2.1. Background of the navigation problem under consideration

We begin with some preliminaries concerning the description of the background for the study of the single-heading navigation problem. Let $(M,h)$ be a Riemannian manifold of dimension $n$ that is locally conformal to $n$-dimensional Euclidean space (conformally flat Riemannian manifold), i.e., the positive definite fundamental metric tensor $h_{ij}$ of the Riemannian metric $h$ satisfies

(1)\begin{equation} h_{ij}(x)=\frac{1}{S^{2}(x)}\delta _{ij}, \quad i,j=1,\ldots,n \end{equation}

for any $x\in M$, where the conformal factor is represented by a positive smooth function $S$ defined on $M$ and $\delta _{ij}$ is the Kronecker delta; see also Catino (Reference Catino2016). Let $(x^{1},\ldots, x^{n})$ be the local coordinates of $x\in M$. The tangent space $T_{x}M$ is spanned by ${\partial }/{\partial x^{1}}, \ldots, {\partial }/{\partial x^{n}}$ and so any $v\in T_{x}M$ can be expressed as $v=\sum _{i=1}^{n} v^{i} ({\partial }/{\partial x^{i}})$, where $v^{1}, \ldots, v^{n}$ are the coordinates of $v$. The length of $v$ with respect to $h$ is

\[ |v|_{h}=\sqrt{\sum_{i,j=1}^{n} h_{ij} v^{i}v^{j}} = \frac{1}{S} \sqrt{\sum_{i=1}^{n}(v^{i})^{2}} \]

or, briefly, $|v|_{h}=({1}/{S}) \sqrt {(v^{i})^{2}}$ is a sum with respect to $i=1, \ldots, n$, since $(v^{i})^{2}=v^{i}v^{i}$. From this point on, in each product where the same index appears twice, it should be read as a sum with respect to this index without writing the explicit symbol $\Sigma$ for the sum.

Let $M\times \mathbb {R}$ be the fibred manifold with a point $(x, t)$ on $M\times \mathbb {R}$, having the local coordinates $(x^{1},\ldots, x^{n},t)$, $t\in \mathbb {R}$. Instead of a smooth curve on $M\times \mathbb {R}$, we simply consider its projection on $M$, namely, a smooth curve $\gamma : [0,1] \subset \mathbb {R}\rightarrow M$, and the variable $t$ is thought of as a parameter to which is ascribed no geometrical significance (Carathéodory, Reference Carathéodory1935; Aldea and Kopacz, Reference Aldea and Kopacz2020). The points of the curve $\gamma$ are denoted by $(x^{1}(t),\ldots, x^{n}(t))$, $t\in [ 0,1]$, and for the tangent vector to $\gamma$ we use the notation

\[ \dot{x}^{i}(t)=\frac{dx^{i}(t)}{dt},\quad i=1,\ldots,n. \]

A time-dependent tangent vector to $M$ can be considered as the projection of a tangent vector to $M\times \mathbb {R}$, e.g., $\xi \in T_{x}M\times \mathbb {R}$ and $\xi =\xi ^{i}(x,t){\partial }/{\partial x^{i}} + {\partial }/{\partial t}$; see also Paláček and Krupková (Reference Paláček and Krupková2012) and Aldea and Kopacz (Reference Aldea and Kopacz2020) in this regard.

The ship will be modelled as a particle moving at variable speed with respect to water (air) on $(M,h)$. Under the influence of perturbation $W$, the ship's resulting velocity will be given by the composed vector $v=W+u$, with the ship's self-velocity $u\neq 0$. Here we consider space- and time-dependent wind $W$, i.e., $W=W^{i}(x,t) {\partial }/{\partial x^{i}} \in T_{x}M$ is the projection of $W^{i}(x,t) {\partial }/{\partial x^{i}}+{\partial }/{\partial t}\in T_{x}M\times \mathbb {R}$. Also, the ship's self-speed is $|u|_{h}=f(x,t)\in (0,1]$ and $W^{i}$, $i=1,\ldots, n$, and $f$ are smooth functions on $M\times \mathbb {R}$. Using the coordinates $(x^{1}(t),\ldots, x^{n}(t))$, the resulting velocity $v$ is the tangent vector to the trajectory, and so the global motion $v=W+u$ can be rewritten in the local coordinates as

(2)\begin{equation} \dot{x}^{i}(t)=W^{i}+u^{i}, \quad i=1,\ldots,n, \end{equation}

which are called the equations of motion.

Let $\cos \theta _{i}(t)$ stand for the directional cosines of the ship's velocity $u(x,t)$. The assumption of local flat conformality allows us to follow an analogous method to that in Arrow (Reference Arrow1949) by considering the functions $\alpha _{i}(x,t) = S(x) \cos \theta _{i}(t)$, $i=1,\ldots, n$, with $\sum _{i}(\alpha _{i})^{2} = S^{2}$. It follows that $u^{i}=f\alpha _{i}$ and then the equations of motion (2) can be rewritten as

(3)\begin{equation} \dot{x}^{i}(t)=W^{i}+f\alpha _{i},\quad i=1,\ldots,n. \end{equation}

The condition $|u|_{h}^{2}=h_{ij}u^{i}u^{j}=f^{2}$ is checked. Further on, under the above setting, the triple $(h,W,f)$ will be called the navigation data, which creates the background for the study of the pseudoloxodromic navigation problem.

