Iterative graph lifting for automatic design of path-complete stability certificates

Léa Ninite; Raphaël M. Jungers

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arXiv · 2607.00637

Iterative graph lifting for automatic design of path-complete stability certificates

Léa Ninite·Raphaël M. Jungers

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Abstract

Stability of switched linear systems under arbitrary switching is a fundamental problem in control theory, closely related to the joint spectral radius (JSR), which characterizes the worst-case growth rate of system trajectories. In this paper, we contribute to the path-complete approach for approximating the JSR. This framework constructs algebraic stability certificates using labeled directed graphs, known as path-complete graphs. These certificates can be computed via an associated optimization problem. We propose an iterative algorithm that refines path-complete graphs in an efficient and parsimonious manner. The algorithm relies on a graph-theoretic analysis of the optimality conditions of the underlying optimization problem. In particular, we derive a sufficient condition under which the exact JSR is attained by a given path-complete graph. When this condition is not satisfied, we identify bottleneck nodes by analyzing the graph induced by the active constraints. We then use this information to refine the path-complete graph via local graph lifting (node splitting), and repeat the procedure. Numerical experiments demonstrate the effectiveness and scalability of the proposed approach, outperforming state-of-the-art methods on all challenging instances tested.

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BibTeX RIS

Léa Ninite, Raphaël M. Jungers. 2026-09-10. Iterative graph lifting for automatic design of path-complete stability certificates. https://arxiv.org/abs/2607.00637

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