Stochastic Control Problems Motivated by Sailboat Trajectory Optimization

Abstract

We develop a mathematical model for sailboat navigation that captures the essential features of the problem and can provide insights that might not be available otherwise. 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 $\sigma > 0$. We formulate two stochastic control/optimal switching problems, in which the objective is to reach a circular upwind target of radius $\eta \geq 0$ as quickly as possible. In the first problem, there is a tacking cost $c > 0$, so this is an impulse control problems, while in the second problem, we assume that $c=0$ and singular controls are needed. We establish the viability of both models (assuming that $\eta > 0$ 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. In the second problem, 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: we provide a proof via the Yamada-Watanabe argument.