  We propose a general method for optimizing periodic input waveforms for
global entrainment of weakly forced limit-cycle oscillators based on phase
reduction and nonlinear programming. We derive averaged phase dynamics of the
oscillator driven by a weak periodic input and optimize the Fourier
coefficients of the input waveform to maximize prescribed objective functions.
The proposed method can be applied to various optimization problems that cannot
be solved analytically using the calculus of variations. As an illustration, we
consider two optimization problems, one for achieving fast global convergence
of the oscillator to the entrained state and the other for realizing prescribed
global phase distributions in a population of identical uncoupled noisy
oscillators. We show that the proposed method can successfully yield optimal
input waveforms to realize the desired states in both cases.
