  In this paper, we use tools from sheaf theory to model and analyze optimal
network control problems and their associated discrete relaxations. We consider
a general problem setting in which pieces of equipment and their causal
relations are represented as a directed network, and the state of this
equipment evolves over time according to known dynamics and the presence or
absence of control actions. First, we provide a brief introduction to key
concepts in the theory of sheaves on partial orders. This foundation is used to
construct a series of sheaves that build upon each other to model the problem
of optimal control, culminating in a result that proves that solving our
optimal control problem is equivalent to finding an assignment to a sheaf that
has minimum consistency radius and restricts to a global section on a
particular subsheaf. The framework thus built is applied to the specific case
where a model is discretized to one in which the state and control variables
are Boolean in nature, and we provide a general bound for the error incurred by
such a discretization process. We conclude by presenting an application of
these theoretical tools that demonstrates that this bound is improved when the
system dynamics are affine.
