Optimal Control from a Fluid Dynamics Perspective
J. Pratt, M. Schneider, A. Perloff
TL;DR
This work addresses making Hamilton-Jacobi-Bellman (HJB) optimal-control problems tractable by recasting them as fluid-dynamics systems. It develops a Madelung transform to convert a Schrödinger-type formulation into Euler-like equations with Bellman density $\rho_{\mathsf{B}}$, Bellman velocity $\mathbf{u}_{\mathsf{B}}$, and Bellman pressure $\mu$, embedding optimization into a fluid potential. It then proposes a two-fluid formulation for Zermelo’s navigation problem, treating the ship as a passive second fluid coupled to a background flow and solved via general CFD workflows (evolving the background, computing $\mu$, then advancing the Bellman-fluid equations). The approach connects optimal-control theory with practical fluid-control methods, including fluid animation techniques, and suggests scalable pathways for global-scale routing through evolving fields within existing CFD infrastructure and Earth-system models.
Abstract
An optimal control problem described by the Hamilton-Jacobi-Bellman equation can be developed into a problem that can be solved by general computational fluid dynamics packages. We describe how this formulation would allow a classical problem in optimal control, Zermelo's problem, to be treated as a multi-fluid problem. This approach has the advantage of allowing optimal navigation problems to be conducted over large areas, as well as to include moderately larger numbers of ships. We draw comparisons between this approach and the field of fluid control for fluid animations in movies.