Zermelo Navigation Problem Research Articles

Balancing the Quantum Speed Limit and Instantaneous Energy Cost in Adiabatic Quantum Evolution

Adiabatic time-optimal quantum controls are extensively used in quantum technologies to break the constraints imposed by short coherence times. However, practically it is crucial to consider the trade-off between the quantum evolution speed and instantaneous energy cost of process because of the constraints in the available control Hamiltonian. Here, we experimentally show that using a transmon qubit that, even in the presence of vanishing energy gaps, it is possible to reach a highly time-optimal adiabatic quantum driving at low energy cost in the whole evolution process. This validates the recently derived general solution of the quantum Zermelo navigation problem, paving the way for energy-efficient quantum control which is usually overlooked in conventional speed-up schemes, including the well-known counter-diabatic driving. By designing the control Hamiltonian based on the quantum speed limit bound quantified by the changing rate of phase in the interaction picture, we reveal the relationship between the quantum speed limit and instantaneous energy cost. Consequently, we demonstrate fast and high-fidelity quantum adiabatic processes by employing energy-efficient driving strengths, indicating a promising strategy for expanding the applications of time-optimal quantum controls in superconducting quantum circuits.

On Funk's parabolas

This study aims to analyze parabolas in the two dimensional unit disk equipped with a Funk metric. The analysis leads to four types of parabolas are obtained, due to the non-reversibility of the Funk metric. Each one with applications to physics in the Zermelo navigation problem. In addition, we identify that two of the four parabolas obtained are in well known Euclidian conics, and the remaining two are characterized by irreducible quartics.

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.

Zermelo's navigation problem for some special surfaces of rotation

<abstract><p>In this paper, we investigate Zermelo's navigation problem for some special rotation surfaces. In this respect, we find some Randers-type metrics for these rotation surfaces. Furthermore, we get the H-distortion for the metric induced by surfaces.</p></abstract>

Kerr-Newman-Jacobi geometry and the deflection of charged massive particles

In this paper, we investigate the deflection of a charged particle moving in the equatorial plane of Kerr-Newman spacetime, focusing on weak field limit. To this end, we use the Jacobi geometry, which can be described in three equivalent forms, namely the Randers-Finsler metric, the Zermelo navigation problem, and the ($n+1$)-dimensional stationary spacetime picture. Based on Randers data and Gauss-Bonnet theorem, we utilize the osculating Riemannian manifold method and the generalized Jacobi metric method, respectively, to study the deflection angle. In the ($n+1$)-dimensional spacetime picture, the motion of charged particle follows the null geodesic, and thus we use the standard geodesic method to calculate the deflection angle. The three methods lead to the same second-order deflection angle, which is obtained for the first time. The result shows that the black hole spin $a$ affects the deflection of charged particles both gravitationally and magnetically at the leading order [$\mathcal{O}([M{]}^{2}/{b}^{2})$]. When $qQ/E<2M$, $a$ will decrease (or increase) the deflection of prograde (or retrograde) charged signal. If $qQ/E>2M$, the opposite happens, and the ray is divergently deflected by the lens. We also show that the effect of the magnetic charge of the dyonic Kerr-Newman black hole on the deflection angle is independent of the particle's charge.

Um problema de navegação de Zermelo: Métrica de Funk

O artigo aborda um modelo específico de Geometria Não-Euclidiana, cujo disco aberto unitário centrado na origem do plano cartesiano é dotado de uma métrica de Randers, que modela o Problema da navegação de Zermelo. Com isso, é gerada a "Geometria de Funk sobre o disco unitário", para qual a distância não é simétrica. Nesse sentido, o estudo apresenta as expressões para distância de ponto a ponto - de ponto a uma linha reta, e de uma linha reta a um ponto; e caracteriza as circunferências nesse tipo de geometria. Exemplos explícitos são incluídos.