2.2. Time-optimal paths

Now, we recall Zermelo's navigation problem,Footnote ¹ where the aim is to find the time-minimal trajectories, or rather the corresponding steering angles (optimal controls), of a ship (an aerial vehicle) that sails or flies in a space $M$, under the influence of a perturbation represented by a vector field $W$, thought of as a wind or a current. The problem was initially formalised and solved by Ernst Zermelo in 1929–1931 for Euclidean spaces of low dimensions, i.e., $\mathbb {R}^{2}$ (Zermelo, Reference Zermelo1930) and $\mathbb {R}^{3}$ (Zermelo, Reference Zermelo1931). Its new link to Finsler geometry has caused purely geometric investigations recently, with the background model created by a Riemannian manifold $(M, h)$, where $h$ is an arbitrary Riemannian metric (Bao et al., Reference Bao, Robles and Shen2004; Chern and Shen, Reference Chern and Shen2005). However, the Finslerian studies were developed under the assumptions that the wind $W$ is time-independent and only weak or critical, i.e., $|W|_{h}^{2}=h(W,W) \leq 1$, as well as a ship's self-speed is constant, $f=1$; see also Yoshikawa and Sabau (Reference Yoshikawa and Sabau2013), Brody and Meier (Reference Brody and Meier2015), Brody et al. (Reference Brody, Gibbons and Meier2015), Caponio et al. (Reference Caponio, Javaloyes and Sánchez2015), Aldea and Kopacz (Reference Aldea and Kopacz2017a, Reference Aldea and Kopacz2017b), Kopacz (Reference Kopacz2017a, Reference Kopacz2017b), Javaloyes and Vitório (Reference Javaloyes and Vitório2018) and Kopacz (Reference Kopacz2019) for some recent and generalised investigations in differential geometry and physics.

In the field of optimal control, the Zermelo problem is usually taken with application of Pontryagin's maximum principle (Pontryagin et al., Reference Pontryagin, Boltyanskii, Gamkrelidze and Mishchenko1962). In what follows we proceed via the Lagrangian and this way is in general equivalent to the discussion on the navigation problem via the Hamiltonian; see Bijlsma (Reference Bijlsma2001, Reference Bijlsma2009), Hull (Reference Hull2009), Techy and Woolsey (Reference Techy and Woolsey2009), Bijlsma (Reference Bijlsma2010), Techy (Reference Techy2011), Jardin and Bryson (Reference Jardin and Bryson2012), Burns (Reference Burns2013), Li et al. (Reference Li, Xu, Teo and Chu2013) and Marchidan and Bakolas (Reference Marchidan and Bakolas2016). However, which of them is more convenient in the sense of solution and computational complexity depends in fact on the specific navigation data. Following the general principles from Levi-Civita (Reference Levi-Civita1931), De Mira Fernandes (Reference De Mira Fernandes1932) and Arrow (Reference Arrow1949), however, in the more general case due to the extended navigation data $(h,W,f)$, recently we have studied the Lagrangian $L(x(\tau ),\dot {x}(\tau ),t(\tau ))$ in Kopacz (Reference Kopacz2018b) and Aldea and Kopacz (Reference Aldea and Kopacz2019a), which is a positive root of the following equation:

(4)\begin{equation} \lambda L^{2}+2pL-|v|_{h}^{2}=0, \end{equation}

where $\lambda =f^{2}-|W|_{h}^{2}$,

\[ p=S^{{-}2}\sum_{i}W^{i}\dot{x}^{i}, \quad |W|_{h}^{2}=S^{{-}2}\sum_{i}(W^{i})^{2}, \quad |v|_{h}^{2}=S^{{-}2}\sum_{i}(\dot{x}^{i})^{2} \]

and $L(x(t),\dot {x}(t),t)=1$ along the curves that satisfy Equation (2). Note that the case $\lambda \neq 0$ (namely, Equation (4) is of degree two) refers to a weak wind ($\lambda >0)$ or a strong wind ($\lambda <0)$, while $\lambda =0$ (namely, Equation (4) is of degree one, under the restriction $p\neq 0)$ refers to a critical wind. Since $L(x(\tau ),\dot {x}(\tau ),t(\tau ))$ is homogeneous of degree one with respect to $\dot {x}$, it allows us to compute the length, i.e., $T=\int _{0}^{1}L(x(\tau ),\dot {x}(\tau ),t(\tau ))\,d\tau$, which is interpreted here as the physical time necessary to travel along the extremals from a starting point $(x^{1}(0),\ldots, x^{n}(0))$ to a destination point$(x^{1}(1),\ldots, x^{n}(1))$. The first variation of $T$, under the condition $dt=L(x(\tau ), \dot {x}(\tau ), t(\tau ))\,d\tau$, leads to the Euler–Lagrange equations (see, e.g., Levi-Civita, Reference Levi-Civita1931), that is

(5)\begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}^{i}}\right) -\frac{\partial L}{\partial{x}^{i}} =\frac{\partial L}{\partial t}\frac{\partial L}{\partial \dot{x}^{i}}, \quad i=1,\ldots,n. \end{equation}

Applying Equation (5) to the Lagrangian $L$ given by Equation (4), we have recently obtained the conditions for time-extremal navigation considered on conformally flat Riemannian manifolds. Namely, the general solution to time-extremal navigation problem on $(M,h)$ is represented by Equation (3) and the condition

(6)\begin{align} \frac{d\alpha _{i}}{dt} &={-}S^{2}\left(\frac{\partial f}{\partial x^{i}}+\frac{1}{ S^{2}}\frac{\partial W^{j}}{\partial x^{i}}\alpha _{j}\right)+\frac{1}{S^{2}}\frac{ \partial W^{k}}{\partial x^{j}}\alpha _{i}\alpha _{j}\alpha _{k}+\frac{ \partial f}{\partial x^{j}}\alpha _{i}\alpha _{j}-Sf\frac{\partial S}{ \partial x^{i}}\nonumber\\ &\quad +\frac{1}{S}\frac{\partial S}{\partial x^{j}}(W^{j}+2f\alpha_{j})\alpha _{i}, \quad i=1,\ldots,n, \end{align}

under the restriction $\lambda +p\neq 0$, where $\lambda +p=f^{2}-|W|_{h}^{2}+p$. Note that the inequality $\lambda +p> 0$ can hold for any wind, but if a wind is weak or critical, then $\lambda +p> 0$. Also, if $\lambda +p\leq 0$, then a wind can only be strong. The curves $x(t)$ that satisfy Equation (3) and $\lambda +p=0$ are called anomalous (or abnormal) and Equation (6) may hold; see, e.g., Agrachev and Sachkov (Reference Agrachev and Sachkov2004) and Carathéodory (Reference Carathéodory1935) in this regard.

