We consider how small delays affect the integral-input-to-state stability (iiss) property for a system. Our result is similar to the input-to-state stability (iss) result obtained in [1]: the iiss property will be preserved in a practical and semi-global manner if the delay interval is small enough. However, since the iiss quantifies the robust stability in terms of a generalized \(L_1\) norm of the inputs instead of a generalized \(L_\infty \) norm of the inputs for the iss case, the techniques and proofs for the iss case do not apply to the iiss case directly. While the proofs in [1] are based on the Lyapunov-Razumikhin approach, our proofs are based on the iiss-Lyapunov functions for the zero-delay system. In addition to the interest by its own in showing how the iiss property is affected by small delays, the result also serves to the study of the iiss property for singularly perturbed systems.
