Optimal solution and asymptotic properties of a stochastic control problem arising in sailboat trajectory optimization

We study the optimal strategy for a sailboat to reach an upwind island under the hypothesis that the wind direction fluctuates according to a Brownian motion and the wind speed is constant. The work is motivated by a concrete problem which typically arises during sailing regattas, namely finding the best tacking strategy to reach the upwind buoy as quickly as possible. We assume that there is no loss of time when tacking. We first guess an optimal strategy and then we establish its optimality by using the dynamic programming principle. The Hamilton Jacobi Bellmann equation obtained is a parabolic PDE with Neumann boundary conditions. Since it does not admit a closed form solution, the proof of optimality involves an intricate estimate of derivatives of the value function. We explicitly provide the asymptotic shape of the value function. In order to do so, we prove a result on large time behavior for solutions to time dependent parabolic PDE using a coupling argument. In particular, a boat far from the island approaches the island at $\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%$ of the boat's speed.

Analysis and numerical simulation of viscoelastic flows

Andrea Bonito

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.

EPFL2006

Stochastic Control and Free Boundary Problems for Sailboat Trajectory Optimization

Laura Vinckenbosch

The topic of this thesis is the study of several stochastic control problems motivated by sailing races. The goal is to minimize the travel time between two locations, by selecting the fastest route in face of randomly changing weather conditions, such as wind direction. When a sailboat is travelling upwind, the key is to decide when to tack. Since this maneuver slows down the yacht, it is natural to model this time lost by a "tacking penalty" which places the problem in the context of optimal stochastic control problems with switching costs. An objective of this work is to propose and to study mathematical models that capture some of the features of a sailing race, but which remain amenable to an explicit rigorous solution that can be proved to be optimal. We consider three different models in which the wind direction is described by a stochastic process. In the first model, we consider a wind that changes randomly only once. In the second model, the wind oscillates between two possible directions according to a continuous-time Markov chain. We exhibit a free boundary problem for the value function involving hyperbolic partial differential equations of Klein-Gordon type. The last model considers the wind direction as a Brownian motion. We prove the existence of a finite value function and exhibit a free boundary problem involving parabolic partial differential equations with non-constant coefficients. In these three models, the optimal solution consists of a partition of the state space into a region where it is optimal to tack immediately and a region where it is optimal to continue on the current tack. The boundaries between these regions are given by one or more "switching curves" and in the cases where we have been able to exhibit them, the optimality of the solution is established by a verification theorem based on the martingale method. We also solve two other control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle by controlling its drift and subject to a switching penalty. In each problem, the value function is written as the solution of a second order ordinary differential equations problem whose unknown boundaries are found by applying the principle of smooth fit. For both problems, we exhibit a candidate strategy as a function of the switching cost and we prove its optimality as well as its generic uniqueness.

EPFL2012

Stochastic Control and Free Boundary Problems for Sailboat Trajectory Optimization

Laura Vinckenbosch

EPFL2012