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Lyapunov Stability-Aware Stackelberg Game for Low-Altitude Economy: A Control-Oriented Pruning-Based DRL Approach

Yue Zhong, Jiawen Kang, Yongju Tong, Hong-Ning Dai, Dong In Kim, Abbas Jamalipour, Shengli Xie

TL;DR

The paper tackles stability-aware resource management in the LAE by coupling control stability with communication latency through Lyapunov theory, enabling explicit latency bounds for UAV-user interactions. It introduces a Stackelberg game where UAVs set prices to satisfy Lyapunov-based stability while users bid for bandwidth, and solves it with a lightweight pruning-based PPO tailored for edge-equipped UAVs. Key contributions include a unified SC^3 closed-loop model, a Lyapunov-to-latency mapping for stability guarantees, a multi-agent Stackelberg formulation with closed-form best responses, and a dynamic structured pruning approach that yields fast convergence with reduced on-board computation. Empirical results demonstrate improved stability and higher system utility under dynamic LAE conditions, with the pruning-based PPO achieving favorable convergence and robustness compared to baselines, making the approach practically deployable on energy-constrained UAV platforms.

Abstract

With the rapid expansion of the low-altitude economy, Unmanned Aerial Vehicles (UAVs) serve as pivotal aerial base stations supporting diverse services from users, ranging from latency-sensitive critical missions to bandwidth-intensive data streaming. However, the efficacy of such heterogeneous networks is often compromised by the conflict between limited onboard resources and stringent stability requirements. Moving beyond traditional throughput-centric designs, we propose a Sensing-Communication-Computing-Control closed-loop framework that explicitly models the impact of communication latency on physical control stability. To guarantee mission reliability, we leverage the Lyapunov stability theory to derive an intrinsic mapping between the state evolution of the control system and communication constraints, transforming abstract stability requirements into quantifiable resource boundaries. Then, we formulate the resource allocation problem as a Stackelberg game, where UAVs (as leaders) dynamically price resources to balance load and ensure stability, while users (as followers) optimize requests based on service urgency. Furthermore, addressing the prohibitive computational overhead of standard Deep Reinforcement Learning (DRL) on energy-constrained edge platforms, we propose a novel and lightweight pruning-based Proximal Policy Optimization (PPO) algorithm. By integrating a dynamic structured pruning mechanism, the proposed algorithm significantly compresses the neural network scale during training, enabling the UAV to rapidly approximate the game equilibrium with minimal inference latency. Simulation results demonstrate that the proposed scheme effectively secures control loop stability while maximizing system utility in dynamic low-altitude environments.

Lyapunov Stability-Aware Stackelberg Game for Low-Altitude Economy: A Control-Oriented Pruning-Based DRL Approach

TL;DR

The paper tackles stability-aware resource management in the LAE by coupling control stability with communication latency through Lyapunov theory, enabling explicit latency bounds for UAV-user interactions. It introduces a Stackelberg game where UAVs set prices to satisfy Lyapunov-based stability while users bid for bandwidth, and solves it with a lightweight pruning-based PPO tailored for edge-equipped UAVs. Key contributions include a unified SC^3 closed-loop model, a Lyapunov-to-latency mapping for stability guarantees, a multi-agent Stackelberg formulation with closed-form best responses, and a dynamic structured pruning approach that yields fast convergence with reduced on-board computation. Empirical results demonstrate improved stability and higher system utility under dynamic LAE conditions, with the pruning-based PPO achieving favorable convergence and robustness compared to baselines, making the approach practically deployable on energy-constrained UAV platforms.

Abstract

With the rapid expansion of the low-altitude economy, Unmanned Aerial Vehicles (UAVs) serve as pivotal aerial base stations supporting diverse services from users, ranging from latency-sensitive critical missions to bandwidth-intensive data streaming. However, the efficacy of such heterogeneous networks is often compromised by the conflict between limited onboard resources and stringent stability requirements. Moving beyond traditional throughput-centric designs, we propose a Sensing-Communication-Computing-Control closed-loop framework that explicitly models the impact of communication latency on physical control stability. To guarantee mission reliability, we leverage the Lyapunov stability theory to derive an intrinsic mapping between the state evolution of the control system and communication constraints, transforming abstract stability requirements into quantifiable resource boundaries. Then, we formulate the resource allocation problem as a Stackelberg game, where UAVs (as leaders) dynamically price resources to balance load and ensure stability, while users (as followers) optimize requests based on service urgency. Furthermore, addressing the prohibitive computational overhead of standard Deep Reinforcement Learning (DRL) on energy-constrained edge platforms, we propose a novel and lightweight pruning-based Proximal Policy Optimization (PPO) algorithm. By integrating a dynamic structured pruning mechanism, the proposed algorithm significantly compresses the neural network scale during training, enabling the UAV to rapidly approximate the game equilibrium with minimal inference latency. Simulation results demonstrate that the proposed scheme effectively secures control loop stability while maximizing system utility in dynamic low-altitude environments.
Paper Structure (24 sections, 1 theorem, 38 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 24 sections, 1 theorem, 38 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Service Provisioning and Mission Planning in LAE
  4. Incentive Mechanism for LAE
  5. Pruning-based DRL Algorithm
  6. System Model and Problem Formulation
  7. Control Model and Latency Model
  8. Control Model
  9. Latency Model
  10. Stackelberg Game Between UAVs and Users
  11. Utilities of UAVs
  12. Utilities of Users
  13. Backward Induction Method
  14. Algorithm for Strategies Design
  15. MAMDP Formulation for the Stackelberg Game
  16. ...and 9 more sections

Key Result

Proposition 1

(Control-to-Communication Mapping): To satisfy the stability condition that is defined in (eq:c1) with a decay rate $\rho_n$, the minimum bandwidth $\kappa_i^{min}$ required for user $i$ is given by where $T_\text{budget}$ is the allowable communication latency margin, and $\text{SNR}_{i,n}=\frac{P_ih_0d_{i,n}^{-\alpha}}{\sigma^2}$ denotes the signal-to-noise ratio (SNR) from user $i$ to UAV $n$r

Figures (6)

  • Figure 1: Illustration of the integrated $\text{SC}^3$-based and Lyapunov stability-aware Stackelberg solved by lightweight pruning-based MADRL for the low-altitude economy framework. The framework illustrates the integration of heterogeneous user demands, the Lyapunov-driven fusion of $\text{SC}^3$ dynamics, and the Stackelberg game formulation, ultimately solved by the dynamic structured pruning-based PPO algorithm.
  • Figure 2: Comparative analysis of network topology and bandwidth allocation strategies under different load conditions. The top row (a)-(d) presents the scenario with $3$ UAVs and $5$ users, while the bottom row (e)-(h) depicts the scenario with $3$ UAVs and $8$ users. The columns illustrate: (a/e) spatial distribution and association; (b/f) correlation between bandwidth, channel quality, and priority; (c/g) aggregate bandwidth usage per UAV; and (d/h) decomposition into stability-required and utility-maximizing bandwidth.
  • Figure 3: Test reward comparison of the proposed pruning-based PPO algorithm with other algorithms, i.e., PPO, greedy, and random algorithms.
  • Figure 4: Test reward comparison of the proposed pruning-based PPO algorithm with other algorithms in different pruning epochs.
  • Figure 5: Stackelberg strategies of UAVs and users are output by the proposed pruning-based PPO algorithm.
  • ...and 1 more figures

Theorems & Definitions (3)

  • proof
  • Proposition 1
  • proof