Equation (6) is the generalisation of the classical formula of Zermelo, namely, the condition for optimal navigation. We remark that by time-optimal solutions to the navigation problem we understand both minimum and maximum time. So, in this sense, we may consider Zermelo's problem as the particular case of the navigation problem. The solutions of least time are of main interest in practice and theoretical studies, but for a complete exposition concerning optimal navigation in arbitrary winds it is necessary to take into consideration also time-maximal paths. This is due to the fact that the same Equation (6) together with Equation (3) can generate both types of time extremals in strong winds. Thus, they need to be distinguished.

Additional conditions for the time-minimal (time-maximal) solutions are provided by the study of the second variation of $T$. Namely, some classification results on both types of extremal paths with respect to the navigation data, so in particular the wind force $|W|_h$, are included in the following proposition (Kopacz, Reference Kopacz2018b; Aldea and Kopacz, Reference Aldea and Kopacz2019a).

2.3. Time-optimal generalised loxodromic navigation

By a generalised loxodromic navigation we call a navigation with a constant direction of the resultant velocity $v$ during the entire travel on $(M,h)$ between a starting point and a terminal point, i.e., ${d\tilde {\beta }_{i}}/{dt}=0$, for all $i=1,\ldots, n$, where $\tilde {\beta } _{i} := \cos \tilde {\theta }_{i}(t)$ are the directional cosines of the ship's resultant velocity $v\neq 0$. The corresponding resulting trajectory on $M$ under the above-mentioned conditions will be called a generalised loxodrome. Also, by $(h,W,f, \tilde {\beta }_{i}=\textrm {const.})$, $i=1,\ldots, n$, we mean the loxodromic (navigation) data. In this subsection we recall some results from Aldea and Kopacz (Reference Aldea and Kopacz2019b) and Kopacz (Reference Kopacz2018b). Expressing the ship's resultant velocity $v\neq 0$ in the form

(7)\begin{equation} \dot{x}^{i}=g(x,t)S\tilde{\beta}_{i},\quad i=1,\ldots,n, \end{equation}

where $\dot {x}^{i}=v^{i}$ and $g=|v|_{h}$ (a non-zero smooth function on $M\times \mathbb {R}$) is not known a priori like $f=|u|_{h}$, and combining Equations (7) and (3), we obtain

(8)\begin{equation} \frac{d\alpha _{i}}{dt}={-}\frac{1}{f}\left(\frac{\partial W^{i}}{\partial t}+ \frac{\partial f}{\partial t}\alpha _{i}\right) -\frac{gS}{f}\left(\frac{ \partial W^{i}}{\partial x^{j}}+\frac{\partial f}{\partial x^{j}}\alpha _{i}\right) \tilde{\beta}_{j}+\frac{S}{f}\frac{dg}{dt}\tilde{\beta}_{i}+ \frac{g^{2}S}{f}\frac{\partial S}{\partial x^{j}}\tilde{\beta}_{i}\tilde{ \beta}_{j}, \end{equation}

with $i=1,\ldots, n$, where

(9)\begin{align} \frac{dg}{dt}&=g\rho \frac{\partial W^{k}}{\partial t}\alpha _{k}+g\rho S^{2} \frac{\partial f}{\partial t}+g^{2}\rho S\frac{\partial W^{k}}{\partial x^{j} }\alpha _{k}\tilde{\beta}_{j}+g^{2}\rho S^{3}\frac{\partial f}{\partial x^{j} }\tilde{\beta}_{j}\nonumber\\ &\quad +g^{2}\left(f\rho S^{2}-1\right) \frac{\partial S}{ \partial x^{j}}\tilde{\beta}_{j}, \end{align}

and $\rho := {f}/(S^{2}(\lambda +p))$, under the loxodromic data and $\lambda +p\neq 0$.

For short, we also write ‘a loxodrome’ instead of ‘a generalised loxodrome’ when no confusion can arise.

3.1. Generalisation of single-heading navigation

Definition 3.1. A generalised single-heading (or pseudoloxodromic) navigation is a navigation with a constant direction of the ship's self-velocity $u$ during a travel on $(M,h)$ between a starting point and a terminal point, i.e.,

(10)\begin{equation} \forall\ i=1,\ldots,n \quad \theta _{i}(t)=\textrm{const.} \end{equation}

By analogy to widely applied rhumb line navigation in practice and the occurrence of the well-known notion of a loxodrome in different theoretical studies, a path on $M$ satisfying Equation (10) will be called a pseudoloxodrome or a generalised single-heading trajectory.

To investigate $u$ with constant direction, we denote by $\beta _{i}(t)$ the directional cosines of the ship's velocity $u(x,t)$, i.e., $\beta _{i}(t):=\cos \theta _{i}(t)$, $i=1,\ldots, n$. This leads to $\sum _{i}(\beta _{i})^{2}=1$ and

(11)\begin{equation} \alpha _{i}=S(x)\beta _{i}(t). \end{equation}

Hence, the equivalent conditions for the pseudoloxodromic navigation are

(12)\begin{equation} \forall\ i=1,\ldots,n \quad \frac{d\beta _{i}}{dt}=0. \end{equation}

By $(h,W,f,\beta _{i}=\textrm {const.})$, $i=1,\ldots, n$, we call this the pseudoloxodromic navigation data.

Now, we aim to find the generalised single-heading trajectories. Thus, we have the following.

