An entropy-based bound for the computational complexity of a switched system

Beno\^it Legat; Pablo A. Parrilo; Rapha\"el M. Jungers

arxiv: 1907.00655 · v1 · pith:WLJMV7FXnew · submitted 2019-07-01 · 🧮 math.OC

An entropy-based bound for the computational complexity of a switched system

Beno\^it Legat , Pablo A. Parrilo , Rapha\"el M. Jungers This is my paper

Pith reviewed 2026-05-25 12:16 UTC · model grok-4.3

classification 🧮 math.OC

keywords joint spectral radiusswitched systemssum of squaresp-radiusconstrained switchingLyapunov functionsstability analysisentropy of languages

0 comments

The pith

The gap between the sum-of-squares upper bound on the joint spectral radius and the true value is controlled by the p-radius and the entropy of the allowed switching sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for switched systems with constraints on allowed switching sequences, the popular sum-of-squares method for finding Lyapunov functions yields an upper bound on the joint spectral radius whose distance to the true value can be bounded explicitly. The bound uses two quantities: the p-radius, which measures growth under a weighted norm, and the entropy of the language of permitted switches, which captures how many sequences are allowed. This guarantee matters because it turns an otherwise heuristic approximation into one with a computable error term. The authors also give a reduction that converts the joint spectral radius problem for low-rank matrices into a constrained joint spectral radius problem on matrices of lower dimension.

Core claim

For constrained switched systems the sum-of-squares relaxation supplies an upper bound on the joint spectral radius whose accuracy is guaranteed by the p-radius of the system together with the entropy of the language of allowed switching sequences; the same framework yields a dimension-reduction procedure that converts the joint spectral radius of low-rank matrices into the constrained joint spectral radius of matrices of strictly smaller size.

What carries the argument

The p-radius combined with the entropy of the language of allowed switching sequences, which together quantify the gap between the sum-of-squares upper bound and the true joint spectral radius.

If this is right

The sum-of-squares method supplies a provably accurate approximation whose error shrinks when the entropy of allowed switches is small.
Stability certificates obtained via sum of squares can be accompanied by an explicit numerical guarantee on their conservatism.
The joint spectral radius of any set of low-rank matrices can be recovered from the constrained joint spectral radius of a smaller set obtained by the reduction.
Constrained joint spectral radius computations become a primitive that solves a wider class of unconstrained problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the entropy can be computed or bounded for common constraint languages, the method yields practical a-priori error estimates without solving the joint spectral radius exactly.
The reduction step suggests that rank-deficient matrix sets arising in applications can be handled by first projecting onto a lower-dimensional constrained problem.
The same entropy-based accounting might extend to other convex relaxations beyond sum of squares, provided an analogous radius quantity exists.

Load-bearing premise

The p-radius and the entropy of the allowed switching language are treated as known or computable quantities that can be used to quantify the gap.

What would settle it

A concrete constrained switched system for which the observed gap between the sum-of-squares upper bound and the true joint spectral radius exceeds the value predicted from its p-radius and switching-language entropy.

Figures

Figures reproduced from arXiv: 1907.00655 by Beno\^it Legat, Pablo A. Parrilo, Rapha\"el M. Jungers.

**Figure 2.** Figure 2: Representation of the solutions to Program 1 with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Result of Example 4 for d = 1, 2, 3, 4, 5, 6; the value of d is given in the horizontal axis. The exact value of the CJSR found in [30] is represented by the horizontal line. The upper and lower bounds given by Lemma 2 using the 2d-radius are denoted “2d-radius”. The upper bound found by Program 1 with polynomials of degree 2d is denoted “Upper Bound”. From this upper bound, three lower bounds can be obtai… view at source ↗

**Figure 4.** Figure 4: Simple example of the low rank reduction. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the p-radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies an entropy-and-p-radius bound on the SOS relaxation gap for constrained JSR plus a reduction of low-rank JSR to a lower-dimensional constrained problem; the reduction is the more immediately usable piece.

read the letter

The main new elements are the stated guarantee that ties the gap between the sum-of-squares upper bound and the true constrained joint spectral radius to the p-radius and the entropy of the allowed switching language, and the reduction that converts JSR computation for low-rank matrices into a constrained JSR problem on matrices of smaller dimension. The reduction looks like the part that could see direct use; if the algebra checks out, it gives a concrete way to handle certain low-rank instances by moving to a different but lower-dimensional constrained setting. The guarantee itself is a quantitative link that had not been written down in exactly this form, so it adds a modest but specific piece to the existing SOS literature on switched systems. The paper does a reasonable job of situating both results against prior work on JSR and convex relaxations. The soft spot is the practical reach of the guarantee. It is only as useful as the p-radius and the language entropy are accessible. If those two quantities turn out to require effort comparable to the original JSR problem, the bound stays non-constructive for most applications. The abstract leaves that question open, and the stress-test note is right to flag it; the body needs to show either an independent way to obtain or bound them or to demonstrate that the guarantee remains informative even when they are not cheap. The low-rank reduction does not appear to inherit the same issue, which is why it stands out. This paper is for readers already working inside hybrid control and JSR approximation methods. Someone outside that subfield will not find a broad new technique or a reshaping of the general problem. A serious referee should see it because the claims are narrow enough to be checked directly and the reduction is the sort of technical step that can be verified or refuted on its own terms. I would send it out rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes the accuracy of sum-of-squares (SOS) Lyapunov relaxations for upper-bounding the constrained joint spectral radius (JSR) of switched systems. It claims a new a-priori guarantee on the gap between the SOS upper bound and the true constrained JSR, expressed in terms of the p-radius of the matrix set and the topological entropy of the admissible switching language. The paper also presents a reduction showing that the (unconstrained) JSR of a set of low-rank matrices can be recovered from the constrained JSR of a related set of matrices of strictly smaller dimension.

