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arxiv: 2605.19709 · v1 · pith:YF5ZL2RGnew · submitted 2026-05-19 · 🧮 math.OC

Feedback Stabilization of Switched Systems: Memory is not needed

Thiago Alves Lima (ITA) , Matteo Della Rossa (Polito) , Antoine Girard (L2S) This is my paper

Pith reviewed 2026-05-20 04:17 UTC · model grok-4.3

classification 🧮 math.OC
keywords switched linear systemsfeedback stabilizationmemoryless controllersfull-information controllershomogeneous of degree one
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The pith

For switched linear systems, any stabilizing full-history controller can be replaced by a memoryless homogeneous one.

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

The paper aims to show that memory is unnecessary for stabilizing switched linear systems. If there exists a controller that stabilizes the system using the complete history of states and switching signals, then there also exists a memoryless controller that is homogeneous of degree one and achieves stabilization. In mode-independent settings, this controller depends solely on the current state, while in mode-dependent settings it depends on the current state and the currently active mode. This is significant for control design because it means that more complicated dynamic controllers with memory do not offer any advantage in terms of stabilization capability over simpler static ones.

Core claim

We prove by construction that if a switched linear system admits a stabilizing full-information controller, with access to the entire history of states and switching signals, then a memoryless and homogeneous of degree one stabilizing controller also exists. Specifically, in the mode-independent setting the controller can be chosen to depend only on the current state, and in the mode-dependent setting only on the current state and active mode. Our results thus show that dynamic controllers offer no additional stabilizing capability for switched linear systems.

What carries the argument

The explicit construction that converts any full-information stabilizing controller into a memoryless homogeneous-of-degree-one controller.

If this is right

  • Stabilizing controllers for switched linear systems can always be chosen memoryless.
  • Dynamic or history-dependent controllers add no stabilization power.
  • Mode-independent stabilization reduces to static state feedback.
  • Mode-dependent stabilization reduces to static feedback depending on the active mode.

Where Pith is reading between the lines

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

  • Designers can restrict attention to simple static controllers when synthesizing stabilizers for switched linear systems.
  • The same reduction from history-dependent to memoryless controllers may be worth testing on nonlinear switched systems.
  • Synthesis algorithms and software for switched systems can safely limit the controller class to memoryless homogeneous maps.

Load-bearing premise

The system switches between linear dynamics in each mode and any stabilizing controller can be taken to be homogeneous of degree one.

What would settle it

A switched linear system that can be stabilized by some full-information controller but cannot be stabilized by any memoryless homogeneous of degree one controller.

read the original abstract

A long-standing assumption in the literature on switched linear systems is that static, homogeneous of degree one feedbacks form the most general class of controllers necessary and sufficient for stabilization. In this paper, we provide a rigorous justification. More specifically, we prove by construction that if a switched linear system admits a stabilizing full-information controller, with access to the entire history of states and switching signals, then a memoryless and homogeneous of degree one stabilizing controller also exists. Specifically, in the modeindependent setting the controller can be chosen to depend only on the current state, and in the mode-dependent setting only on the current state and active mode. Our results thus show that dynamic controllers offer no additional stabilizing capability for switched linear systems, formally validating this folklore claim.

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 / 3 minor

Summary. The paper proves by explicit construction that for switched linear systems, the existence of a stabilizing full-information controller (with access to the full history of states and switching signals) implies the existence of a memoryless homogeneous-of-degree-one stabilizing controller. In the mode-independent case the resulting feedback depends only on the current state; in the mode-dependent case it depends on the current state and the active mode. This shows that dynamic controllers with memory provide no additional stabilizing capability beyond static homogeneous feedbacks for this class of systems.

Significance. If the construction is correct, the result formally justifies a long-standing assumption in the switched-systems literature that static homogeneous feedbacks suffice for stabilization. The explicit reduction from full-information to memoryless controllers is a strength, as it supplies a concrete mechanism rather than an existence argument and directly addresses the question of whether memory or dynamics add value.

minor comments (3)
  1. [Abstract] Abstract: the compound word 'modeindependent' should be hyphenated as 'mode-independent' for standard mathematical English.
  2. [Main theorem section] The proof construction in the main theorem would benefit from an explicit statement of the homogeneity degree and the precise norm used to define the memoryless map, to make the reduction step easier to verify.
  3. [Discussion or concluding remarks] A short remark on how the construction behaves under arbitrary (non-dwell-time) switching signals would clarify the scope, even if the result already covers the general case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report accurately captures the main result: an explicit construction showing that any stabilizing full-information controller for a switched linear system can be reduced to a memoryless homogeneous-of-degree-one controller (mode-independent or mode-dependent as appropriate). No specific major comments or requested changes were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; result via explicit construction

full rationale

The paper's central claim is a mathematical implication proved by explicit construction: any stabilizing full-information controller for a switched linear system can be reduced to a memoryless homogeneous-of-degree-one controller. This is a direct proof establishing the result rather than a re-expression or fit of prior quantities. No load-bearing self-citations, self-definitional steps, or fitted inputs called predictions appear in the provided abstract or description. The derivation is self-contained and supplies independent content through the reduction for linear dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of switched linear systems and the class of homogeneous degree-one controllers; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The system is a switched linear system with a finite number of modes, each governed by linear dynamics.
    Standard modeling assumption for the class of systems under study, invoked throughout the abstract.
  • domain assumption Stabilizing controllers are sought within the class of memoryless homogeneous-of-degree-one feedback laws.
    The paper restricts attention to this controller class when constructing the memoryless stabilizer.

pith-pipeline@v0.9.0 · 5658 in / 1241 out tokens · 41929 ms · 2026-05-20T04:17:21.290346+00:00 · methodology

discussion (0)

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Reference graph

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