Towards Optimal Control and Algorithmic Structure of Decompression Schedules

TL;DR

This paper formulates decompression planning as an optimal control problem, deriving structural properties and computational methods for efficient and feasible decompression schedules.

Contribution

It introduces a formal optimal control framework for decompression, proving bang-bang control structure and providing dynamic programming solutions with error bounds.

Findings

Proved existence of solutions under relaxed conditions.

Derived bang-bang control structure for vertical rate.

Provided explicit discretisation error bounds for algorithms.

Abstract

We formalise decompression planning as an optimal control problem with gas feasibility windows (ppO$_2$, END), affine ceilings, and convex penalties in normalised oversaturation. The depth trajectory is constrained to be a monotone ascent, matching operational decompression practice. In this setting we prove relaxed existence, derive bang-bang structure for the vertical rate control, and obtain nonsmooth dwell time KKT conditions. For finite stop grids we give resource constrained dynamic programming and label setting formulations with explicit discretisation error bounds, while also stating the tissue state quantisation or label growth assumptions needed for pseudo-polynomial complexity. The time risk attainable set is generally nonconvex because gas, stop, and switching choices are discrete. We also isolate the precise scope of the two segment exchange argument. It orders terminal tissue tension under monotone inert fraction ordering, but it does not prove that re-descents are dominated for the oversaturation only penalty used here.