  To design control strategies that predictably manipulate a system's behavior,
it is first necessary to understand how the system's structure relates to its
response. Many complex systems can be represented as multilayer networks whose
response and control can be studied in the framework of network control theory.
It remains unknown how a system's layered architecture dictates its control
properties, particularly when control signals can only access the system
through a single input layer. Here, we use the framework of linear control
theory to probe the control properties of a duplex network with directed
interlayer links. We determine the manner in which the structural properties of
layers and relative interlayer arrangement together dictate the system's
response. For this purpose, we calculate the exact expression of optimal
control energy in terms of layer spectra and the relative alignment between the
eigenmodes of the input layer and the deeper target layer. For a range of
numerically constructed duplex networks, we then calculate the control
properties of the two layers as a function of target-layer densities and layer
topology. The alignment of layer eigenmodes emerges as an important parameter
that sets the cost and routing of the optimal energy for control. We understand
these results in a simplified limit of a single-mode approximation, and we
build metrics to characterize the routing of optimal control energy through the
eigenmodes of each layer. Our analytical and numerical results together provide
insights into the relationships between structure and control in duplex
networks. More generally, they serve as a platform for future work designing
optimal \emph{interlayer} control strategies.
