Stochastic Control and Free Boundary Problems for Sailboat Trajectory Optimization
Stochastic Control and Free Boundary Problems for Sailboat Trajectory Optimization
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帆船轨迹优化的随机控制和自由边界问题
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发表时间：
2012
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通讯作者：
Laura Vinckenbosch
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作者：
Laura Vinckenbosch
本论文的主题是研究帆船比赛引发的几个随机控制问题。其目标是在风向等随机变化的天气条件下选择最快的路线，从而最大限度地减少两个地点之间的旅行时间。当帆船逆风航行时，关键是决定何时开航。由于这种操作会减慢游艇的速度，因此很自然地会对这次失败的情况进行建模，将问题置于具有切换成本的最优随机控制问题的背景下。这项工作的一个目标是提出和研究数学模型，这些模型捕捉了帆船比赛的一些特征，但仍然服从于可以被证明是最优的显式严格解。我们考虑了三种不同的模型，在这些模型中，风向是用随机过程来描述的。在第一个模型中，我们考虑的是只随机变化一次的风。在第二个模型中，风按照连续时间马尔科夫链在两个可能的方向之间振荡。研究了Klein-Gordon型双曲型偏微分方程值函数的自由边界问题。最后一个模型将风向视为布朗运动。我们证明了一个有限值函数的存在性，并展示了一个涉及变系数抛物型偏微分方程解的自由边界问题。在这三个模型中，最优解包括将状态空间划分为立即调整最优的区域和继续当前调整最优的区域。这些区域之间的边界是由一条或多条“切换曲线”给出的，在我们能够展示它们的情况下，解的最优性是由基于鞅方法的验证定理建立的。我们还解决了另外两个控制问题，其中玩家试图通过控制布朗粒子的漂移并受到切换惩罚来最小化或最大化离开布朗粒子的区间的退出时间。在每个问题中，值函数被写成一个二阶常微分方程组问题的解，该问题的未知边界通过光滑拟合原理得到。对于这两个问题，我们给出了一个候选策略作为切换成本的函数，并证明了它的最优性以及它的一般唯一性。
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.