Lyapunov stability of the Euler method

We extend the Lyapunov stability criterion to Euler discretizations of differential inclusions. It relies on a pair of Lyapunov functions, one in continuous time and one in discrete time. In the context of optimization, this yields sufficient conditions for the stability of nonisolated local minima when using the Bouligand subgradient method.

💡 Research Summary

The paper “Lyapunov stability of the Euler method” addresses a gap in the stability theory of numerical discretizations for differential inclusions. Classical Lyapunov stability, originally formulated for ordinary differential equations, guarantees that trajectories starting sufficiently close to an equilibrium remain within a prescribed neighborhood for all future time. Extending this notion to set‑valued dynamics (\dot x\in F(x)) is straightforward, but doing so for the explicit Euler scheme
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