I think the questions needs clarifying. What kind of profile do you refer to? Do you mean temporal or physical profile?
As Amin Ezazi explained, transistor-type pulse generator can produce controllable pulses with controlled electrical profile. As far as I know, only square waves have been attempted.
As for physical shape, that really depends on gap conditions.
In EDM process we can control the profile of sparks. I need to know the used type of process energy source (generator?) and the regulation parameter (most cases the average gap voltage) of the respective EDM system. Due to the geometry and the electrodes surfaces and the changing in the gap contamination, there will be different forms of discharge in spite of process control.
Process control can be various other process analyzes are performed, e.g. voltage and current levels for arc and/or anormal sparks, state diagramms of ignition delays.