  Optimal entrainment of limit-cycle oscillators by strong periodic inputs is
studied on the basis of the phase-amplitude reduction and Floquet theory. Two
methods for deriving the input waveforms that keep the system state close to
the original limit cycle are proposed, which enable the use of strong inputs
for entrainment. The first amplitude-feedback method uses feedback control to
suppress deviations of the system state from the limit cycle, while the second
amplitude-penalty method seeks an input waveform that does not excite large
deviations from the limit cycle in the feedforward framework. Optimal
entrainment of the Van der Pol and Willamowski-R\"ossler oscillators with real
or complex Floquet exponents are analyzed as examples. It is demonstrated that
the proposed methods can achieve much faster entrainment and provide wider
entrainment ranges than the conventional method that relies only on phase
reduction.
