<p>This paper investigates the numerical approximation of an optimal control problem (OCP) constrained by the one-dimensional thin film epitaxy equation with slope selection. The control is of distributed type and subject to pointwise bound constraints. A cubic B-spline finite element method (FEM) combined with discontinuous Galerkin time-stepping scheme is applied to approximate the state and co-state variables. First and second order optimality conditions are derived. A priori error estimates for the state, co-state, and control variables are obtained through delicate analytic techniques when the control is discretized by piecewise constant functions. In the end, numerical experiments are performed to verify the theoretical findings.</p>