The purpose of this paper is to present a constructive procedure to obtain an approximate optimal control of the controllability problem for a distributed parameter system of parabolic type. The problem of optimal control determination is formulated as the problem of minimizing a functional $J(f)$ which measures the distance between the terminal state $u(T)$ and a given one $u_T $. Controllability is not sufficient for the existence of a minimizing solution of $J(f)$. It only assures $\inf _f J(f) = 0$. For a given $\eta > 0$ the elements which satisfy $J(f) \leqq \eta $ generate the closed convex subset $X_\eta $. It is natural that the optimal control sequence is defined by the sequence such that each element has minimum norm in $X_\eta $ for each $\eta $. An approximation method by regularization is presented to obtain constructively the optimal control sequence. The method gives uniquely the elements of the optimal control sequence depending continuously on the initial data $u_0 $ and the terminal one...