  In this paper we investigate a relaxed concept of controllability, known in
the literature as herdability, namely the capability of a system to be driven
towards the(interior of the) positive orthant. Specifically, we investigate
herdability for linear time-invariant systems, both from an algebraic
perspective and based on the graph representing the systems interactions. In
addition, we focus on linear state-space models corresponding to matrix pairs
(A;B) in which the matrix B is a selection matrix that determines the leaders
in the network, and we show that the weights that followers give to the leaders
do not affect the herdability of the system. We then focus on the herdability
problem for systems with a single leader in which interactions are symmetric
and the network topology is acyclic, in which case an algorithm for the leader
selection is provided. In this context, under some additional conditions on the
mutual distances, necessary and sufficient conditions for the herdability of
the overall system are given.
