On the Reinhardt Conjecture and Formal Foundations of Optimal Control

TL;DR

This paper reformulates Reinhardt's 1934 conjecture on convex packings as an optimal control problem, analyzing its structure, solutions, and connections to reinforcement learning and formal foundations.

Contribution

It introduces a novel optimal control formulation of Reinhardt's conjecture, explores its Hamiltonian structure, and links it to reinforcement learning formalizations.

Findings

Hamiltonian and Lax pair structures identified

Explicit solutions for constant control provided

Connection to multi-dimensional Fuller control system established

Abstract

We describe a reformulation (following Hales (2017)) of a 1934 conjecture of Reinhardt on pessimal packings of convex domains in the plane as a problem in optimal control theory. Several structural results of this problem including its Hamiltonian structure and Lax pair formalism are presented. General solutions of this problem for constant control are presented and are used to prove that the Pontryagin extremals of the control problem are constrained to lie in a compact domain of the state space. We further describe the structure of the control problem near its singular locus, and prove that we recover the Pontryagin system of the multi-dimensional Fuller optimal control problem (with two dimensional control) in this case. We show how this system admits logarithmic spiral trajectories when the control set is the circumscribing disk of the 2-simplex with the associated control performing an infinite number of rotations on the boundary of the disk in finite time. We also describe formalization projects in foundational optimal control viz., model-based and model-free Reinforcement Learning theory. Key ingredients which make these formalization novel viz., the Giry monad and contraction coinduction are considered and some applications are discussed.