Proposition 3.2. Let $(M,h)$ be a conformally flat Riemannian manifold of dimension $n$, where $h$ is given by Equation (1). Let $(h, W, f, \beta _{i} = \textrm {const.})$, $i=1,\ldots, n$, be the pseudoloxodromic navigation data. If there exist the pseudoloxodromes on $(M,h)$, then they are the solutions of the system of Equations (3) and

(13)\begin{equation} \frac{d\alpha _{i}}{dt}=\frac{1}{S}\frac{\partial S}{\partial x^{j}} (W^{j}+f\alpha _{j})\alpha _{i},\quad i=1,\ldots,n. \end{equation}

Proof. From Equation (11) it results that

\[ \frac{d\alpha_{i}}{dt} = \frac{\partial S}{\partial x^{j}}\dot{x}^{j}\beta_{i} + S\frac{d\beta_{i}}{dt}. \]

Next, we require the direction of $u$ to be constant, i.e., Equation (12) holds. This yields

\[ \frac{d\alpha _{i}}{dt} = \frac{\partial S}{\partial x^{j}}\dot{x}^{j}\beta _{i}, \]

which together with Equation (3) leads to Equation (13).  ■

Roughly speaking, the existence of the generalised single-heading paths depends on a wind, so in particular its ‘force’ $|W|_h$.

Note that the spherical coordinates denoted by $\varphi _{l}$, $l=1,\ldots, n-1$, are more suitable for the navigational description of a ship's motion (steering, controls) than the angles $\theta _{i}$, $i=1,\ldots, n$. The latter fit better to the geometrical description of the motion determined by the vectors of the ship's velocities. Therefore, a coordinate system in an $n$-dimensional Euclidean space can be defined such that the coordinates consist of a radial coordinate, $r$, and $n-1$ angular coordinates, $\varphi _{1},\ldots, \varphi _{n-1}$, where $\varphi _{2}, \ldots, \varphi _{n-1} \in [ -{\pi }/{2}, {\pi }/{2}]$ and $\varphi _{1} \in [ 0,2\pi )$. If $x^{i}$, $i=1,\ldots, n$, represent the Cartesian coordinates, then they can be computed from $r$ and $\varphi _{1},\ldots, \varphi _{n-1}$. Next, we can find the relation between the directional cosines $\beta _{i}=\cos \theta _{i}$ and $\varphi _{l}$. Taking into consideration that $\beta _{i}$ is the component of the versor of a given vector, so $r=1$ and $x^{i}=\beta _{i}$, we have the following relation for the ship's velocity $u$:

(14)\begin{equation} {\beta}_{k}=\prod_{i=k}^{n}\cos {\varphi}_{i}\sin {\varphi}_{k-1}, \end{equation}

where $k=1,\ldots, n$, ${\varphi }_{0}={\pi }/{2}$ and ${\varphi }_{n}=0$. Making use of the spherical coordinates we can prove the following.

Regarding the above restriction excluding $\pm {\pi }/{2}$ see, for instance, dimension three. Namely, take $\theta _{1}=\theta _{2}={\pi }/{2}=\textrm {const.}$, and so $\theta _{3}=0=\textrm {const.}$ Then $\varphi _{2}={\pi }/{2}=\textrm {const.}$, but $\varphi _{1}$ is arbitrary, so not necessarily constant. Thus, in dimension three, by application of the spherical coordinates the three angles of the directional cosines, i.e., $\theta _{1}$, $\theta _{2}$ and $\theta _{3}$, are reduced by Equation (14) to two spherical coordinates. These are the angular controls, i.e., the heading $\varphi _{1}:=\varphi$ and the elevation $\varphi _{2}:=\vartheta$. The navigation with $\varphi _{1}(t)=\textrm {const.}$ and $\varphi _{2}(t)\neq \textrm {const.}$ is not considered here as the generalised single-heading navigation, although the (common) heading is constant in time. This is due to the fact that the (three-dimensional) direction of the ship's velocity $u$ is varying. In higher dimensions, our reasoning is analogous. We can also consider a piecewise pseudoloxodromic route (flight), i.e., with different pseudoloxodromic legs between the waypoints.

In dimension two, the heading is $\varphi :=\theta _{1}$. Since $\theta _{1}+\theta _{2}={\pi }/{2}$, we get $\beta _{1}=\cos \varphi$ and $\beta _{2}=\sin \varphi$, and so $\alpha _{1}=S\cos \varphi$ and $\alpha _{2}=S\sin \varphi$. In this case Equation (13) means that $-\dot {\varphi }\sin \varphi =0$ and $\dot {\varphi }\cos \varphi =0$, which yield $\dot {\varphi }=0$ and thus $\varphi =\varphi _{0}=\textrm {const.}$ The paths with $\varphi _{0}=\textrm {const.}$ are the solutions of the system

(15)\begin{equation} \dot{x}^{1} = W^{1}+fS\cos \varphi _{0}, \quad \dot{x}^{2} = W^{2}+fS\sin \varphi _{0}. \end{equation}

3.2. Time-optimal pseudoloxodromes

When the single-heading navigation (in dimension two) was introduced for the first time, it was taken for granted that the resulting trajectories themselves were extremals and that, consequently, along these trajectories the time of navigation would assume an extreme value. Although, in general, the time of navigation is shorter along a single-heading trajectory than along the geometrically shortest route, it is only in fields of flow with a particular structure that the single-heading trajectories are time-minimising or time-maximising extremals. The single-heading trajectories in $\mathbb {R}^{2}$, which at the same time are also extremals, were called ‘single-heading extremals’ in De Jong (Reference De Jong1956). We would like to mention that the planar examples of time-minimal trajectories that are single-heading have already been emphasised in Zermelo (Reference Zermelo1930), Levi-Civita (Reference Levi-Civita1931), Arrow (Reference Arrow1949) and Carathéodory (Reference Carathéodory1935) in the very first studies on the navigation problem.

We thus arrive at the following theorem.

Theorem 3.4. Let $(M, h)$ be a conformally flat Riemannian manifold of dimension $n$, where $h$ is given by Equation (1). Let $(h,W,f)$ be the navigation data and $\lambda +p\neq 0$. The necessary conditions for a pseudoloxodromic time-extremal navigation on $(M,h)$ are

(16)\begin{equation} \left(\frac{\partial W^{k}}{\partial x^{j}}\beta _{k}+\frac{\partial (fS)}{ \partial x^{j}}\right) \beta _{ij}=0,\quad i=1,\ldots,n, \end{equation}

where $\beta _{ij}:=\delta _{ij}-\beta _{i}\beta _{j}$.

