  Control theory uses `signal-flow diagrams' to describe processes where
real-valued functions of time are added, multiplied by scalars, differentiated
and integrated, duplicated and deleted. These diagrams can be seen as string
diagrams for the PROP FinRel_k, the strict version of the category of
finite-dimensional vector spaces over the field of rational functions k = R(s)
and linear relations, where the variable s acts as differentiation and the
monoidal structure is direct sum rather than the usual tensor product of vector
spaces. Control processes are also described by controllability and
observability---whether the input can drive the process to any state, and
whether any state can be determined from later outputs. For any field k we give
a presentation of FinRel_k in terms of generators of the free PROP of
signal-flow diagrams together with the equations that give FinRel_k its
structure. The `cap' and `cup' generators, missing when the morphisms are
linear maps, make it possible to model feedback. The relations say, among other
things, that the 1-dimensional vector space k has two special commutative
dagger-Frobenius structures, such that the multiplication and unit of either
one and the comultiplication and counit of the other fit together to form a
bimonoid. This sort of structure, but with tensor product replacing direct sum,
is familiar from the `ZX-calculus' obeyed by a finite-dimensional Hilbert space
with two mutually unbiased bases. In order to address controllability and
observability, we construct the PROP Stateful_k and relate it back to the PROP
of signal-flow diagrams. This provides a way to graphically express
controllability and observability for linear time-invariant processes.

Category Theory;Optimization and Control