In this paper, we investigate a hybrid dynamical system which incorporates flow swap process, green-time proportion swap process, and flow divergence for a general network with multiple Origin-Destination (OD) pairs and multiple routes, where flow swap process is specified in which traffic swaps from more costly to less costly input links, green-time proportion swap process is specified in which green time at each intersection swaps from less pressurized stages to more pressurized stages, flow may diverge at each intersection from one OD pair to other OD pairs. Unlike the dynamical system model, where bottleneck delays need to be intentionally constructed to yield the equilibrium flow vector and green-time proportion vector, we propose a novel control policy to fill the gap by only adjusting the green-time proportion vector. We derive a sufficient condition for the existence of equilibrium of the dynamical system under the mild constraints that 1) the travel cost function and stage pressure function should be continuous functions and 2) the flow and green-time proportion swap processes project all flow and green-time proportion vectors on the boundary of the feasible region onto itself. We derive the condition of unique equilibrium for fixed green-time proportion vector and show that with varying green-time proportion vector, the set of equilibria is a compact, non-convex set, and with the same partial derivative of travel cost function with respect to the flow and green-time proportion vectors. Finally, we prove the stability of the proposed dynamical system by using Lyapunov stability analysis.

中文翻译：

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使用具有多个 OD 对的路由选择和信号控制的网络容量最大化

在本文中，我们研究了一个混合动力系统，该系统结合了流量交换过程、绿色时间比例交换过程和具有多个起点 - 目的地（OD）对和多条路线的一般网络的流量分流，其中流量交换过程在哪个交通从成本较高的输入链路切换到成本较低的输入链路，指定绿灯时间比例交换过程，其中每个路口的绿灯时间从低压阶段切换到高压阶段，每个路口的流量可能从一个 OD 对到另一个 OD对。与需要有意构建瓶颈延迟以产生平衡流向量和绿灯时间比例向量的动态系统模型不同，我们提出了一种新的控制策略，仅通过调整绿灯时间比例向量来填补空白。我们推导出了在温和约束下动力系统平衡存在的充分条件：1) 行程成本函数和阶段压力函数应该是连续函数 2) 流量和绿灯时间比例交换过程投影所有流量和绿灯- 可行区域边界上的时间比例向量到自身。我们推导出固定绿色时间比例向量的唯一均衡条件，并表明随着绿色时间比例向量的变化，均衡集是一个紧凑的非凸集，并且具有相同的旅行成本函数的偏导数到流量和绿灯时间比例向量。最后，我们通过使用李雅普诺夫稳定性分析证明了所提出的动力系统的稳定性。