Proof. Taking into consideration the assumption for the constant ‘headings’, i.e., Equation (13) with $\alpha _{i}=S\beta _{i}$ and inserting this into Equation (6), we obtain

(17)\begin{align} \frac{\partial S}{\partial x^{j}}(W^{j}+fS\beta _{j})\beta _{i}&={-}S^{2}\left(\frac{\partial f}{\partial x^{i}}+\frac{1}{S}\frac{\partial W^{j}}{\partial x^{i}}\beta _{j}\right) +S\frac{\partial W^{k}}{\partial x^{j}}\beta _{i}\beta _{j}\beta _{k}+S^{2}\frac{\partial f}{\partial x^{j}} \beta _{i}\beta _{j} \nonumber\\ &\quad -Sf\frac{\partial S}{\partial x^{i}}+\frac{\partial S}{\partial x^{j}} (W^{j}+2fS\beta _{j})\beta _{i}, \end{align}

which gives Equation (16).  ■

Note that $\beta _{ij}$, $i,j=1,\ldots, n$, is not an invertible matrix, but it is symmetric ($\beta _{ij}=\beta _{ji}$, $i,j=1,\ldots, n)$ and $\beta _{ij}\beta _{i}=0$, $j=1,\ldots, n$. Equations (3) and (16) determine the time-optimal pseudoloxodromes, if they exist.

Now, taking together Proposition 2.1 and Theorem 3.4, we have the next result.

3.3. Which winds yield the pseudoloxodromic time-optimal navigation?

Recall that, in the time-extremal navigation problem, one normally aims to find the resulting trajectory or the corresponding optimal control for given navigation data including wind. However, we can also look at the problem from a different point of view. Namely, we can look for the vector fields $W$ in the presence of which the time-extremal solutions have some special properties. For instance, all of them are pseudoloxodromic or loxodromic, or both. We should mention that, under the considered navigation data (with given $W$) by Proposition 2.2 and Theorem 3.4, one can obtain some pseudoloxodromic and loxodromic paths among the family of time-extremal trajectories, respectively, if they exist. Namely, not all of them are of such special kind.

To extend the study, further on we ask about the types of perturbations in which all existing solutions of extreme time are generalised single-heading or loxodromic, or both at the same time. Thus, we can reformulate Theorem 3.4 in the following way.

In particular, we have the following remarks. Namely, if $f=\textrm {const.}$, i.e., the standard version of Zermelo's problem (with constant maximum ship's self-speed) or if $S=\textrm {const.}$, i.e., the background metric is Euclidean, then the PDE system admits simplified forms. Considering $(\mathbb {R}^{n},\delta _{ij})$ with $f=1$, then $W=W(t)$ or $W^{i}=\textrm {const.}$, $i=1,\ldots, n$, are the solutions of Equation (16). Thus, in this last case all existing time-extremal paths are pseudoloxodromic.

In dimension two, since the necessary and sufficient condition for a constant direction of velocity $u$ is $\varphi _{0}=\varphi (t)=\textrm {const.}$, the PDE system (16) is reduced to

(18)\begin{align} 0&={-}\frac{\partial W^{1}}{\partial x^{2}}\cos ^{2}\varphi _{0}+\left(\frac{ \partial W^{1}}{\partial x^{1}}-\frac{\partial W^{2}}{\partial x^{2}}\right) \sin \varphi _{0}\cos \varphi _{0}+\frac{\partial W^{2}}{\partial x^{1}}\sin ^{2}\varphi _{0} \nonumber\\ &\quad +\frac{\partial (fS)}{\partial x^{1}}\sin \varphi _{0}-\frac{\partial (fS) }{\partial x^{2}}\cos \varphi _{0}, \end{align}

for any $\varphi _{0}=\textrm {const.}$ and unknown wind components $W^{1}$ and $W^{2}$. So, the last equation leads to the types of perturbations under which the solutions to the time-extremal problem are single-heading. In particular, if $f=\textrm {const.}$ and $S=\textrm {const.}$, then Equation (18) yields the condition

(19)\begin{equation} \frac{\partial W^{1}}{\partial x^{2}}\cos ^{2}\varphi _{0}-\left(\frac{ \partial W^{1}}{\partial x^{1}}-\frac{\partial W^{2}}{\partial x^{2}}\right) \sin \varphi _{0}\cos \varphi _{0}-\frac{\partial W^{2}}{\partial x^{1}}\sin ^{2}\varphi _{0}=0. \end{equation}

This is in fact the classical navigation formula of Zermelo (Reference Zermelo1930, Reference Zermelo1931) considered for the constant heading in $\mathbb {R}^{2}$, $\varphi :=\varphi _{0}$. The condition (19) implies that in $\mathbb {R}^{2}$ the only stationary winds $W$ in which all the solutions to the Zermelo problem ($f=\textrm {const.}$) are single-heading are given by the radial, negative radial or constant vector fields. Thus, the above leads to the result from De Jong (Reference De Jong1956), that is, ‘all extremals are single-heading extremals in convergent or divergent and in uniform rectilinear fields of flow’. Such a theorem was proved by means of Hamilton's PDEs (see De Jong (Reference De Jong1956, Reference De Jong1974) for more details and the investigation on the planar vector fields in which the single-heading extremals exist).

Now taking into consideration Proposition 2.2 we have the following.

