  Linear and nonlinear resonant states can be restrictive: they exist at
particular discrete states in frequency and/or elasticity, under particular
(e.g., simple-harmonic) waveforms. In forced oscillators, this restrictiveness
is an obstacle to system design and control modulation: altering the system
elasticity, or modulating the response, would both appear to necessarily incur
a penalty to efficiency. In this work, we describe an approach for bypassing
this obstacle. Using novel work-loop techniques, we prove and illustrate how
certain classes of resonant optimisation problem lead to non-unique solutions.
In a structural optimisation context, several categories of
energetically-optimal elasticity are non-unique. In an optimal control context,
several categories of energetically-optimal frequency are non-unique. For these
classes of non-unique optimum, we can derive simple bounds defining the optimal
region. These novel theoretical results have practical implications for the
design and control of a range of biomimetic propulsion systems, including
flapping-wing micro-air-vehicles: using these results, we can generate
efficient forms of wingbeat modulation for flight control.
