## 1 Introduction  
Satellite optical networks (SONs) constitute critical infrastructure for global connectivity in the globalization era [1, 2], addressing escalating demands through large-scale deployment. Unlike bandwidth-limited microwave-based systems, SONs leverage laser communication for high-capacity, low-latency transmission with inherent atmospheric interference resistance [3, 4]. A pivotal transformation is underway as edge artificial intelligence (AI) evolves from terrestrial applications to space-based deployments, fundamentally redefining SON operational paradigms [5]. This transition is driven by two critical factors. First, mega-constellations are increasingly leveraging machine learning. They rely on it for dynamic resource allocation and interference management [6]. Second, multi-access edge computing platforms have been deployed on orbital nodes. These platforms enable the operation of latency-sensitive applications. Such applications include real-time Earth observation and distributed learning frameworks [7, 8]. This computational migration introduces profound energy challenges. Satellite systems operate within stringent power constraints, alternating between solar-charging phases and battery-dependent eclipse periods [9]. Edge-AI workloads, and particularly compute-intensive operations like adaptive beamforming and channel prediction, generate stochastic power demands. These power demands induce deep discharge cycles during eclipse operations [10]. Frequent high-depth discharges accelerate lithium-ion battery degradation through irreversible electrochemical mechanisms, including cathode lattice destabilization and solid-electrolyte interphase growth [11]. The number of charge and discharge cycles of a battery is limited and is referred to as the cycle life. As the cycle life is consumed, the reliability of the power supply diminishes progressively. Once the reliability of a battery falls below a certain threshold, the satellite fails to operate correctly, thus jeopardizing the stability of the network. Therefore, minimizing the consumption of battery cycle life is crucial for extending the operational lifespan of satellite networks [12; 13; 14; 15; 16]. Consequently, exploring the balance between communication efficiency and energy efficiency for sustainable edge AI in SONs represents a significant scientific challenge. Current energy optimization approaches exhibit fundamental limitations in addressing edge AI workloads. Device-level schemes successfully reduce computation energy but disregard LISL power dynamics during AI data exchange [17; 18; 19]. Network-level routing protocols effectively balance battery usage yet remain oblivious to task criticality distinctions--failing to differentiate between latency-sensitive operations like disaster response and delay-tolerant activities like model training [20; 21; 22; 23; 24]. Topology control mechanisms lack computation-aware dormancy strategies, deactivating links without consideration for AI workflow phases. This landscape reveals a critical research gap: the absence of integrated solutions that jointly optimize computation offloading, DoD-resilient routing, and battery-aging-aware topology control specifically for edge AI sustainability. To address these challenges, we propose green task-aware routing planning (GreenRP), a task-aware routing plan designed to enable sustainable edge AI through synergistic mechanisms. The framework dynamically offloads computation-intensive tasks from energy-constrained satellites to resource-rich neighbors, strategically redistributing workloads. A classic validated battery aging model is also employed to enhance the credibility of the proposed scheme, given its capability to accurately capture the dynamic degradation characteristics of satellite batteries under varying operational conditions [21].