In this paper, we discuss an hp-discontinuous Galerkin finite
element method (hp-DGFEM) for the laser surface hardening of
steel, which is a constrained optimal control problem governed by a
system of differential equations, consisting of an ordinary
differential equation for austenite formation and a semi-linear
parabolic differential equation for temperature evolution. The space
discretization of the state variable is done using an hp-DGFEM,
time and control discretizations are based on a discontinuous
Galerkin method. A priori error estimates are developed at
different discretization levels. Numerical experiments
presented justify the theoretical order of convergence obtained.