The paper formulates conditions under which the roots closest to the imaginary axis (critical roots) of the characteristic equation of a linearized system are real for the maximum possible degree of stability of the closed-loop control system of a technological process with pure delay. For the parameters of the controllers corresponding to the maximum degree of stability, these roots are multiples. Their multiplicity order is one more than the number of coefficients in the transfer function of the controller. It is demonstrated that for a typical technological control object, these conditions are satisfied for all “serial” control laws. This allowed for obtaining analytical expressions for optimal settings and limiting degrees of stability as functions of object parameters for typical dynamic characteristics of technological processes. The paper considers the problem of robust stability for control systems with an object containing pure delay. It has been proven that in the maximum stability problem, the operations of maximizing over controller parameters and minimizing over the set of possible object parameters can be interchanged. Therefore, selecting robust settings amounts to determining the minimum of the maximum stability over the set of possible object parameter values. Controllers with such settings are suitable, without modification, for a whole class of technological processes.