Links in practical systems, such as satellite--terrestrial integrated networks, exhibit distinct delay distributions, intermittent availability, and heterogeneous energy costs. These characteristics pose significant challenges to maintaining timely and energy-efficient status updates. While link availability restricts feasible transmission routes, routing decisions determine the actual delay and energy expenditure. This paper tackles these challenges by jointly optimizing sampling and routing decisions to minimize monotonic, non-linear Age of Information (AoI). The proposed formulation incorporates key system features, including multiple routes with correlated random delays, stochastic link availability, and route-dependent energy consumption. We model the problem as an infinite-horizon Constrained Semi-Markov Decision Process (CSMDP) with a hybrid state--action space and develop an efficient nested algorithm, termed Bisec-\textsc{ReaVI}, to solve this problem. We analyze the structural properties of the solution and reveal a well-defined jointly optimal policy structure: (i) For general monotonic penalty functions, the optimal sampling policy is a piecewise linear waiting policy with at most $N$ breakpoints given $N$ routes; and (ii) under a derived Expected Penalty Ordering condition, the optimal routing policy is a monotonic threshold-based handover policy characterized by at most $\binom{N}{2}$ thresholds. Numerical experiments in a \textit{satellite--terrestrial} integrated routing scenario demonstrate that the proposed scheme efficiently balances energy usage and information freshness, and reveal a counter-intuitive insight: \textit{even routes with higher average delay, higher delay variance or lower availability can still play a critical role in minimizing monotonic functions of AoI}.

翻译：实际系统（如星地融合网络）中的链路具有独特的延迟分布、间歇可用性以及异构的能量成本。这些特性对维持及时且能量高效的状态更新构成了显著挑战。链路可用性限制了可行的传输路径，而路由决策则决定了实际的延迟与能量消耗。本文通过联合优化采样与路由决策，以最小化单调非线性信息年龄（AoI），应对上述挑战。所提出的建模框架融合了关键系统特征，包括具有相关随机延迟的多条路径、随机链路可用性以及路径依赖的能量消耗。我们将该问题建模为一个具有混合状态-动作空间的无限时域约束半马尔可夫决策过程（CSMDP），并开发了一种高效的嵌套算法（称为Bisec-\textsc{ReaVI}）来求解。我们分析了解决方案的结构特性，揭示了一个定义清晰的联合最优策略结构：（i）对于一般的单调惩罚函数，给定$N$条路径，最优采样策略是一个至多具有$N$个断点的分段线性等待策略；（ii）在推导出的期望惩罚排序条件下，最优路由策略是一个单调的基于阈值的切换策略，其特征由至多$\binom{N}{2}$个阈值刻画。在\textit{星地融合}路由场景中的数值实验表明，所提方案能有效平衡能量使用与信息新鲜度，并揭示了一个反直觉的见解：\textit{即使平均延迟更高、延迟方差更大或可用性更低的路径，在最小化AoI的单调函数方面仍可发挥关键作用}。