Stochastic Control Problems Motivated by Sailboat Trajectory Optimization

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 $\sigma > 0$. We formulate two stochastic control 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 $c > 0$, while in the second problem, we assume that $c=0$. 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. 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.