  Optimal control is ubiquitous in many fields of engineering. A common
technique to find candidate solutions is via Pontryagin's maximum principle. An
unfortunate aspect of this method is that the dimension of system doubles. When
the system evolves on a Lie group and the system is invariant under left (or
right) translations, Lie-Poisson reduction can be applied to eliminate half of
the dimensions (and returning the dimension of the problem to the back to the
original number).
  Hybrid control systems are an extension of (continuous) control systems by
allowing for sudden changes to the state. Examples of such systems include the
bouncing ball - the velocity instantaneously jumps during a bounce, the
thermostat - controls switch to on or off, and a sailboat undergoing tacking.
The goal of this work is to extend the idea of Lie-Poisson reduction to the
optimal control of these systems. If $n$ is the dimension of the original
system, $2n$ is the dimension of the system produced by the maximum principle.
In the case of classical Lie-Poisson reduction, the dimension drops back down
to $n$. This, unfortunately, is impossible in hybrid systems as there must be
an auxiliary variable encoding whether or not an event occurs. As such, the
analogous hybrid Lie-Poisson reduction results in a $n+1$ dimensional system.
The purpose of this work is to develop and present this technique.