Significance. If the central guarantee is correct and the p-radius together with the switching entropy can be obtained independently, the result supplies a concrete, non-asymptotic error bound for a widely used convex relaxation; this would be a useful theoretical tool for stability certification of constrained hybrid systems. The reduction for low-rank matrices is a self-contained algorithmic contribution that may lower the effective dimension of certain JSR instances. Both contributions are stated at a level of generality that could apply beyond the SOS setting.

major comments (2)

[Abstract] Abstract, paragraph 3: the claimed guarantee is expressed solely in terms of the p-radius and the entropy of the admissible language, yet the manuscript supplies neither an algorithm nor a reduction showing that either quantity is computable by a procedure whose complexity is strictly lower than that of the original constrained-JSR problem; without such a demonstration the bound remains non-constructive and does not tighten the practical analysis of SOS relaxations.
[Final section] The reduction for low-rank matrices (final section) is stated without an explicit statement of the dimension reduction factor or the precise relation between the original and the auxiliary constrained system; this relation is load-bearing for the claim that the method reduces computational complexity.

minor comments (1)

Notation for the constrained JSR and the admissible language is introduced without a dedicated preliminary section, making it difficult to track the precise dependence on the switching constraint throughout the derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 3: the claimed guarantee is expressed solely in terms of the p-radius and the entropy of the admissible language, yet the manuscript supplies neither an algorithm nor a reduction showing that either quantity is computable by a procedure whose complexity is strictly lower than that of the original constrained-JSR problem; without such a demonstration the bound remains non-constructive and does not tighten the practical analysis of SOS relaxations.

Authors: The central contribution is an explicit a-priori bound on the gap between the SOS upper bound and the true constrained JSR, expressed in terms of the p-radius and the topological entropy of the admissible language. We do not claim or demonstrate that these two quantities are always computable by a strictly cheaper procedure; the bound is therefore theoretical rather than a fully constructive certificate. In many structured cases (regular languages, known entropy bounds from automata theory, or easily computable p-radius upper bounds) the quantities can be obtained independently of the full JSR. We will revise the abstract and the opening of Section 1 to state this limitation explicitly and to clarify that the result quantifies the accuracy of SOS relaxations when the auxiliary quantities are known or bounded, rather than supplying a new computational shortcut. revision: partial
Referee: [Final section] The reduction for low-rank matrices (final section) is stated without an explicit statement of the dimension reduction factor or the precise relation between the original and the auxiliary constrained system; this relation is load-bearing for the claim that the method reduces computational complexity.

Authors: We agree that the final section would benefit from greater precision. The reduction shows that the (unconstrained) JSR of a set of matrices of maximum rank r can be recovered exactly from the constrained JSR of an auxiliary system whose matrices have dimension strictly smaller than the original (the precise factor is linear in r). In the revised manuscript we will add an explicit theorem stating the dimension-reduction factor and the exact algebraic relation between the two JSR values, thereby making the complexity claim rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent standard quantities

full rationale

The paper derives an upper-bound guarantee for the SOS relaxation of the constrained JSR expressed in terms of the p-radius and the topological entropy of the allowed switching language; both quantities are standard, independently defined objects in the switched-systems literature and are not shown to be computed from the same optimization as the target JSR. The final section supplies an explicit reduction from low-rank unconstrained JSR to constrained JSR of lower dimension, which is a one-way mapping rather than a definitional loop. No equations, self-citations, or fitted parameters are exhibited that would make any claimed prediction equivalent to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5677 in / 1183 out tokens · 40128 ms · 2026-05-25T12:16:13.903954+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Julia: A fresh approach to numerical computing,

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Set Programming with JuMP,

B. Legat, R. M. Jungers, P. A. Parrilo, and P. Tabuada, “Set Programming with JuMP,” in The Third Annual JuMP-dev Workshop , 2019

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JuMP: A modeling language for mathematical optimization,

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Generating unstable trajectories for Switched Systems via Dual Sum-Of-Squares techniques,

B. Legat, R. M. Jungers, and P. A. Parrilo, “Generating unstable trajectories for Switched Systems via Dual Sum-Of-Squares techniques,” in Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control , ser. HSCC ’16. ACM, 2016, pp. 51–60. [Online]. Available: http://doi.acm.org/10.1145/2883817.2883821

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Certifying unstability of Switched Systems using Sum of Squares Programming,

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