Определение траекторий наибыстрейшего движения материальной точки в горизонтальном векторном поле

Предложено решение известной навигационной задачи Цермело классическими вариационными методами. Классическая задача Цермело в рамках теории оптимального управления формулируется следующим образом. Корабль должен пройти через область сильных течений, величина и направление скорости течения задаются как функции фазовых переменных. При этом задается относительная скорость корабля, модуль которой во время движения остается постоянным. Нужно найти такое оптимальное управление, которое обеспечивает прибытие корабля в заданную точку за минимальное время, то есть следует определить управление кораблем по быстродействию. Рассмотрено брахистохронное движение материальной точки в плоском векторном поле подвижной жидкости, для которого сформулирована классическая вариационная задача поиска экстремальных траекторий. Целью исследования является получение уравнений экстремальных траекторий движения, вдоль которых материальная точка перемещается от заданной стартовой точки к заданной финишной за кратчайшее время. Решение поставленной задачи осуществлялось с помощью классических методов теории вариационного числа. Для заданного варианта граничных условий установлены алгебраические уравнения экстремалей движения материальной точки в виде отрезков степеневого ряда. Проведен сравнительный анализ быстродействия как по экстремальным траекториям, так и альтернативным путем — по прямой линии, соединяющей две заданные точки старта и финиша. Анализ результатов показал, что рассматриваемая вариационная задача имеет две решения, которые отличаются лишь знаком. Однако только одно решение обеспечивает минимальное время перемещения материальной точки между двумя заданными. Также установлено, что экстремальная траектория брахистохронного движения точки не прямой, а имеет колебательный характер.
 Запропоновано розвʼязання відомої навігаційної задачі Цермело класичними варіаційними методами. Класична задача Цермело в рамках теорії оптимального керування формулюється таким чином. Корабель повинен пройти через область сильних течій, величина і напрямок швидкості течії задаються як функції фазових змінних. При цьому задається відносна швидкість корабля, модуль якої під час руху залишається сталим. Потрібно знайти таке оптимальне керування, яке забезпечує прибуття корабля в задану точку за мінімальний час, тобто слід визначити керування кораблем за швидкодією. Розглянуто брахістохронний рух матеріальної точки в плоскому векторному полі рухомої рідини, для якого сформульовано класичну варіаційну задачу пошуку екстремальних траєкторій. Метою дослідження є отримання рівнянь екстремальних траєкторій руху, уздовж яких матеріальна точка переміщується від заданої стартової точки до заданої фінішної за найкоротший час. Розвʼязання поставленої задачі здійснювалося за допомогою класичних методів теорії варіаційного числення. Для заданого варіанту граничних умов встановлені алгебраїчні рівняння екстремалей руху матеріальної точки у вигляді відрізків степеневого ряду. Проведено порівняльний аналіз швидкодії як за екстремальними траєкторіями, так і альтернативним шляхом — за прямою лінією, яка зʼєднує дві задані точки старту і фінішу. Аналіз результатів показав, що розглянута варіаційна задача має два розвʼязки, які відрізняються лише знаком. Однак тільки одне рішення забезпечує мінімальний час переміщення матеріальної точки між двома заданими. Також встановлено, що екстремальна траєкторія брахістохронного руху точки не є прямою, а має коливальний характер.

A Zermelo navigation problem with a vortex singularity

Helhmoltz–Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.

Sharp uncertainty principles on general Finsler manifolds

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg, and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fullycharacterizesthe nature of the Finsler manifold in terms of three non- Riemannian quantities, namely, itsreversibilityand the vanishing of theflag curvatureandSS-curvatureinduced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

The holonomy group of locally projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of the circle.

On Generalised Single-Heading Navigation

Introducing the notion of a pseudoloxodrome, we generalise a single-heading navigation to conformally flat Riemannian manifolds, under the action of a perturbing vector field (wind, current) of arbitrary force. The findings are applied to time-optimal navigation with the use of the Euler–Lagrange equations. We refer to the Zermelo navigation problem admitting space and time dependence of both a perturbation and a ship's speed. The necessary conditions for single-heading time-optimal navigation are obtained and the pseudoloxodromes of minimum and maximum time are discussed. Furthermore, we describe winds which yield the pseudoloxodromic and loxodromic time extremals. Our research is also illustrated with the examples in dimension two emphasising the single-heading solutions among the time-optimal trajectories in the presence of some space-dependent winds.

