Scaled Relative Graphs in Normed Spaces

arXiv:2604.02407 · math.OC, cs.SY, eess.SY · Submitted 2026-04-02 · Read on arXiv

math.OC, cs.SY, eess.SY

Submitted: 2026-04-02

Updated: 2026-09-21

Comments: To appear in the Proceedings of the 65th IEEE Conference on Decision and Control (CDC 2026)

Project page: https://fbullo.github.io/ctds

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

The gist: The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to normed spaces.

Terminology

Hilbert Space — 10× in this paper · explained in 2 episodes
Scaled Relative Graph — 10× in this paper
Homogeneity — 4× in this paper · explained in 2 episodes
Additivity — 1× in this paper · explained in 1 episode
Alignment — 1× in this paper · explained in 1 episode
Lyapunov function — 1× in this paper · explained in 2 episodes
Markov Decision Process — 1× in this paper
Regularization — 1× in this paper · explained in 3 episodes
Reinforcement Learning — 1× in this paper · explained in 4 episodes
Reward Function — 1× in this paper · explained in 2 episodes
Sample Complexity — 1× in this paper · explained in 15 episodes

Abstract

The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to normed spaces. Our extension replaces the inner product with a regular pairing, whose asymmetry gives rise to directional angles and, in turn, directional SRGs. Directional SRGs are shown to provide geometric containment tests certifying key operator properties, including contraction and monotonicity. Calculus rules for SRGs under scaling, inversion, addition, and composition are also derived. The theory is illustrated by numerical examples, including a graphical analysis of the contractivity of Bellman operators using directional SRGs.

Scaled Relative Graphs in Normed Spaces

Terminology

Abstract

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