In this paper, we develop several necessary conditions of turnpike property for generalized linear–quadratic (LQ) optimal control problem in infinite-dimensional setting. The term ‘generalized’ here means that both quadratic and linear terms are considered in the running cost. The turnpike property reflects the fact that over a sufficiently large time horizon, the optimal trajectories and optimal controls stay for most of the time close to a steady state of the system. We show that the turnpike property is strongly connected to certain system theoretical properties of the control system. We provide suitable conditions to characterize the turnpike property in terms of the detectability and stabilizability of the system. Subsequently, we show the equivalence between the exponential turnpike property for generalized LQ and LQ optimal control problems.

在本文中，我们为无限维设置下的广义线性二次（LQ）最优控制问题开发了收费公路特性的几个必要条件。这里的术语“广义”意味着在运行成本中考虑二次项和线性项。收费公路特性反映了这样一个事实：在足够长的时间范围内，最优轨迹和最优控制在大部分时间都保持在接近系统的稳定状态。我们表明，收费公路特性与控制系统的某些系统理论特性密切相关。我们提供合适的条件来表征收费公路在系统可检测性和稳定性方面的特性。随后，我们展示了广义 LQ 和 LQ 最优控制问题的指数收费公路性质之间的等价性。