  We derive a-priori error estimates for the finite-element approximation of a
distributed optimal control problem governed by the steady one-dimensional
Burgers equation with pointwise box constraints on the control. Here the
approximation of the state and the control is done by using piecewise linear
functions. With this choice, an $L^2$ superlinear order of convergence for the
control is obtained; moreover, under a further assumption on the regularity
structure of the optimal control this error estimate can be improved to
$h^{3/2}$. The theoretical findings are tested experimentally by means of
numerical examples.