A Neural Network-Based Policy Iteration Algorithm with Global H^2-Superlinear Convergence for Stochastic Games on Domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H^2-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

On the definition and examples of cones and Finsler spacetimes

A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics $L$ is carried out. As a link between both notions, cone triples $(\Omega,T, F)$, where $\Omega$ (resp. $T$) is a 1-form (resp. vector field) with $\Omega(T)\equiv 1$ and $F$, a Finsler metric on $\ker (\Omega)$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $\C$ are bijectively associated with classes of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic} is introduced consistently with both structures. As a non-relativistic application, the {\em time-dependent} Zermelo navigation problem is posed rigorously, and its general solution is provided.

Finslerian geometrization of quantum mechanics in the hydrodynamical representation

We consider a Finslerian type geometrization of the non-relativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation the Schr\"{o}dinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similar to its classical counterpart. After the reparametrization of the Lagrangian it turns out that the Finsler metric describing the geometric properties of quantum hydrodynamics is a Kropina metric. We present and discuss in detail the metric and the geodesic equations describing the geometric properties of the quantum motion in the presence of electromagnetic fields. As an application of the obtained formalism we consider the Zermelo navigation problem in a quantum hydrodynamical system, whose solution is given by a Kropina metric. The case of the Finsler geometrization of the quantum hydrodynamical motion of spinless particles in the absence of electromagnetic interactions is considered in detail, and the Zermelo navigation problem for this case is also discussed.

On generalization of Zermelo navigation problem on Riemannian manifolds

With the extended navigation data, we consider the generalized Zermelo navigation on Riemannian manifolds, admitting a space-dependent ship’s speed in the presence of perturbation determined by a weak velocity vector field, with application of Finsler metric of Randers type. The approach is shown via indicatrix and inner product. We also compare our findings in the context of conformality for the cases of weak and critical winds. The study is illustrated with the example in dimension 2.

A note on time-optimal paths on perturbed spheroid

We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's heading $\varphi(t)$ (steering direction). As the example we present the solutions to the problem on an oblate ellipsoid with acting infinitesimal rotation.

A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation

AbstractWe generalize the Zermelo navigation on Riemannian manifolds (M; h), admitting a space dependence of a ship's speed 0 &lt; |u(x)|h≤ 1 in the presence of a perturbation W̃ determined by a strong (critical) velocity vector field satisfying |W̃ (x)|h= |u(x)|h, with application of Finsler metric of Kropina type.

Wind Riemannian spaceforms and Randers–Kropina metrics of constant flag curvature

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

Generalized Zermelo Navigation on Hermitian Manifolds with a Critical Wind

We study the Zermelo navigation problem on Hermitian manifolds (M, h) in the presence of a critical perturbation W determined by a complex velocity vector field such that \(0<||W(z)||_{h}=||u(z)||_{h}\le 1\), admitting space dependence of a ship’s relative speed \(||u(z)||_{h}\), with application of complex Finsler metric of complex Kropina type. We discuss the geodesics corresponding to the critical solutions to Zermelo’s problem and we find the necessary and sufficient conditions for the obtained locally projectively flat solutions.

Generalized Zermelo navigation on Hermitian manifolds under mild wind

We generalize and study the Zermelo navigation problem on Hermitian manifolds in the presence of a perturbation W determined by a mild complex velocity vector field ‖W(z)‖h<‖u(z)‖h, with application of complex Finsler metric of complex Randers type. By admitting space-dependence of ship's relative speed ‖u(z)‖h≤1 we discuss the projectively related complex Finsler metrics, the geodesics corresponding to the solutions of Zermelo's problem, in particular the conformal case and the connections between the corresponding background Hermitian metric h, new Hermitian metric a and resulting complex Randers metric F. Moreover, we present some necessary and sufficient conditions for the obtained locally projectively flat solutions. Our findings are also illustrated with several examples.