Uniformly continuous dependence of a solution to a controlled functional operator equation on a shift of control

We establish sufficient conditions for the uniform (with respect to the set of admissible controls) continuous dependence of a solution to a controlled functional operator equation on a shift of control along a vector of independent variables. The shift of control may mean, in particular, some time delay (or outstripping) of the control. We illustrate the use of general results by an example of a mixed boundary-value problem associated with a wave equation.