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arxiv: 2604.26455 · v1 · submitted 2026-04-29 · 🧮 math.OC

Fixed-Time and Arbitrarily Fast Exponential Stabilization of Discrete-Time Switched Linear Systems

Picchiotti Flavio (L2S) , Thiago Alves Lima (ITA) , Girard Antoine (L2S) This is my paper

Pith reviewed 2026-05-07 11:25 UTC · model grok-4.3

classification 🧮 math.OC
keywords switched linear systemsfixed-time stabilizationexponential stabilizationdiscrete-time controlstate feedbackgeometric approachstructural decompositionmode-dependent control
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The pith

Switched linear systems can be fixed-time stabilized if and only if they allow arbitrarily fast exponential stabilization.

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

This paper derives geometric conditions for stabilizing discrete-time switched linear systems in any prescribed number of steps, independent of the switching sequence. It gives constructive methods to find the required state feedback gains when the controller knows the current mode and when it does not. The key step is a structural decomposition of the system that makes analysis easier and proves that fixed-time stabilizability is exactly equivalent to the ability to achieve exponential stability at any desired speed. A reader would care because this links two stability notions in switched systems, which appear in many engineering applications with mode changes.

Core claim

Using a geometric approach, conditions are derived under which discrete-time switched linear control systems can be stabilized within a prescribed number of steps independently of the switching sequence. Constructive procedures compute the stabilizing state-feedback gains for mode-dependent and mode-independent cases. A structural decomposition is introduced that equates fixed-time stabilizability with arbitrarily fast exponential stabilizability.

What carries the argument

Structural decomposition of switched systems that equates fixed-time stabilizability to arbitrarily fast exponential stabilizability and simplifies controller design.

Load-bearing premise

The switched linear system admits a structural decomposition that simplifies stabilizability analysis and controller design.

What would settle it

A discrete-time switched linear system that is fixed-time stabilizable but not arbitrarily fast exponentially stabilizable, or the reverse, would disprove the equivalence.

Figures

Figures reproduced from arXiv: 2604.26455 by Girard Antoine (L2S), Picchiotti Flavio (L2S), Thiago Alves Lima (ITA).

Figure 1
Figure 1. Figure 1: Closed-loop trajectory under mode-dependent controller. The trajectory reaches the origin in 3 view at source ↗
Figure 2
Figure 2. Figure 2: Closed-loop trajectories under mode-dependent controller. One trajectory exhibits exponential view at source ↗
Figure 3
Figure 3. Figure 3: Time-scale plot of the closed-loop trajectory under mode-independent controller. view at source ↗
read the original abstract

In this paper we first study the fixed-time stabilizability of discrete-time switched linear control systems. Using a geometric approach, we derive conditions under which such systems can be stabilized within a prescribed number of steps, independently of the switching sequence. We address both the mode-dependent case, where the controller has access to the active mode, and the mode-independent case, where a common feedback law must be employed. For each setting, we present constructive procedures to compute the stabilizing state-feedback gains. Building on these results, we then introduce a structural decomposition of switched systems, which serves to simplify stabilizability analysis and controller design. This allows us to establish the equivalence between fixed-time stabilizability and arbitrarily fast exponential stabilizability. The effectiveness of the proposed methods is illustrated through a numerical example.

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.

Referee Report

0 major / 2 minor

Summary. The paper studies fixed-time stabilizability of discrete-time switched linear control systems via a geometric approach, deriving conditions and constructive procedures for computing stabilizing state-feedback gains in both mode-dependent and mode-independent settings. It introduces a structural decomposition to simplify the analysis and establishes the equivalence between fixed-time stabilizability and arbitrarily fast exponential stabilizability, with the results illustrated through a numerical example.

Significance. The equivalence between fixed-time and arbitrarily fast exponential stabilization, if the structural decomposition holds generally, would be a meaningful contribution to switched system control theory. The constructive nature of the methods adds to its practical significance, allowing for direct computation of controllers.

minor comments (2)
  1. [Abstract] The abstract could briefly preview the key properties of the structural decomposition to better contextualize how it enables the equivalence result.
  2. [Numerical example] The numerical example section should explicitly report the system dimensions, the chosen fixed time, and the achieved exponential rate to allow readers to assess the practical tightness of the bounds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The referee's description accurately reflects the paper's focus on fixed-time stabilizability via geometric methods, the constructive controller design for both mode-dependent and mode-independent cases, the structural decomposition, and the equivalence result. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper first derives conditions for fixed-time stabilizability of discrete-time switched linear systems via a geometric approach, with explicit constructive procedures for state-feedback gains in both mode-dependent and mode-independent cases. It then introduces a structural decomposition of the switched system to simplify analysis, and applies this decomposition to prove an equivalence between fixed-time stabilizability and arbitrarily fast exponential stabilizability. No load-bearing step reduces by definition or construction to its own inputs, no parameter is fitted to data and then relabeled as a prediction, and no uniqueness or ansatz is smuggled via self-citation. The derivation chain is constructive and self-contained against the stated geometric and decomposition-based arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identifiable. The geometric approach likely relies on standard assumptions from linear algebra and switched systems theory.

pith-pipeline@v0.9.0 · 5441 in / 987 out tokens · 53398 ms · 2026-05-07T11:25:03.925359+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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