Foundations of Machine-Checked Control Theory in Lean

We introduce an open-source library for machine-checked control theory in the interactive proof assistant Lean to lay foundations for the verification of cyber-physical systems.

To this end, as representative theorems, we present formalizations of Lyapunov stability theory and the small-gain theorem.

First, the machinery employed for formalizing Lyapunov stability, i.e., neighborhood filters, allows stating a Lyapunov theorem that covers both points and sets and applies to continuous, discrete, and hybrid systems.

Second, the small-gain theorem is proved via stating input-output systems as relations without the usual well-posedness assumption.

The Lean formalization of each of these theorems is then presented.

We conclude by discussing the library architecture and mentioning some of the other system theoretic results that are formalized in the library along with future plans.