2-stability and 2-instability of linear time-invariant distributed feedback systems perturbed by a small delay in the loop

We consider a continuous-time, single-input, single-output, linear time-invariant distributed feedback system perturbed by a small delay in the loop. Conditions are given under which the L²- stability and L²-instability of this feedback system can be deduced from those of the unperturbed system (obtained by setting the delay equal to zero). These conditions require the examination of a high-frequency approximation to the unperturbed system. The precise results are stated in Theorem 1. Our condition for neglecting the small delay in the L²-stability problem is stricter than the well-posedness condition given by Willems.