  This paper considers the notion of herdability, a set-based reachability
condition, which asks whether the state of a system can be controlled to be
element-wise larger than a non-negative threshold. The basic theory of herdable
systems is presented, including a necessary and sufficient condition for
herdability. This paper then considers the impact of the underlying graph
structure of a linear system on the herdability of the system, for the case
where the graph is represented as signed and directed. By classifying nodes
based on the length and sign of walks from an input, we find a class of
completely herdable systems as well as provide a complete characterization of
nodes that can be herded in systems with an underlying graph that is a directed
out-branching rooted at a single input.
