed Lyapunov-based conditions to ensure input-to-state stability (ISS) of nonlinear time-delay systems. Their strength lies in the fact that the dissipation rate of the Lyapunov-Krasovskii functional (LKF) involves only the current value of solution’s norm rather than the LKF itself. The additional requirement takes the form of a growth condition between the dissipation rate and its maximal increase along the system’s solutions. We show through examples that the obtained conditions are more general than existing techniques, including the strictification method through the addition of an exponential term in the integral kernel of the LKF, whose limitations are highlighted through a counter-example.