In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval \([0,\, T]\) and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals \([0, \, T]\) give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form

is of the order \(1/\min \{t^n, (T-t)^n\}\). We also show that a similar result holds for \(\epsilon \)-optimal solutions of the optimal control problems if \(\epsilon >0\) is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

本文针对最优控制问题研究了收费公路现象，该问题同时出现了时间点状态和控制约束。我们假设在目标函数中，出现一个跟踪项，该跟踪项在时间间隔\（[0，\，T] \）上作为积分给出，并测量到所需稳态的距离。在最佳控制问题中，规定了初始和期望的终端状态。如果时间跨度足够长，我们假设该系统在抽象意义上是完全可控的。我们表明，在时间间隔\（[0，\，T] \）上相应的最优控制问题在一定意义上产生了收费区间结构，即对于自然数n，如果T足够大，则目标函数对以下形式的[0，T ]的子间隔的贡献 

的顺序为\（1 / \ min \ {t ^ n，（Tt）^ n \} \）。我们还表明，如果选择（\ epsilon> 0 \）足够小，则\（\ epsilon \）-最优控制问题的最优解具有相似的结果。在本文的最后，我们介绍了两种由常微分方程控制的系统和由偏微分方程控制的系统，在这些系统中可以应用结果。