随着城市空中交通（UAM）系统向高密度运营扩展，管理自主无人机（UAV）流量需要兼具可处理性与安全关键性的控制框架。本文提出了一种基于最优控制理论的多目的地 UAM 网络路由基础，该网络受垂直起降场容量和流量约束。我们首先将网络建模为条件于目的地的连续时间马尔可夫链（CTMC），以捕捉排队、服务和飞行状态之间的随机转移。为确保可处理性，我们采用平均场流体近似，并将底层系统动力学推导为一组耦合常微分方程。本研究的关键贡献在于对无队列状态空间正不变性的形式化证明。我们表明，在特定的轻载条件下，初始无队列的系统将永久保持无队列状态。这一结果使我们能够将复杂的无限维连续时间最优控制问题转化为可处理的有限维代数优化问题。所得框架在确保全网稳定性的同时，联合优化旅行时间和多跳效率。我们通过表征稳态流平衡并为大规模移动系统中的安全、无拥堵运行提供充分条件来验证该方法。

As Urban Air Mobility (UAM) systems scale toward high-density operations, managing autonomous Unmanned Aerial Vehicle (UAV) traffic requires control frameworks that are both tractable and safety-critical. This paper presents a principled optimal control-theoretic foundation for routing in multi-destination UAM networks subject to vertiport capacity and flow constraints. We first model the network as a destination-conditioned Continuous-Time Markov Chain (CTMC) to capture the stochastic transitions between queueing, service, and flight states. To ensure tractability, we employ a mean-field fluid approximation and derive the underlying system dynamics as a set of coupled ordinary differential equations. A key contribution of this work is the formal proof of the positive invariance of the queue-free state space. We demonstrate that under specific underloaded conditions, a system initialized without queues will remain queue-free indefinitely. This result allows us to transform a complex, infinite-dimensional continuous-time optimal control problem into a tractable, finite-dimensional algebraic optimization. The resulting framework jointly optimizes for travel time and multi-hop efficiency while ensuring network-wide stability. We validate the approach by characterizing the steady-state flow equilibria and providing sufficient conditions for safe, congestion-free operation in large-scale mobility systems.