Proposition 3.7. Let $(M, h)$ be a conformally flat Riemannian manifold of dimension $n$, where $h$ is given by Equation (1). Let $u(x,t)$ be a ship's self-velocity with $|u|_{h}=f(x,t)\in (0,1]$ and the directional cosines $\tilde {\beta }_{i}=\textrm {const.}$, $i=1,\ldots, n$. Then the wind $W=W^{i}(x,t) {\partial }/{\partial x^{i}} \in T_{x}M$, with $0\leq |W|_{h}$ and $\lambda +p\neq 0$, under which all time-extremal paths yielded by Equations (3) and (6) are loxodromes, is the solution of the following PDE system:

(20)\begin{align} &\left(\frac{\partial W^{k}}{\partial t}+gS\frac{\partial W^{k}}{\partial x^{j}}\tilde{\beta}_{j}\right) \tilde{\beta}_{ik}-fS\left(\frac{\partial W^{k}}{\partial x^{j}}\beta _{k}+\frac{\partial (fS)}{\partial x^{j}}\right) \beta _{ij}\nonumber\\ &\quad =g\rho S^{3}\left(\frac{\partial f}{\partial t}+\frac{\partial (fS)}{\partial x^{j}}\tilde{\beta}_{j}\right) \beta _{ik}\tilde{\beta}_{k}, \end{align}

$i=1,\ldots, n$, where $\tilde {\beta }_{ik}:=\delta _{ik}-g\rho S^{2}\tilde { \beta }_{i}\beta _{k}$.

Proof. Substituting Equations (8) and (9) into the system of Equations (3) and (6) in which the wind $W$ is unknown, results in Equation (20).  ■

Note that $\tilde {\beta }_{ik}\beta _{i}=0$, $\tilde {\beta }_{ik}\tilde {\beta }_{k}=0$ and $\beta _{ik}\tilde {\beta }_{k}=\tilde {\beta }_{i}-({1}/(g\rho S^{2}))\beta _{i}$, $k=1,\ldots, n$. Also $\tilde {\beta }_{ik}$, $i,k=1,\ldots, n$, is not an invertible matrix. Comparing the PDE systems (16) and (20), under the assumption $fS=\textrm {const.}$, we can emphasise the following fact. If the components of $W$ are arbitrary time-dependent functions, then such wind is the solution of Equation (16). However, it may not solve Equation (20). Therefore, there exist time-extremal trajectories such that they are pseudoloxodromic, but they are not loxodromic.

Finally, by Propositions 3.6 and 3.7 we obtain the following corollary.

Corollary 3.8. Let $(M,h)$ be a conformally flat Riemannian manifold of dimension $n$, where $h$ is given by Equation (1). Let $u(x,t)$ be a ship's velocity with $|u|_{h}=f(x,t)\in (0,1]$ and the directional cosines $\tilde {\beta }_{i}=\textrm {const.}$, $i=1,\ldots, n$. Then the wind $W=W^{i}(x,t) {\partial }/{\partial x^{i}} \in T_{x}M$, with $0\leq |W|_{h}$ and $\lambda +p\neq 0$, under which all time-extremal paths yielded by Equations (3) and (6) are both pseudoloxodromic and loxodromic, is the solution of the PDE system of Equations (16) and

(21)\begin{equation} \left(\frac{\partial W^{k}}{\partial t}+gS\frac{\partial W^{k}}{\partial x^{j}}\tilde{\beta}_{j}\right) \tilde{\beta}_{ik}=g\rho S^{3}\left(\frac{ \partial f}{\partial t}+\frac{\partial (fS)}{\partial x^{j}}\tilde{\beta} _{j}\right) \beta _{ik}\tilde{\beta}_{k}, \quad i=1,\ldots,n. \end{equation}

Proof. Equation (21) results from Equations (16) and (20).  ■

In order to emphasise some solutions to Equations (16) and (21), we consider the following cases. First, if $\textit {fS}=\textrm {const.}$, then the wind $W$ with $W^{i}=\textrm {const.}$, $i=1,\ldots, n$, is the solution of Equations (16) and (21), and so all time-extremal trajectories are pseudoloxodromic and loxodromic, if they exist. Second, also under the assumption $\textit {fS}=\textrm {const.}$, for the wind $W$ with $W^{i}=ax^{i}+b_{i}$, with $a,b_{i}=\textrm {const.}$, $i=1,\ldots, n$, all time-extremal trajectories are pseudoloxodromic and loxodromic. Indeed, since ${\partial W^{k}}/{\partial t}=0$ and ${\partial W^{k} }/{\partial x^{j}}=\delta _{jk}$, which imply

\[ \frac{\partial W^{k}}{\partial x^{j}}\beta_{k}\beta_{ij}=\beta_{j}\beta_{ij}=0 \quad \mbox{and} \quad \frac{\partial W^{k}}{\partial x^{j}}\tilde{\beta}_{j}\tilde{\beta}_{ik} =\tilde{\beta}_{j} \tilde{\beta}_{ij}=0, \]

we have Equations (16) and (21) identically checked. This last case is actually the generalisation of the two-dimensional result from De Jong (Reference De Jong1956), i.e., the wind is radial ($a>0$), negative radial ($a<0$) or constant ($a=0$).

4.1. Euclidean plane with linear (shear) wind

First, let $(M, h)$ be $(\mathbb {R}^2, \delta _{ij})$ and let the perturbation be given by the shear river-type vector field, i.e., $W = (x^2, 0)$. Thus, the current of the ‘river’ increases as a linear function of $x^2$, reaching its minimum force in the midstream. We mention that Zermelo described the problem in the presence of such perturbation (actually he used $W = (-x^2, 0)$) as ‘das einfachste nicht-triviale Beispiel unserer Theorie’ (‘the simplest nontrivial example of our theory’) in his first paper (Zermelo, Reference Zermelo1930), where the navigation problem was formalised initially, with $f=1=\textrm {const.}$ An example with the field and the use of the Hamiltonian formalism was also investigated in Carathéodory (Reference Carathéodory1935). In particular, if $|W|_h = |x^2|<1$, then the ship can arrive at any point of destination on the planar sea, so the solution exists, since the current is weak. Now, however, let $f(t,x^1, x^2)=0{\cdot }95\sin ^2x^2+0{\cdot }01$, so the ship's speed is varying in space and the restriction $0<f\leq 1$ is checked. Applying Equation (6) we get the time-optimal trajectories from the system

(22)\begin{equation} \dot{\varphi}={-}\cos^2\varphi-0{\cdot}95\sin2x^2\cos\varphi, \quad \dot{x}^1=x^2+f\cos\varphi, \quad \dot{x}^2=f\sin\varphi. \end{equation}

Having solved the last system, we present the time extremals starting from $(0; -0{\cdot }5)$ for different headings $\varphi (0)=\varphi _0\in [0; 360^{\circ })$, i.e., with the increments $\Delta \varphi _0=15^{\circ }$. They are shown in Figure 1. Since we search the pseudoloxodromic extremals, by Equation (19) we get $\cos ^2\varphi _0+0{\cdot }95\sin 2x^2\cos \varphi _0=0$. This implies that the single-heading navigation of minimum time is possible only if $\varphi _0=90^{\circ }$ or $\varphi _0=270^{\circ }$; $\lambda +p=f(f+x^2_0\cos \varphi _0)>0$. Thus, the strategy is to sail orthogonally to the flowing current $W$ continuously. The single-heading paths of least time are presented in red and blue, respectively.

