# Regular path-constrained time-optimal control problems in three-dimensional flow fields

## TL;DR

This paper develops a computational approach for solving three-dimensional time-optimal control problems with state constraints in steady flow fields, using an indirect maximum principle method that relies on regularity conditions.

## Contribution

It introduces a regularity-based numerical method for solving complex flow field control problems, ensuring continuity of multipliers and stable extremal computations.

## Key findings

- Method successfully computes extremals in various flow fields.
- Regularity condition guarantees multiplier continuity and numerical stability.
- Approach applicable to diverse three-dimensional flow control scenarios.

## Abstract

This article concerns a class of time-optimal state constrained control problems with dynamics defined by an ordinary differential equation involving a three-dimensional steady flow vector field. The problem is solved via an indirect method based on the maximum principle in Gamkrelidze's form. The proposed computational method essentially uses a certain regularity condition imposed on the data of the problem. The property of regularity guarantees the continuity of the measure multiplier associated with the state constraint, and ensures the appropriate behavior of the corresponding numerical procedure which, in general, consists in computing the entire field of extremals for the problem in question. Several examples of vector fields are considered to illustrate the computational approach.