偏微分方程、ResNets及其他问题最优控制中的收费公路
Turnpike in optimal control of PDEs, ResNets, and beyond
摘要：
当代宏观经济学中的\emph{turnpike property}认为，如果一个经济规划者试图将一个经济体从一个资本水平转移到另一个资本水平，那么只要规划者有足够的时间，最有效的路径就是迅速将库存转移到接近最佳静态或恒定路径的水平，然后允许资本沿着这条路径发展，直到接近预期的期限，此时，股票应该被转移到最终目标。在过去的十年中，由于其作为资源分配策略的性质，收费公路性质也被证明适用于力学中出现的几类偏微分方程。当数学形式化时，收费公路理论证实了经济学的见解：对于有限时间范围内的最优控制问题，最优控制和相应的状态在大多数时间内（除了接近初始和最终时间）接近（通常是指数）相关静态最优控制问题的最优控制和相应的状态。特别是，前者大多是随着时间的推移而恒定的。这一事实提供了一个严格的意义，渐近简化，一些最优控制问题似乎享受在很长的时间间隔，允许考虑相应的平稳问题的计算和应用。我们回顾了过去十年中开发的理论的一部分-底层系统的可控性是一个重要的组成部分，甚至可以用来设计简单的类似收费公路的策略，这些策略几乎是最优的-并提出了几个新的应用，包括Hamilton-Jacobi-Bellman渐近的表征，以及通过残差神经网络进行深度学习的稳定性估计。
英文摘要：
The \emph{turnpike property} in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.
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