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Towards Optimal Control and Algorithmic Structure of Decompression Schedules

Benjamin Marsh

TL;DR

The first formal existence and bang-bang structure proof under mixed gas feasibility windows is provided, and pseudo-polynomial DP and label-setting algorithms with a priori error bounds are given.

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. We prove existence, a monotone no re-descent structure and bang-bang ascents under a mild monotonicity assumption on inert fraction, and establish dwell time KKT conditions. We give pseudo-polynomial DP and label-setting algorithms with a priori error bounds, derive Lipschitz regularity of the online value function, and discuss multi-species extensions. The efficient frontier is continuous and generally nonconvex. We provide the first formal existence and bang-bang structure proof under mixed gas feasibility windows.

Towards Optimal Control and Algorithmic Structure of Decompression Schedules

TL;DR

The first formal existence and bang-bang structure proof under mixed gas feasibility windows is provided, and pseudo-polynomial DP and label-setting algorithms with a priori error bounds are given.

Abstract

We formalise decompression planning as an optimal control problem with gas feasibility windows (ppO, END), affine ceilings, and convex penalties in normalised oversaturation. We prove existence, a monotone no re-descent structure and bang-bang ascents under a mild monotonicity assumption on inert fraction, and establish dwell time KKT conditions. We give pseudo-polynomial DP and label-setting algorithms with a priori error bounds, derive Lipschitz regularity of the online value function, and discuss multi-species extensions. The efficient frontier is continuous and generally nonconvex. We provide the first formal existence and bang-bang structure proof under mixed gas feasibility windows.
Paper Structure (25 sections, 25 theorems, 67 equations)

This paper contains 25 sections, 25 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Model and well-posedness
  4. Assumptions, primitives, and initialisation
  5. Measurable selections
  6. Well-posedness
  7. Structural results
  8. Two-segment exchange and no re-descents
  9. Pontryagin maximum principle and bang-bang
  10. Finite stop grid suffices
  11. Dwell-time optimality and KKT equalisation
  12. Complexity
  13. Algos
  14. Frontier geometry and monotonicity
  15. Numerical methods and a-priori error bounds
  16. ...and 10 more sections

Key Result

Proposition 3.8

For every $\varepsilon>0$, there exists a feasible profile composed of finitely many constant-depth holds, separated by max rate ascents, and with piecewise constant gas on each hold, whose cost $J_\lambda$ is within $\varepsilon$ of the optimal value of $(\mathbf P_\lambda)$ (and similarly for $(\m

Theorems & Definitions (51)

  • Remark 3.6: Justification of modelling choices
  • Definition 3.7: Scalarised and capped problems
  • Proposition 3.8: Finite segmented profiles suffice up to $\varepsilon$
  • proof : Proof sketch
  • Lemma 3.9: Measurable gas selection
  • proof
  • Lemma 3.10: Existence via relaxation and chattering
  • proof
  • Lemma 4.2: Two-segment exchange
  • proof
  • ...and 41 more