An Explicit Minimum-Time Solution for the Simple Harmonic Oscillator

Abstract

It is well known that the minimum time control of a simple harmonic oscillator subjected to a bounded control input is a bang-bang control where the input cannot remain constant for more than half of the period of the oscillator. The necessary switching times for implementing such bang-bang control are usually determined by geometric means in the state-space, thus making it difficult to obtain the switchings and the final time for an arbitrary initial condition. An explicit solution for the switching times and the final time for a motion starting from an arbitrary initial state subjected to an arbitrary but constant input bound is developed for the simple harmonic oscillator. The explicit solution matches exactly the graphical solution (Athans and Falb, 1966).