Machine Learning for Lyapunov Function Synthesis: A Comprehensive Review

Abstract

Lyapunov functions underpin the stability analysis and controller synthesis for aerospace vehicles, from spacecraft attitude control to autonomous flight systems and UAV swarm coordination. This review discusses the development of computational Lyapunov function synthesis methods. Within this scope, it traces the evolution from traditional Sum-of-Squares (SOS) programming to artificial intelligence symbolic discovery technologies, prompted by the increasing complexity of nonlinear systems. Specifically, the following stages can be identified: computational relaxation methods constrained by polynomial templates; data-driven paradigms leveraging neural network-based empirical approximations; and the fusion of machine learning fitting and formal verification through the Counter-Example Guided Inductive Synthesis (CEGIS) framework. Subsequently, researchers developed construction-based network architectures and hybrid scalable verification mechanisms to address bottlenecks in computational verification. Interestingly, the use of generative models and reinforcement learning to explore analytically expressible expressions with physical interpretability is emerging as an active research area in this field. To date, these tools have been extended to various dynamic scenarios, such as stochastic systems, decentralized multi-agent topologies (e.g., unmanned aerial vehicle swarms and satellite constellations), safe reinforcement learning for autonomous flight control, and partial differential equation (PDE) solving. This review evaluates the aforementioned methodological stages, with a focus on analyzing the trade-offs among model expressivity, computational scalability, and the rigor of formal guarantees across different technical routes.