For a controlled system with mixed equality- and inequality-type constraints and a geometric constraint that is a nonempty closed convex set, sufficient conditions for the existence of feasible positional controls are obtained in terms of first derivatives of mappings that define the mixed constraints. In addition, in terms of the first and second derivatives of these mappings, sufficient conditions for the existence of feasible positional controls are found that are also applicable in the case of degeneration of the first derivatives of the mappings.