Traffic conflict points (e.g., intersections, work-zones) cause travel delay, stop-and-go traffic, and excessive energy consumption. Efforts have been taken to improve traffic conflict point performance via trajectory control of connected automated vehicles (CAV) as the CAV technology emerges. One major challenge to these efforts is the complexity in optimization of CAV trajectories, particularly with joint signal timing optimization. This challenge poses barriers to real-time application requirements, scaling them up to address network level problems and drawing analytical insights into problem structures. To overcome this challenge, this paper aims to seek for an efficient and analytical solution to a joint vehicle trajectory and signal timing optimization problem. This problem simultaneously optimizes CAV trajectories and signal timing to minimize travel delay and energy consumption at a conflicting point with two traffic approaches. This study modifies the original complex formulation in two ways. First, the vehicle trajectory shape is simplified into a piece-wise quadratic function with no more than five segments. Second, instead of using the highly non-linear instantaneous fuel consumption function, a simplified macroscopic measure is proposed to approximate fuel consumption as an analytical quadratic function of signal red interval. These simplifications provide elegant theoretical properties that enable solving an analytical exact solution to this complex problem with parsimonious analytical insights. Numerical examples reveal that the proposed model can significantly reduce travel delay and fuel consumption. Moreover, it is demonstrated that the presented algorithm is highly efficient and appropriate for real-world traffic applications.

交通冲突点（例如交叉路口，工作区）会导致旅行延误，走走停停的交通以及过多的能源消耗。随着CAV技术的出现，人们已经通过连接自动车（CAV）的轨迹控制来改善交通冲突点性能。这些努力的一个主要挑战是CAV轨迹优化的复杂性，尤其是联合信号定时优化。这一挑战对实时应用程序要求构成了障碍，将其扩展以解决网络级问题并将分析见解吸引到问题结构中。为了克服这一挑战，本文旨在寻求一种有效的分析解决方案来解决联合的车辆轨迹和信号定时优化问题。此问题同时优化了CAV轨迹和信号定时，以在两种交通方式的冲突点将行驶延迟和能耗降至最低。这项研究以两种方式修改了原始的复杂配方。首先，将车辆的轨迹形状简化为不超过五个线段的分段二次函数。其次，代替使用高度非线性的瞬时燃料消耗函数，提出了一种简化的宏观测量方法，将燃料消耗近似为信号红色间隔的分析二次函数。这些简化提供了优雅的理论特性，使您能够利用简约的分析见解来解决这一复杂问题的精确解析解决方案。数值算例表明，提出的模型可以显着减少旅行延误和燃油消耗。此外，证明了所提出的算法是高效的并且适合于现实世界中的交通应用。