Title: Optimal Input Placement in Lattice Graphs,
Abstract: The control of dynamical, networked systems continues to receive much
attention across the engineering and scientific research fields. Of particular
interest is the proper way to determine which nodes of the network should
receive external control inputs in order to effectively and efficiently control
portions of the network. Published methods to accomplish this task either find
a minimal set of driver nodes to guarantee controllability or a larger set of
driver nodes which optimizes some control metric. Here, we investigate the
control of lattice systems which provides analytical insight into the
relationship between network structure and controllability. First we derive a
closed form expression for the individual elements of the controllability
Gramian of infinite lattice systems. Second, we focus on nearest neighbor
lattices for which the distance between nodes appears in the expression for the
controllability Gramian. We show that common control energy metrics scale
exponentially with respect to the maximum distance between a driver node and a
target node.