Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A non-parametric Zermelo navigation equation for strictly convex control sets

Matteo Della Rossa, Lorenzo Freddi, Mattia Pinatto

TL;DR

This paper extends Zermelo's navigation problem by allowing the admissible velocity set $U$ to be strictly convex and compact, and studies minimum-time control under a current field $s$. Using Pontryagin's maximum principle and convex analysis, it proves existence under weak currents and shows that strict convexity yields smooth optimal controls; in two dimensions it derives a non-parametric navigation equation that generalizes the classical Zermelo equation and recovers it when $U$ is a ball. It also analyzes constant and linear currents, providing explicit 2D navigation formulas, and demonstrates how sail-assisted or asymmetric propulsion geometries influence optimal trajectories. The work thereby connects convex-analytic tools with non-parametric optimal control to yield both theoretical insights and practical strategies for maritime and aerial navigation under realistic velocity constraints.

Abstract

We study a generalized version of Zermelo's navigation problem in which the admissible set of control velocities is a strictly convex compact set, rather than the classical spherical or ball-shaped one. After establishing existence results under the natural assumption of weak currents, we derive necessary optimality conditions via Pontryagin's maximum principle and convex analysis. In particular, we prove that strictly convex control sets ensure smoothness of optimal controls. In dimension two, this regularity allows us to eliminate the adjoint variables and obtain a second-order differential equation for the optimal control, which extends the classical Zermelo navigation equation to strictly convex control sets in a non-parametric setting. We also develop the case of an affine current, with a particular emphasis on the constant one where optimal trajectories reduce to straight lines. The results are illustrated with examples relevant to ship routing with asymmetric or sail-assisted propulsion.

A non-parametric Zermelo navigation equation for strictly convex control sets

TL;DR

This paper extends Zermelo's navigation problem by allowing the admissible velocity set to be strictly convex and compact, and studies minimum-time control under a current field . Using Pontryagin's maximum principle and convex analysis, it proves existence under weak currents and shows that strict convexity yields smooth optimal controls; in two dimensions it derives a non-parametric navigation equation that generalizes the classical Zermelo equation and recovers it when is a ball. It also analyzes constant and linear currents, providing explicit 2D navigation formulas, and demonstrates how sail-assisted or asymmetric propulsion geometries influence optimal trajectories. The work thereby connects convex-analytic tools with non-parametric optimal control to yield both theoretical insights and practical strategies for maritime and aerial navigation under realistic velocity constraints.

Abstract

We study a generalized version of Zermelo's navigation problem in which the admissible set of control velocities is a strictly convex compact set, rather than the classical spherical or ball-shaped one. After establishing existence results under the natural assumption of weak currents, we derive necessary optimality conditions via Pontryagin's maximum principle and convex analysis. In particular, we prove that strictly convex control sets ensure smoothness of optimal controls. In dimension two, this regularity allows us to eliminate the adjoint variables and obtain a second-order differential equation for the optimal control, which extends the classical Zermelo navigation equation to strictly convex control sets in a non-parametric setting. We also develop the case of an affine current, with a particular emphasis on the constant one where optimal trajectories reduce to straight lines. The results are illustrated with examples relevant to ship routing with asymmetric or sail-assisted propulsion.
Paper Structure (10 sections, 11 theorems, 101 equations, 4 figures)

This paper contains 10 sections, 11 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Existence of a solution
  4. Pontryagin necessary optimality conditions
  5. Navigation equations for a strictly convex control set
  6. Particular cases and examples
  7. The case of a constant current
  8. The case of a linear current
  9. Conclusions and future research
  10. Convex analysis main tools

Key Result

Lemma 3.1

Suppose that Assumptions assump:AB and assump:Convexity be satisfied together with (s1) and (s2). Assume, moreover, that the following permanence condition holds: For every $x_0\in A$ such that ${\mathcal{T}}(x_0)<+\infty$ the inf in the definition def:MTF of ${\mathcal{T}}(x_0)$ is minimum.

Figures (4)

  • Figure 1: Navigation between two points with an egg-shaped control set $U$.
  • Figure 2: Navigation from a starting line $A$ to a point $B$ with an egg-shaped control set $U$.
  • Figure 3: Optimal trajectory (in black) and suboptimal trajectory with the constant control \ref{['eq:constcontrl_ci']} (in green).
  • Figure 4: Optimal trajectory (in black) and suboptimal trajectory with constant control \ref{['eq:constcontrl']} (in green).

Theorems & Definitions (47)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4: case of a weak current
  • proof
  • Remark 3.5
  • Lemma 4.1: PMP
  • proof
  • ...and 37 more