This paper addresses the nature and multiplicity of an optimally designed electric field for controlling quantum-dynamical processes. A rather general cost functional is considered, with only mild conditions called for amongst the various operators involved. An explicit upper bound on the magnitude of the controlling electric field is attained in terms of the norms of various operators entering into the control cost functional. An earlier work employing first-order perturbation theory arguments showed that, under rather mild assumptions, a denumerably infinite number of control-field solutions exists for the optimal control problem. In the present work, it is shown that through a bound on the remainder of the nonlinear terms in the expansion, this same conclusion concerning the control-field multiplicity continues to hold.