  Controllability and observability are properties that establish the existence
of full-state controllers and observers, respectively. The notions of output
controllability and functional observability are generalizations that enable
respectively the control and estimation of part of the state vector. These
generalizations are of utmost importance in applications to high-dimensional
systems, such as large-scale networks, in which only a target subset of
variables (nodes) are sought to be controlled or estimated. Although the
duality between controllability and observability is well established, the
characterization of the duality between their generalized counterparts remains
an outstanding problem. Here, we establish both the weak and the strong duality
between output controllability and functional observability. Specifically, we
show that functional observability of a system implies output controllability
of a dual system (weak duality), and that under a certain condition the
converse also holds (strong duality). As an application of the strong duality
principle, we derive a necessary and sufficient condition for target control
via static feedback. This allow us to establish a separation principle between
the design of a feedback target controller and the design of a functional
observer in closed-loop systems. These results generalize the well-known
duality and separation principles in modern control theory.
