Zermelo Navigation Problem Research Articles

Application of planar Randers geodesics with river-type perturbation in search models

We consider remodeling the planar search patterns, in the presence of the river-type perturbation represented by the weak vector field, basing on the time-optimal paths as Finslerian solutions to the Zermelo navigation problem via Randers metric.

The stochastic reach-avoid problem and set characterization for diffusions

In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems—the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation (PDE) in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

Time-optimal control of a self-propelled particle in a spatiotemporal flow field

ABSTRACTWe address a minimum-time problem that constitutes an extension of the classical Zermelo navigation problem in higher dimensions. In particular, we address the problem of steering a self-propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field. Furthermore, we assume that the norm of the rate of change of the particle's velocity relative to the flow is upper bounded by an explicit upper bound. To address the problem, we first employ Pontryagin's minimum principle to parameterise the set of candidate time-optimal control laws in terms of a parameter vector that belongs to a compact set. Subsequently, we develop a simple numerical algorithm for the computation of the minimum time-to-come function that is tailored to the particular parametrisation of the set of the candidate time-optimal control laws of our problem. The proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation problem, which can be computationally intense, especially when one is interested in characterising the optimal synthesis of the minimum-time problem. Numerical simulations that illustrate the theoretical developments are presented.

Solution to the quantum Zermelo navigation problem.

The solution to the problem of finding a time-optimal control Hamiltonian to generate a given unitary gate, in an environment in which there exists an uncontrollable ambient Hamiltonian (e.g., a background field), is obtained. In the classical context, finding the time-optimal way to steer a ship in the presence of a background wind or current is known as the Zermelo navigation problem, whose solution can be obtained by working out geodesic curves on a space equipped with a Randers metric. The solution to the quantum Zermelo problem, which is shown here to take a remarkably simple form, is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a Randers metric on the space of unitary operators. The result reveals that the optimal control in a sense "goes along with the wind."

Application of codimension one foliation in Zermelo's problem on Riemannian manifolds

We research the Zermelo navigation problem on Riemannian manifolds in dim⁡(R×M)=3 under the force representing the action of the perturbing “wind” distribution modeled by the vector field on manifold M. We consider a fibered manifold π:R×M→R where M represents the “sea”, R represents the time and π is the first canonical projection. We perturb it by a time-(in)dependent vector field on the manifold M with a codimension one foliation. We aim to research the behavior of the paths in reference to their optimization. Geometrically we find the deviation of geodesics under the action of a time-dependent vector field. We present corresponding simulation with the whirlpool perturbation. We also consider a new concept of real application in the navigational decision support systems.

Route optimisation and solving Zermelo's navigation problem during long distance migration in cross flows

The optimum path to follow when subjected to cross flows was first considered over 80 years ago by the German mathematician Ernst Zermelo, in the context of a boat being displaced by ocean currents, and has become known as the 'Zermelo navigation problem'. However, the ability of migrating animals to solve this problem has received limited consideration, even though wind and ocean currents cause the lateral displacement of flyers and swimmers, respectively, particularly during long-distance journeys of 1000s of kilometres. Here, we examine this problem by combining long-distance, open-ocean marine turtle movements (obtained via long-term GPS tracking of sea turtles moving 1000s of km), with a high resolution basin-wide physical ocean model to estimate ocean currents. We provide a robust mathematical framework to demonstrate that, while turtles eventually arrive at their target site, they do not follow the optimum (Zermelo's) route. Even though adult marine turtles regularly complete incredible long-distance migrations, these vertebrates primarily rely on course corrections when entering neritic waters during the final stages of migration. Our work introduces a new perspective in the analysis of wildlife tracking datasets, with different animal groups potentially exhibiting different levels of complexity in goal attainment during migration.

신경회로망을 이용한 선박의 최단시간 궤적제어

위치에 따라 변화하는 해류가 있는 해상에서 선박이 일정속도로 움직이면서 진행각도를 제어하여 특정목표점에 최단시간으로 도착하고자 한다. 이러한 문제는 Zermelo의 항해문제라고 불리고 있으며 calculus of variation에 의해 최단시간제어의 근사해를 구할 수 있지만 매우 복잡한 비선형방정식의 역관계를 테이블 형태로 이용하여야 하므로 그 정확도가 높지 않다. 본 논문에서는 이러한 근사해의 정확도를 높이기 위해 신경회로망을 이용하여 비선형 방정식의 역관계식을 표현하였고, 신경회로망의 보간 기능으로 인해 높은 정확도의 최단시간제어가 가능하였다. 여러 가지 항해 상황에 대한 컴퓨터 시뮬레이션을 통해서 본 논문에서 제안한 방법이 기존의 방법보다 우수함을 확인할 수 있었다. A ship is intended to reach a specified target point in the minimum-time when it travels with a constant speed through a region of strong currents and its heading angle is the control variable. This is called the Zermelo's navigation problem. Its approximate solution for the minimum-time control may be found using the calculus of variation. However, the accuracy of its approximate solution is not high since the solution is based on a table form of inverse relations for some complicated nonlinear equations. To enhance the accuracy, this paper employs the neural network to represent the inverse relation of the complicated nonlinear equations. The accurate minimum-time control is possible with the interpolation property of the neural network. Through the computer simulation study we have found that the proposed method is superior to the conventional ones.

Stationary metrics and optical Zermelo-Randers-Finsler geometry

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painlev\'e-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalization of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.

On Zermelo-Like Problems: Gauss–Bonnet Inequality and E. Hopf Theorem

The aim of this paper is to describe the Zermelo navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method of evaluating its control curvature. We will show that, up to the change of the Riemannian metric on the manifold, the control curvature of the Zermelo problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss–Bonnet formula in the form of an inequality. This Gauss–Bonnet inequality allows one to generalize the Zermelo problems and obtain a theorem of E. Hopf that establishes the flatness of Riemannian tori without conjugate points.

On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem

The goal of this paper is to extend the notion of Gaussian curvature of two-dimensional surfaces to nonlinear time-optimal control systems in the plane by applying the moving frame method. This notion of curvature was introduced earlier by A. A. Agrachev and R. V. Gamkrelidze by means of Jacobi curves. Here we give a self-contained presentation of its two-dimensional version and apply the results to the well-known Zermelo navigation problem.

Randers metrics on CP2

The complex projective space CP2 is a classical example of Einstein metric in Riemannian geometry. Moreover, beside this property, it has other interesting geometrical properties: it is a symmetric space, and a C manifold. We would like to know whether there is an Einstein metric of Randers type on CP2 with similar properties. Based on some the generalization of Zermelo navigation problem for Finsler manifolds we construct such Randers metric on CP2 and study some of its geometrical properties.