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With the development of space-air-ground integrated networks, Low Earth Orbit (LEO) satellite networks are envisioned to play a crucial role in providing data transmission services in the 6G era. However, the increasing number of connected devices leads to a surge in data volume and bursty traffic patterns. Ensuring the communication stability of LEO networks has thus become essential. While Lyapunov optimization has been applied to network optimization for decades and can guarantee stability when traffic rates remain within the capacity region, its applicability in LEO satellite networks is limited due to the bursty and dynamic nature of LEO network traffic. To address this issue, we propose a robust Lyapunov optimization method to ensure stability in LEO satellite networks. We theoretically show that for a stabilizable network system, traffic rates do not have to always stay within the capacity region at every time slot. Instead, the network can accommodate temporary capacity region violations, while ensuring the long-term network stability. Extensive simulations under various traffic conditions validate the effectiveness of the robust Lyapunov optimization method, demonstrating that LEO satellite networks can maintain stability under finite violations of the capacity region.