  The duality between controllability and observability enables methods
developed for full-state control to be applied to full-state estimation, and
vice versa. In applications in which control or estimation of all state
variables is unfeasible, the generalized notions of output controllability and
functional observability establish the minimal conditions for the control and
estimation of a target subset of state variables, respectively. Given the
seemly unrelated nature of these properties, thus far methods for target
control and target estimation have been developed independently in the
literature. Here, we characterize the graph-theoretic conditions for target
controllability and target observability (which are, respectively, special
cases of output controllability and functional observability for structured
systems). This allow us to rigorously establish a weak and strong duality
between these generalized properties. When both properties are equivalent
(strongly dual), we show that efficient algorithms developed for target
controllability can be used for target observability, and vice versa, for the
optimal placement of sensors and drivers. These results are applicable to
large-scale networks, in which control and monitoring are often sought for
small subsets of nodes.
