In this article we generalize the Popov criterion to the class of strongly stable infinite-dimensional linear systems; the semigroup is strongly stable and the input to state, state to output and input to output maps are all bounded on the infinite-time interval. One application is to show that integral control can be used to track constant reference signals for positive-real strongly stable systems in the presence of sectorial non-linearities. A second application is to show the robustness of asymptotic stability of positive-real strongly stable systems to a large class of non-linear perturbations. Systems satisfying the assumptions in this paper include dissipative systems with collocated actuators and sensors.