  The linear quadratic regulator is a famous application of optimal control
theory. This class of control systems has linear dynamics (in both the state
and control), while minimizing a quadratic cost. Upon application of
Pontryagin's maximum principle, the co-states can be fully decoupled from the
state which results in a matrix Riccati equation. As such, solutions can be
found by solving this matrix equation backwards. The purpose of this work is to
extend this analysis to systems with linear state jumps, which are referred to
as linear hybrid systems. The extension of the maximum principle to these
systems results in the ``hybrid maximum principle.'' However, successful
application of this theory requires many subtle properties which are usually
ignored - specifically beating/blocking and Zeno. It turns out that these
phenomena occur in linear hybrid systems and as such, the hybrid maximum
principle is not immediately applicable to these seemingly simple systems. We
show that for spatially triggered linear hybrid systems, beating always occurs
while blocking and Zeno can be successfully avoided if a certain
controllability assumption is satisfied. For these trivially blocking systems,
we develop conditions for optimality for the two cases of when beating is
excluded and present. This work concluded with an example.
