Stochastic Control Problems Motivated by Sailboat Trajectory Optimization

TL;DR

The paper constructs a stochastic model for upwind sailboat navigation driven by a Brownian wind on a circle and formulates two optimal control problems: an impulse-control variant with tacking costs and a singular-control variant with zero cost. It proves the viability of both problems, deriving finite upper bounds on the associated value functions using a constructive strategy, and proposes a candidate optimal feedback strategy for the zero-cost case. The authors provide rigorous existence and uniqueness arguments for the controlled SDEs, employing Yamada–Watanabe techniques to handle discontinuities and degeneracy, and lay the groundwork for their companion papers which establish optimality and extend the analysis to η=0. Overall, the work offers a mathematically rich framework for stochastic control in sailing that parallels Black–Scholes-style abstraction in finance, with potential extensions to wind dynamics and sailboat specifics. This yields both theoretical insights and practical implications for understanding sailing times under wind variability.

Abstract

We develop a mathematical model for sailboat navigation that can play the same role that the Black and Scholes model plays in mathematical finance: it captures essential features of sailboat navigation, it can provide insights that might not be available otherwise, and it is a source of interesting mathematical problems. In our model, the motion of the sailboat, which would travel at speed $v>0$ in a constant wind, is the solution of a system of two stochastic differential equations driven by a Brownian motion on a circle with speed $σ> 0$. We formulate two stochastic control problems, in which the objective is to reach a circular upwind target of radius $η\geq 0$ as quickly as possible. In the first problem, there is a tacking cost $c > 0$, while in the second problem, we assume that $c=0$. We establish the viability of both models (assuming that $η> 0$ in the second model), that is, their value functions are finite, and we obtain bounds on these value functions related to the parameters of the problem. The first problem falls into the class of impulse control problems, while the second one involves singular controls. In this second case, since the state equation for the optimally controlled motion has discontinuous coefficients and is driven by a degenerate diffusion, standard results on existence and uniqueness of strong solutions do not apply, and we provide a proof via the Yamada-Watanabe argument.

Figures (9)

• Figure 1: The picture on the left-hand side shows the no go zone. The picture on the right-hand side shows a tacking strategy for a boat beating upwind in a constant wind. Another boat starting at the same point and following any other zigzag path parallel to the laylines would travel the same distance up to the target (dotted trajectory).
• Figure 2: Example of a polar diagram for a 20 knot wind speed (on the left-hand side) and our simplified butterfly model (on the right-hand side) where the optimal angle for an upwind (resp. downwind) route is fixed at $45^o$ (resp. $135^o$).
• Figure 3: The picture on the left-hand side uses the geographic reference frame to show the trajectory of a boat starting at position $0$ and sailing on port tack. The wind direction $\beta_t$ in the geographic frame is given by $\beta_t = \gamma_1 \, 1_{[\tau_1, \tau_2[}(t) + \gamma_2\, 1_{[\tau_2, \tau_3[}(t)$, where $0 < \tau_1 < \tau_2 < \tau_3$ and $0 < \gamma_2 < \gamma_1 < \frac{\pi}{4}$. We simplify the notation by using $x_i := x_{\tau_i}$ and $y_i := y_{\tau_i}$, $i=0, 1, 2$. The butterfly-shaped regions correspond to the feasible directions when sailing on port tack (resp. starboard tack). The boat follows the solid line to move from position $0$ to position $1$ between times $0$ and $\tau_1$ while the wind is coming from the North, then, when the wind direction changes to $\beta_{\tau_1} = \gamma_1$, the boat follows the dashed line to move from position $1$ to position $2$ between times $\tau_1$ and $\tau_2$, and when the wind direction changes to $\gamma_2$ (which implies a change of $\gamma_2 - \gamma_1$ from the previous direction), the boat follows the dotted line to move from position $2$ to position $3$ between times $\tau_2$ and $\tau_3$. The picture on the right-hand side shows the corresponding trajectory of the boat in the rotating reference frame attached to the wind direction.
• Figure 4: The sailboat's motion on port tack on the left hand side and on starboard tack on the right hand side. The reference frame rotates with the wind.
• Figure 5: Illustration, in polar coordinates, of the motion of the boat in a constant wind ($\sigma$ set to $0$ in \ref{['eds_proc_controle_polar']}), which corresponds to the flow induced by the drift $(\mu_1(\theta, a), \mu_2(r,\theta, a))$. On the left-hand (respectively right-hand) side, the yacht is on port tack (respectively starboard tack). In our fluctuating wind, in addition to the motion along the flow, the yacht moves as a Brownian motion in a direction parallel to the $\theta$-axis, with a speed that depends on the diffusion coefficient $\sigma$. Note that there is no Brownian component in the radial direction.

Theorems & Definitions (38)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Remark 2.4
• Definition 3.1
• Theorem 3.2
• proof
• Definition 3.3