Figure 1. The only single-heading solutions to the time-optimal navigation problem with variable ship's speed $f(t,x^1, x^2)=0{\cdot }95\sin ^2x^2+0{\cdot }01$ and under the action of the linear vector field $W=(x^2, 0)$ are for the headings $\varphi _0 = 90^{\circ }$ (time-minimal, red) and $270^{\circ }$ (time-minimal, blue). In the background are other optimal paths: time-minimal (solid black) for $\varphi _0 \in (62{\cdot }8^{\circ },297{\cdot }2^{\circ })$ and time-maximal (dashed black) for $\varphi _0 \in [0,62{\cdot }8^{\circ }) \cup (297{\cdot }2^{\circ },360^{\circ })$, to an accuracy of $0{\cdot }1^{\circ }$, obtained with the increments of the initial heading $\Delta \varphi _0=15^{\circ }$. Also, there exist two anomalous paths (dot-dashed black) for $\varphi _0 \in \{62{\cdot }8^{\circ },297{\cdot }2^{\circ }\}$. The starting point $(0; -0{\cdot }5)$ is within the area of strong current; $t\leq 35$.

For comparison, we solve the problem for different ship's self-speed, which has the form $f(t,x^1, x^2)=({1}/{\sqrt {k}})x^1-C$, with $C=\textrm {const.}$ Then the domain is $x^1\in (C\sqrt {k}; (1+C)\sqrt {k}]$ due to the restriction on $f$, and from Equation (19) we get $\cos ^2\varphi _0=({1}/{\sqrt {k}})\sin \varphi _0$. For example, let $k:=16$ and $C:=-0{\cdot }33$, so $x^1\in (-1{\cdot }32; 2{\cdot }68]$. The single-heading time-minimal paths among other time-optimal solutions are shown in Figure 2. Their desired headings are: $62^{\circ }$ (blue) and $118^{\circ }$ (red). We remark that there can also exist single-heading extremals of both types (minimal and maximal), if we shift the initial point slightly, e.g., $x_0:=-0{\cdot }5$. Then the values for both pseudoloxodromic extremals are preserved, but the path with $\varphi _0=62^{\circ }$ becomes time-maximal, since $\lambda +p=f(t_0,x^1_0, x^2_0)(f(t_0,x^1_0, x^2_0)+x^2_0\cos \varphi _0)<0$.

Figure 2. The only single-heading solutions to the time-optimal navigation problem with variable ship's speed $f(t,x^1, x^2)=0{\cdot }25x^1+0{\cdot }33$ and under the action of the linear vector field $W=(x^2, 0)$ are for the headings $\varphi _0 = 62^{\circ }$ (blue) and $118^{\circ }$ (red), both time-minimal. In the background are other optimal paths: time-minimal (solid black) for $\varphi _0 \in (48{\cdot }7^{\circ }, 311{\cdot }3^{\circ })$ and time-maximal (dashed black) for $\varphi _0\in [0, 48{\cdot }7^{\circ }) \cup (311{\cdot }3^{\circ }, 360^{\circ })$, to an accuracy of $0{\cdot }1^{\circ }$, obtained with the increments of the initial heading $\Delta \varphi _0=15^{\circ }$. There also exist two anomalous paths for $\varphi _0\in \{48{\cdot }7^{\circ }, 311{\cdot }3^{\circ }\}$. The starting point $(0; -0{\cdot }5)$ is within the area of strong current; $t\leq 8$.

Finally, we also show the solutions when the ship proceeds with constant maximum speed, i.e., $f=1$. Now the same starting point is within the area of weak wind and there are no time-maximal paths. The corresponding single-heading time-minimal trajectories are parabolic (solid red and solid blue in Figure 3).

Figure 3. The only single-heading solutions of minimum time (parabolic) to the standard navigation problem ($f=1=\textrm {const.}$), under the action of the linear (shear) vector field $W=(x^2, 0)$ are for the headings $\varphi _0 = 90^{\circ }$ (solid red) and $270^{\circ }$ (solid blue). The starting point $(0; -0{\cdot }5)$ is within the area of weak current. In the background are other time-minimal solutions (solid black) generated with the increments of the initial heading $\Delta \varphi _0=15^{\circ }$ (30 paths), $\varphi _0\in [0, 360^{\circ })$. The dashed (red, blue) lines are just the prolongations of the single-heading paths of least time to show their parabolic shape clearly.

