In the paper, a new concept for a set-valued map \(F(\cdot,\cdot):[S,T]\times {\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is introduced; namely, to have bounded variation with respect to the time variable along a given continuous trajectory uniformly over a given set. Several properties for this class of set-valued maps are proved. As applications of this study in optimal control theory, there are obtained sensitivity relations showing how the output of a control system is influenced by a small time delay in the realization of the control.