Set input-to-state stability for nonlinear time-delay systems with disturbances

We propose new results on input-to-state stability (ISS) subject to time delays in the input for compact, invariant sets that contain the origin. First, using nonlinear small-gain theory, we prove a Razumikhin-type theorem that ensures ISS for sets in the context of functional differential equations with delayed disturbances. Next we demonstrate that this theorem can be used to ensure set ISS for nonlinear systems with input delays and disturbances. In comparison to the existing research on robustness of set ISS with respect to time delays at the input, our results are more general, retain the ISS gain, and do not impose constraints on time delayed states. The advantages of the method are illustrated through two case-studies on set-stability for classes of nonlinear oscillators of practical interest.