The 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 insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the 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.

收费公路物业在当代宏观经济学中断言，如果经济计划者试图将经济从一个资本水平转移到另一个资本水平，那么只要计划者有足够的时间，最有效的路径就是迅速将存量移动到接近最优静止的水平或恒定路径，然后允许资本沿着该路径发展，直到接近预期的期限，此时股票应该移动到最终目标。在过去的十年中，部分受其作为资源分配策略的性质的推动，收费公路的特性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，最优控制和相应状态在大多数时间（通常是指数级的）接近于（通常是指数级的），除了接近初始和最终时间之外，最优控制和相关稳态最优控制问题的相应状态。特别是，前者随着时间的推移大部分是恒定的。这一事实为渐近简化提供了严格的含义，一些最优控制问题似乎在很长的时间间隔内享有，从而允许考虑相应的计算和应用的平稳问题。我们回顾了过去十年中发展起来的部分理论——底层系统的可控性是一个重要因素，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——并提出了一些新的应用，包括好多其它的，