# Time-Optimal Control Problem With State Constraints In A Time-Periodic Flow Field

## TL;DR

This paper develops a computational framework using Pontryagin's maximum principle to solve time-optimal control problems with state constraints in time-periodic flow fields, exemplified by tidal path-planning.

## Contribution

It introduces a regularity-based indirect method for solving constrained control problems in time-dependent flows, with new conditions to avoid singular controls.

## Key findings

- Effective computation of extremals in tidal flow scenarios
- Continuity of measure Lagrange multipliers under natural assumptions
- Examples demonstrating optimal solutions in periodic fluid flows

## Abstract

This article contributes to a framework for a computational indirect method based on the Pontryagin maximum principle to efficiently solve a class of state constrained time-optimal control problems in the presence of a time-dependent flow field. Path-planning in a flow with tidal variations is an emblematic real-life task that serves as an example falling in the class of problems considered here. Under rather natural assumptions on the flow field, the problem is regular and the measure Lagrange multiplier, associated with the state constraint, is continuous. These properties (regularity and continuity) play a critical role in computing the field of extremals by solving the two-point boundary value problem given by the maximum principle. Moreover, some sufficient conditions preventing the emergence of singular control processes are given. A couple of examples of time-periodic fluid flows are studied and the corresponding optimal solutions are found.