In this paper, we consider the problem of robustness in the gap metric for infinite dimensional systems. Vy study the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are H∞ functions with continuous boundary values. To this end, the underlying Hankel and related operators, which are important in the gap optimization problem, are studied and relations between their singular values and vectors are established. A computational approach to the optimal robustness problem is developed for single-input/ single-output systems whose transfer function is an inner function in H∞ times a rational function. The procedure is applied to a general first-order delay system and a closed-form formula is obtained for the optimal controller. We present and discuss the frequency response plots of the compensated system for various values of time delay.