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arxiv: 2605.28242 · v1 · pith:4RIW4T6Hnew · submitted 2026-05-27 · 🧮 math.OC

Hautus-Type Criteria for Controllability and Stabilizability of Backward-Structured Stochastic Systems

Jingrui Sun This is my paper

Pith reviewed 2026-06-29 11:00 UTC · model grok-4.3

classification 🧮 math.OC
keywords Hautus testexact controllabilitystabilizabilitystochastic linear systemsLyapunov operatorpositive semidefinite eigenmatricesbackward stochastic systems
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The pith

Exact controllability of backward-structured stochastic linear systems is equivalent to the absence of nonzero positive semidefinite eigenmatrices of the Lyapunov-type operator that are orthogonal to the control directions.

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

The paper proves sharp stochastic versions of the classical Popov-Belevitch-Hautus test for exact controllability and stabilizability. For these systems the obstruction is specifically a positive semidefinite eigenmatrix rather than an arbitrary symmetric one or a left eigenvector. Exact controllability fails precisely when such a nonzero matrix exists that is orthogonal to the controls, and this restriction to the positive semidefinite cone is necessary and sufficient. Stabilizability follows from the same condition applied only to the nonstable part of the spectrum.

Core claim

Exact controllability is equivalent to the absence of nonzero positive semidefinite eigenmatrices orthogonal to the control directions. Stabilizability is characterized by the same cone-restricted Hautus condition imposed only on the nonstable spectral part of the Lyapunov-type operator.

What carries the argument

The cone-restricted Hautus condition: positive semidefinite eigenmatrices of the Lyapunov-type operator that are orthogonal to the control directions.

If this is right

  • Finite-rank and Gramian characterizations of exact controllability hold.
  • A controllability decomposition is established for these systems.
  • Exact controllability implies stabilizability.
  • Stabilizability is characterized by the same restricted condition on the nonstable spectral part.

Where Pith is reading between the lines

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

  • The cone restriction may extend to controllability questions in other classes of stochastic differential equations.
  • Numerical checks for the existence of such eigenmatrices could yield practical tests.
  • The decomposition may reduce higher-dimensional control design to lower-dimensional subsystems.

Load-bearing premise

The backward-structured stochastic linear systems admit a well-defined Lyapunov-type operator whose eigenmatrices can be classified as positive semidefinite or not, and the orthogonality condition with control directions is well-posed under the system class assumptions.

What would settle it

A concrete backward-structured stochastic system that is exactly controllable despite the existence of a nonzero positive semidefinite eigenmatrix orthogonal to the controls, or vice versa.

read the original abstract

This paper develops sharp Hautus-type criteria, stochastic counterparts of the classical Popov-Belevitch-Hautus test, for exact controllability and stabilizability of backwardstructured stochastic linear systems. The main finding is that the stochastic Hautus obstruction is not a left eigenvector, as in deterministic linear systems, nor an arbitrary symmetric eigenmatrix, but a positive semidefinite eigenmatrix of a Lyapunov-type operator. We prove that exact controllability is equivalent to the absence of such nonzero positive semidefinite eigenmatrices that are orthogonal to the control directions. This cone restriction is sharp: excluding all symmetric eigenmatrices with the same orthogonality property is sufficient but not necessary. We further show that stabilizability is characterized by the same cone-restricted Hautus condition imposed only on the nonstable spectral part of the Lyapunov-type operator. Thus the stochastic Hautus theory developed here is governed by a simultaneous spectral restriction and cone restriction. In addition to these criteria, we provide finite-rank and Gramian characterizations underlying exact controllability, establish the corresponding controllability decomposition, and show that exact controllability implies stabilizability.

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. This manuscript develops sharp Hautus-type criteria for exact controllability and stabilizability of backward-structured stochastic linear systems. It proves that exact controllability is equivalent to the absence of nonzero positive semidefinite eigenmatrices of a Lyapunov-type operator that are orthogonal to the control directions. The cone restriction is shown to be sharp by an explicit counterexample involving a symmetric but non-PSD eigenmatrix. Stabilizability is characterized by the same condition restricted to the nonstable spectral part. The paper also supplies Gramian and finite-rank characterizations, establishes a controllability decomposition, and shows that exact controllability implies stabilizability.

Significance. If the derivations hold, the work supplies a precise stochastic counterpart to the classical PBH test, with the positive-semidefinite cone restriction as the central technical contribution. The self-adjointness verification of the Lyapunov operator, the Gramian equivalence in Theorem 3.4, the explicit sharpness counterexample, and the decomposition results constitute concrete strengths. These parameter-free, falsifiable characterizations are likely to be useful for subsequent analysis of stochastic systems.

minor comments (3)
  1. [§2] §2: the precise definition of the backward-structured system class and the domain of the Lyapunov operator L could be stated with an explicit list of standing assumptions on the coefficients to facilitate reading.
  2. The notation for the Frobenius inner product and the orthogonality condition with the control operator is introduced without a dedicated display equation; adding one would improve clarity when the condition is invoked in the main theorems.
  3. The finite-rank characterization is stated after the Gramian result; a forward reference from the Gramian theorem to the finite-rank corollary would help the logical flow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending acceptance. The referee's summary accurately captures the main contributions regarding the cone-restricted Hautus-type criteria for exact controllability and stabilizability of backward-structured stochastic systems.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from the explicit definition of the backward-structured system and the associated Lyapunov-type operator L (self-adjoint w.r.t. the Frobenius product), through the Gramian characterization (Theorem 3.4) and finite-rank conditions, to the cone-restricted Hautus equivalence. All steps are internal to the stated system class; the PSD restriction is shown sharp by an explicit counter-example of a non-PSD eigenmatrix, and no load-bearing step reduces to a fitted parameter, self-citation chain, or definitional renaming. The central claim is therefore self-contained against the paper's own assumptions and constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions about the existence and spectral properties of Lyapunov-type operators for backward-structured stochastic systems, with no free parameters or invented entities introduced.

axioms (1)
  • domain assumption Existence and well-defined spectral properties of Lyapunov-type operators for the given stochastic system class
    Invoked to define the eigenmatrices and their positive semidefinite classification.

pith-pipeline@v0.9.1-grok · 5719 in / 1210 out tokens · 52115 ms · 2026-06-29T11:00:56.728873+00:00 · methodology

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Reference graph

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