In this paper we prove existence and uniqueness of the solutions for a kind of Volterra equation, with an initial condition, in the space of the continuous functions with bounded variation which take values in an arbitrary Banach space. Then we give a parameters variation formula for the solutions of certain kind of linear integral equation. Finally, we prove exact controllability of a particular integral equation using that formula. Moreover, under certain condition, we find a formula for a control steering of a type of system which is studied in the current work, from an initial state to a final one in a prescribed time.