4.2. Prolate ellipsoid under weak rotational wind

Let $\Sigma ^{2}$ be an ellipsoid embedded in the Euclidean space $\mathbb {R}^{3}$, with the Cartesian coordinates $(x,y,z)$, and axes equal to $2r$, $2r$ and $2ar$. The parametrisation of $\Sigma ^{2}$ in the spherical coordinate system $(\tilde {\rho },\phi, \theta )$ leads to the relations $x=r\sin \theta \cos \phi$, $y=r\sin \theta \sin \phi$ and $z=ar\cos \theta$, where the azimuth $\phi \in [0,2\pi )$ and the inclination (colatitude) $\theta \in [0,\pi ]$. The parameter $a>0$ determines the shape of an ellipsoid and as a consequence the flow of the geodesics on a spheroid, which can be oblate ($0<a<1$) or prolate ($a>1$). The line element of $\Sigma ^{2}$ expressed in terms of the spherical coordinates $(\phi, \theta )$ on $\Sigma ^{2}$ is $ds^{2} = r^{2}\sin ^{2}\theta (d\phi )^{2} + r^{2}(\cos ^{2}\theta + a^{2}\sin ^{2}\theta )(d\theta )^{2}$. For simplicity, we assume that $r=1$, so the ellipsoid has the semiaxes $(1,1,a)$. In particular, travelling between two points on $\mathbb {S}^2$, we can reorient the system of coordinates so that the destination point $D$ is located in one of the poles. This can be done by the appropriate rotation of the sphere and adapting $W$. In order to solve the navigation problem on the rotational ellipsoid, we consider the transformation

(23)\begin{equation} x^{1}=\phi, \quad x^{2}=\int \frac{\sqrt{\varepsilon }}{\sin \theta }\,d\theta, \end{equation}

where we have $\varepsilon := \cos ^{2}\theta +a^{2}\sin ^{2}\theta$, $\theta \in (0,\pi )$. This can be rewritten as $ds^{2}=({1}/{S^{2}})[(dx^{1})^{2}+(dx^{2})^{2}]$, with $S=S(x^{2})={1}/(r\sin \theta )$, $\theta \in (0,\pi )$.

Next, we consider the perturbing (rotational) vector field in the following form:

$$\begin{align} W(x,y,z) &=\tilde{W}^{1}(x,y,z)\frac{\partial }{\partial x}+\tilde{W} ^{2}(x,y,z)\frac{\partial }{\partial y}+\tilde{W}^{3}(x,y,z)\frac{\partial }{ \partial z} \\ &=cy\frac{\partial }{\partial x}-cx\frac{\partial }{\partial y}+0\frac{ \partial }{\partial z}, \quad c\in \mathbb{R}, \end{align}$$

which acts on $\Sigma ^{2}$ seen as embedded in $\mathbb {R}^{3}$. After transformation, the expression of $W$ in the base $\{{\partial }/{ \partial \phi },{\partial }/{\partial \theta }\}$ becomes

(24)\begin{equation} W(\phi,\theta)=W^{1}(\phi,\theta)\frac{\partial }{\partial \phi } +W^{2}(\phi,\theta)\frac{\partial }{\partial \theta }={-}c\frac{\partial }{ \partial \phi }, \end{equation}

and $|W(\phi, \theta )|_{h}=|c|\sin \theta$, $\theta \in (0,\pi )$, which includes all types of winds, i.e., $|W|_{h}<f$ (weak), $|W|_{h}=f$ (critical) and $|W|_{h}>f$ (strong). Let $f=1$, $c=\frac {5}{7}$ (weak wind) and the flattening is determined by $a:=\frac {3}{2}$. Thus, the two equations of the spheroidal motion are

(25)\begin{equation} \dot{\phi}=\frac{\cos\varphi}{\sin\theta}-c, \quad \dot{\theta}={-}\frac{\sin\varphi}{\sqrt{\varepsilon}}, \end{equation}

and by Equation (18) it follows easily that the condition for the generalised single-heading optimal navigation is $\varphi _0\in \{90^{\circ }, 270^{\circ }\}$. Consequently, the strategy is to head the self-velocity $u$ towards the destination point(s), i.e., the pole(s). This means that the ship ($u$) follows a meridian pointing north or south; however, the spherical track will look variously. Namely, the angle $\tilde {\varphi }(t)$ between the resulting velocity vector $v$ and the meridians varies. This is investigated in Figure 5 for three paths starting from different points on $\Sigma ^2$, i.e., at three colatitudes: $60^{\circ }$ (blue), $90^{\circ }$ (red) and $120^{\circ }$ (black). If $\tilde {\varphi }(t)$ is taken clockwise from north, then it corresponds to a course over ground in navigation. The corresponding colour-coded trajectories are presented in Figure 4. It is clear that the solutions are not loxodromic, since $\tilde {\varphi }\neq \textrm {const.}$, which Figure 5 shows. However, they are single-heading and time-optimal. We remark that the constant heading $\varphi$ changes sign, if the point of destination becomes the opposite pole. We also mention that several moons of the Solar System approximate prolate spheroids in shape; however, they are actually triaxial ellipsoids. Also, the algorithms for accurate and global navigational calculations correspond to spheroidal (oblate, $a\in (0; 1)$) geometric models; see, e.g., Earle (Reference Earle2006), Pallikaris and Latsas (Reference Pallikaris and Latsas2012), Tseng et al. (Reference Tseng, Earle and Guo2012) and Kopacz (Reference Kopacz2018a) in this regard.

Figure 4. The single-heading (pseudoloxodromic) solutions of minimum time starting from different colatitudes on the prolate ellipsoid ($a:=1{\cdot }5$), i.e., $45^{\circ }$ (solid blue), $90^{\circ }$ (solid black) and $120^{\circ }$ (solid red), among other time-minimal paths (dashed colours, respectively), under weak rotational wind given by Equation (24), with $c:=\frac {5}{7}$; $t\leq 1{\cdot }5$. The corresponding time extremals are generated for different initial headings $\varphi (0)=\varphi _{0}$, i.e., with the steps $\Delta \varphi _{0}=30^{\circ }$; $t\leq 5$. The colour-coded surface of the ellipsoid shows wind ‘force’ represented by the norm $|W|_{h}$ (left, side view; right, top (north) view).

Figure 5. The investigation of varying direction $\tilde {\varphi }(t)$ (course over ground) of the tangent vectors $v$ to the single-heading (pseudoloxodromic) solutions of least time shown in Figure 4, respectively (in colours). The horizontal green line represents the direction of the self-velocity $u$, i.e., the heading $\varphi (t)$, which points towards a pole and is constant (up to modulus), i.e., $|\varphi (t)|=\varphi _0=90^{\circ }=\textrm {const.}$ for each considered path; $t\leq 5$.