  We consider the Selective Harmonic Modulation (SHM) problem, consisting in
the design of a staircase control signal with some prescribed frequency
components. In this work, we propose a novel methodology to address SHM as an
optimal control problem in which the admissible controls are piecewise constant
functions, taking values only in a given finite set. In order to fulfill this
constraint, we introduce a cost functional with piecewise affine penalization
for the control, which, by means of Pontryagin's maximum principle, makes the
optimal control have the desired staircase form. Moreover, the addition of the
penalization term for the control provides uniqueness and continuity of the
solution with respect to the target frequencies. Another advantage of our
approach is that the number of switching angles and the waveform need not be
determined a priori. Indeed, the solution to the optimal control problem is the
entire control signal, and therefore, it determines the waveform and the
location of the switches. We also provide several numerical examples in which
the SHM problem is solved by means of our approach.
