Review on computational methods for Lyapunov functions

By Peter Giesl, Sigurdur Freyr Hafstein. 2015

275 citations recorded by the source corpus.

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expan- sion, linear programming, linear matrix inequalities, collocation methods, al- gebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.

Subfields: Control and Systems Engineering

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Citations in the corpus

Safe Model-based Reinforcement Learning with Stability Guarantees (2017, background): “Moreover, it has recently been shown that it is possible to compute suitable Lyapunov functions [31, 32].”
Safe Control With Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction Methods for Robotics and Control (2022, methodology): “Perhaps the most well known of these certificate functions are Lyapunov functions: pseudo-energy functions that, if shown to be strictly decreasing along trajectories of the system, prove the stability of the system about a fixed point [1].”