  It has been asserted previously by the author that optimal control theory can
be a valuable framework for theoretical studies about the shape that a wind
instrument should have in order to satisfy some optimization criterion, inside
a fairly general class. The purpose of the present work is to develop this new
approach with a look at a specific criterion to be optimized. In this setting,
the Webster horn equation is regarded as a controlled dynamical equation in the
space variable. Pressure is the state, the control being made of two parts: one
variable part, the inside diameter of the duct and one constant part, the
weights of the elementary time-harmonic components of the velocity potential.
Then one looks for a control that optimizes a criterion related to the
definition of an {oscillation regime} as the cooperation of several natural
modes of vibration with the excitation, the {playing frequency} being the one
that maximizes the total generation of energy, as exposed by A.H. Benade,
following H. Bouasse. At the same time the relevance of this criterion is
questioned with the simulation results.
