  This paper focuses on boundary approximate controllability under positivity
constraints of a wide range of infinite-dimensional control systems. We develop
frequency domain controllability criteria. Firstly, we derive a controllability
result under positivity constraints on the control for such systems. Then, and
more importantly, we provide a necessary and sufficient condition for
controllability under positivity constraints on the control and the state.
  The obtained results are applied to the controllability of transportation and
heat conduction networks. In particular, provided that the underlying graph is
strongly connected, the controllability under positivity constraints on the
control/state of transport network systems is fully characterized by a
Kalman-type rank condition. For a system of heat equations with Robin boundary
conditions on a path-like network, we establish approximate controllability
under positivity state-constraint with a single positive input through the
starting node. However, we prove the lack of controllability under unilateral
control-constraint.
