  This paper considers two types of boundary control problems for linear
transport equations. The first one shows that transport solutions on a
subdomain of a domain X can be controlled exactly from incoming boundary
conditions for X under appropriate convexity assumptions. This is in contrast
with the only approximate control one typically obtains for elliptic equations
by an application of a unique continuation property, a property which we prove
does not hold for transport equations. We also consider the control of an
outgoing solution from incoming conditions, a transport notion similar to the
Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen
coefficients in the transport equation, this control may not be possible. In
such situations and by (Fredholm) duality, we obtain the existence of
non-trivial incoming conditions that are compatible with vanishing outgoing
conditions.
