  In this paper, we study the well-posedness and approximate controllability of
a class of network systems having delays and controls at the boundary
conditions. The particularity of this work is that the network system is
defined on infinite metric graphs. This fact offers many difficulties in
applying the usual methods. In fact, the well-posedness of the delay network
system is obtained by using a semigroup approach on product spaces which is
based on the concept of feedback theory of infinite-dimensional linear systems.
This technique allows us to reformulate the delay system into a free-delay
distributed control system. From this transformation, we deduce necessary and
sufficient conditions for the boundary approximate controllability of such
systems. Furthermore, a Rank condition for the approximate controllability is
also obtained. This condition coincides with the usual Kalman controllability
criterion in the case of a simple transport process on a finite graph. Finally,
by applying our approach to a linear Eulerian model (with airborne delays) for
(ATFM), we provide a new algebraic condition for the controllability of such a
model in terms of the generic rank of the so-called extended controllability
matrix.
