  No quantum circuit can turn a completely unknown unitary gate into its
coherently controlled version. Yet, coherent control of unknown gates has been
realised in experiments, making use of a different type of initial resources.
Here, we formalise the task achieved by these experiments, extending it to the
control of arbitrary noisy channels, and to more general types of control
involving higher dimensional control systems. For the standard notion of
coherent control, we identify the information-theoretic resource for
controlling an arbitrary quantum channel on a $d$-dimensional system:
specifically, the resource is an extended quantum channel acting as the
original channel on a $d$-dimensional sector of a $(d+1)$-dimensional system.
Using this resource, arbitrary controlled channels can be built with a
universal circuit architecture. We then extend the standard notion of control
to more general notions, including control of multiple channels with possibly
different input and output systems. Finally, we develop a theoretical
framework, called supermaps on routed channels, which provides a compact
representation of coherent control as an operation performed on the extended
channels, and highlights the way the operation acts on different sectors.
