## Pawel Sobocinski

### A categorical approach to open and interconnected dynamical systems (joint work with Brendan Fong and Paolo Rapisarda)

In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. In this talk, I will argue that these ideas are naturally expressed in the language of symmetric monoidal categories.

In particular, I will present a sound and fully complete string diagram algebra for reasoning about LTI systems, which builds on previous work with Filippo Bonchi and Fabio Zanasi. The string diagrams are essentially the classical notion of signal flow graphs, but in contrast to previous work, we will consider them as multi-input multi-output transducers that process streams with an infinite past as well as an infinite future. Such systems are at the heart of control theory, and allow the study of non-controllable behaviours. At the categorical level, the algebraic characterisation is that of the prop of corelations, instead of linear relations as in previous work with Bonchi and Zanasi.

The string diagrammatic perspective leads to a novel structural characterisation of controllability, and consequently allows a compositional analysis of the controllability of networked and interconnected systems.