  Motivated by applications requiring sparse or nonnegative controls, we
investigate reachability properties of linear infinite-dimensional control
problems under conic constraints. Relaxing the problem to convex constraints if
the initial cone is not already convex, we provide a constructive approach
based on minimising a properly defined dual functional, which covers both the
approximate and exact reachability problems. Our main results heavily rely on
convex analysis, Fenchel duality and the Fenchel-Rockafellar theorem. As a
byproduct, we uncover new sufficient conditions for approximate and exact
reachability under convex conic constraints. We also prove that these
conditions are in fact necessary. When the constraints are nonconvex, our
method leads to sufficient conditions ensuring that the constructed controls
fulfill the original constraints, which is in the flavour of bang-bang type
properties. We show that our approach encompasses and generalises several
works, and we obtain new results for different types of conic constraints and
control systems.
