  The quantification of controllability and observability has recently received
new interest in the context of large, complex networks of dynamical systems. A
fundamental but computationally difficult problem is the placement or selection
of actuators and sensors that optimize real-valued controllability and
observability metrics of the network. We show that several classes of energy
related metrics associated with the controllability Gramian in linear dynamical
systems have a strong structural property, called submodularity. This property
allows for an approximation guarantee by using a simple greedy heuristic for
their maximization. The results are illustrated for randomly generated systems
and for placement of power electronic actuators in a model of the European
power grid.
