About two months ago I gave a talk at the AMS Fall Western Sectional Meeting. While I could have safely rehashed my QPL talk, I decided to push forward instead. That may or may not have been the best idea – the results were certainly less polished. On the other hand, I was able to describe controllability and observability of a control system in terms of string diagrams. This is something that was painfully missing from my QPL talk’s results in July. Seeing the nontrivial constants in the string diagrams the quantum folk were using provided the key insight, and I wanted to capitalize on it as soon as possible.
The punchline of the AMS talk is that the duality between controllability and observability noticed by Kalman in the late 50s and early 60s can be expressed in terms of a PROP, which is a kind of symmetric monoidal category. In particular, this PROP includes a subPROP of finite-dimensional vector spaces and linear relations, which is basically what Paweł Sobociński deals with here under the name of Interacting Hopf monoids. Okay, so the actual punchline is that the duality Kalman noticed six and a half decades ago between controllability and observability? it’s simply time-reversed bizarro duality.
Bizarro is Sobociński’s term (seen in episode 7 of his blog), but I’m kind of partial to it.