  We consider an optimal control problem for a single-spot pulsed laser welding
problem. The distribution of thermal energy is described by a quasilinear heat
equation. Our emphasis is on materials which tend to suffer from hot cracking,
such as aluminum alloys. A simple indicator for the occurrence of hot cracks is
the velocity of the solidification front. We therefore formulate an optimal
control problem whose objective contains a term which penalizes excessive
solidification velocities. The control function to be optimized is the laser
power over time, subject to pointwise lower and upper bounds. We describe the
finite element discretization of the problem and a projected gradient scheme
for its solution. Numerical experiments for material data representing the EN
AW 6082-T6 aluminum alloy exhibit interesting laser pulse patterns which
perform significantly better than standard ramp-down patterns.
