Lyapunov–Barrier Duality: A Structural Perspective

Sasaki, Seigo

doi:10.51094/jxiv.2002

##article.authors##

Sasaki, Seigo Department of Electrical and Electronic Engineering, National Defense Academy of Japan https://orcid.org/0000-0002-7596-5291

DOI:

https://doi.org/10.51094/jxiv.2002

キーワード:

Lyapunov function、 barrier function、 contraction theory、 region-based stability、 nonlinear control、 Hamilton-Jacobi-Bellman (HJB)、 reinforcement learning

抄録

This article proposes a unified region-based paradigm that reconnects historically independent developments in nonlinear control, optimal control, and safety-critical systems. We began with Lyapunov's original insight that stability is fundamentally a property of regions and their deformation under flow, we trace a conceptual lineage through Zubov's characterization (1950s), the variable-gradient method (1962), extended quadratic Lyapunov functions (1997-1998), contraction theory (2014-), and recent advances in safe reinforcement learning and Hamilton-Jacobi-Bellman methods (2020-). We argue that contemporary challenges in learning-based and safety-critical control stem from treating stability as a pointwise property rather than a geometric one. A unified region-based view clarifies the complementary roles of Lyapunov functions (inner contraction), barrier functions (outer invariance), and value functions (performance landscapes within feasible domains). It also highlights constructive methods that jointly design gradients and level sets. By reconstructing these connections, we provide a coherent framework that situates modern tools within Lyapunov's original geometric philosophy—not as a new theory, but as a synthesis for interpreting and integrating existing methods.

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The author declares no conflicts of interest.

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