  We derive optimal periodic controls for entrainment of a self-driven
oscillator to a desired frequency. The alternative objectives of minimizing
power and maximizing frequency range of entrainment are considered. A state
space representation of the oscillator is reduced to a linearized phase model,
and the optimal periodic control is computed from the phase response curve
using formal averaging and the calculus of variations. Computational methods
are used to calculate the periodic orbit and the phase response curve, and a
numerical method for approximating the optimal controls is introduced. Our
method is applied to asymptotically control the period of spiking neural
oscillators modeled using the Hodgkin-Huxley equations. This example
illustrates the optimality of entrainment controls derived using phase models
when applied to the original state space system.
