Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

💡 Research Summary

The paper introduces a novel framework for approximating the joint spectral radius (JSR) of a finite set of matrices by exploiting multiple Lyapunov functions organized through a labeled directed graph, termed a “path‑complete graph.” The authors begin by recalling that the JSR ρ(A) characterizes the maximal asymptotic growth rate of arbitrary products of matrices from a set A={A₁,…,A_m} and that ρ(A)<1 is equivalent to absolute asymptotic stability (AAS) of the switched linear system x_{k+1}=A_{σ(k)}x_k for all switching sequences σ. Classical approaches rely on a single common Lyapunov function (often quadratic) to certify stability, leading to semidefinite programs (SDPs) that provide upper bounds on ρ(A). However, such methods are conservative: the quadratic common Lyapunov bound is within a factor √n of the true JSR, and even sum‑of‑squares (SOS) polynomial Lyapunov functions, while tighter, incur rapidly increasing computational cost.

To overcome these limitations, the authors propose representing Lyapunov inequalities as edges of a directed graph. Each node i carries a continuous, positive‑definite, homogeneous Lyapunov function V_i(x). An edge from node i to node j labeled by a finite word ℓ∈A^* (i.e., a product of matrices) encodes the inequality V_j(ℓx) ≤ V_i(x) for all x. A graph is called path‑complete if every possible finite word over the alphabet A appears as a label of some path in the graph. The central result (Theorem 2.4) shows that if a path‑complete graph admits a feasible set of Lyapunov functions satisfying all edge inequalities with a scaling factor γ>0, then ρ(A) ≤ γ. Consequently, any path‑complete graph yields a valid SDP‑based upper bound on the JSR.

The framework subsumes many existing techniques as special cases. A single‑node graph reproduces the common quadratic (or common SOS) method; a graph whose nodes correspond to a collection of quadratic forms and whose edges are labeled by single matrices captures max‑of‑quadratics or min‑of‑quadratics Lyapunov functions; more elaborate graphs encode path‑dependent quadratic Lyapunov functions studied in earlier works. By varying the graph topology, one obtains a hierarchy of increasingly tight bounds: richer graphs (more nodes, more edges) can represent more intricate relationships among the Lyapunov functions and thus reduce conservatism.

The authors conduct a systematic study of graph families. They analyze De Bruijn graphs and their duals, showing that these graphs provide approximation guarantees that improve the classical √n factor to a constant independent of the system dimension. For the case of two matrices (m=2) and two nodes, they enumerate all possible directed labeled graphs, classify them according to a partial order reflecting their relative performance, and identify the optimal configurations (Proposition 4.2). They also prove a duality theorem (Theorem 5.1) stating that transposing all matrices in A corresponds to swapping a graph with its dual, preserving the JSR bound.

A constructive converse theorem (Theorem 6.1) is established for max‑of‑quadratics Lyapunov functions: given any desired relative error ε, there exists a worst‑case bound on the number of quadratic functions required (on the order of (1/ε)^n) to achieve an ε‑approximation of the JSR. This result demonstrates that the max‑of‑quadratics approach is not only expressive but also theoretically complete. Additionally, Theorem 6.2 introduces a new class of LMIs that involve matrix products of varying lengths, providing further flexibility and tighter bounds.

Numerical experiments illustrate the practical impact. The authors apply their graph‑based SDP relaxations to three disparate problems: (i) the growth rate of overlap‑free words, (ii) the Euler ternary partition function, and (iii) continuity properties of wavelet functions. In each case, the path‑complete graph methods outperform traditional common quadratic or SOS relaxations, achieving faster computation times and tighter upper bounds on the JSR.

In conclusion, the paper delivers a unifying, graph‑theoretic perspective on Lyapunov‑based JSR approximation. By translating Lyapunov inequalities into the language of automata and path‑completeness, it creates a systematic way to generate, analyze, and compare a wide spectrum of SDP relaxations. The results bridge gaps between previously disparate methods, provide rigorous approximation guarantees, and open avenues for future work such as automated graph synthesis, extensions to continuous‑time or nonlinear switched systems, and distributed SDP implementations for large‑scale applications.