Let X be the Banach space underlying a linear discrete-time system, and let A_m^k denote its associated evolution operator from time k to time m. The system is uniformly exponentially stable when it satisfies the corresponding uniform exponential decay condition. Barbashin-type characterization. For every linear discrete-time system, the following statements are equivalent: The system is uniformly exponentially stable. There exist constants B≥ 1 and b>0 such that ∑_k=0^me^b(m-k)lVert A_m^k x rVert≤ B lVert x rVert for all (m,x)∈N× X. There exists a constant B≥ 1 such that ∑_k=0^mlVert A_m^k x rVert≤ B lVert x rVert for all (m,x)∈N× X. The result would extend the preceding operator-norm characterization by replacing lVert A_m^k rVert with pointwise estimates on A_m^k x. The implication from condition (iii) to condition (i) is presented as an open problem, while the other direction follows from the stated characterization.

Barbashin-type characterization of uniform exponential stability

At least 12 years old · documented by 2013

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$ . The system is uniformly exponentially stable when it satisfies the corresponding uniform exponential decay condition. Barbashin-type characterization. For every linear discrete-time system, the following statements are equivalent:

The system is uniformly exponentially stable.
There exist constants $B\geq 1$ and $b>0$ such that

\sum_{k=0}^{m}e^{b(m-k)}\lVert A_m^k x\rVert\leq B\lVert x\rVert

for all $(m,x)\in\mathbb{N}\times X$ . 3. There exists a constant $B\geq 1$ such that

\sum_{k=0}^{m}\lVert A_m^k x\rVert\leq B\lVert x\rVert

for all $(m,x)\in\mathbb{N}\times X$ .

The result would extend the preceding operator-norm characterization by replacing $\lVert A_m^k\rVert$ with pointwise estimates on $A_m^k x$ . The implication from condition (iii) to condition (i) is presented as an open problem, while the other direction follows from the stated characterization.

References

Primary source

Ioan-Lucian Popa, Traian Ceausu and Mihail Megan, “On exponential stability for linear discrete-time systems in Banach spaces”, arXiv:1305.2036 (2013).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-submitted construction claims the criterion is false in general, but the construction has not been independently checked.

Popa, Ceaușu, and Megan posed the question in 2013: whether the pointwise bounded-sum condition forces uniform exponential stability. They proved the corresponding operator-norm characterization and left the pointwise implication open.

Known results

The operator-norm conditions with $\lVert A_m^k\rVert$ characterize uniform exponential stability.
Uniform exponential stability implies the pointwise bounded-sum condition; the converse was explicitly recorded as open.

Community submission (unverified), August 26, 2026

A submitted counterexample takes $X=\ell^1(\mathbb{N}_0)$ and $A(n)x=x_{n-1}e_n$ . It argues that $\sum_{k=0}^{m}\lVert A_m^k x\rVert_1\leq 2\lVert x\rVert_1$ , while $\lVert A_m^k\rVert=1$ for $k<m$ , so the system is not uniformly exponentially stable. An invertible weighted-shift variant is also sketched.

Current status (as of August 2026): The implication $(iii)\Rightarrow(i)$ is claimed false by an unverified community counterexample; no independently verified resolution is recorded.

Sources

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A counterexample to the pointwise Barbashin criterion

Result

The conjecture in Section 7 of Popa--Ceaușu--Megan, On exponential stability for linear discrete-time systems in Banach spaces, is false for general Banach spaces.

The source considers a sequence A(n) of bounded operators and defines

A_m^k=A(m)\cdots A(k+1)\quad(k<m),\qquad A_m^m=I.

It asks whether the existence of B≥1 such that

\tag{1} \sum_{k=0}^{m}\lVert A_m^k x\rVert\le B\lVert x\rVert

for every m and x forces uniform exponential stability.

Counterexample

Take X=ℓ¹(ℕ₀) with unit vectors e_0,e_1,…. Put A(0)=0, and for n≥1 define

A(n)x=x_{n-1}e_n.

These are bounded rank-one operators of norm one.

For 0≤k<m,

\tag{2} A_m^k x=x_ke_m.

Indeed, A(k+1) first maps x to x_ke_{k+1}, and every subsequent factor moves that same coefficient forward by one coordinate. Since A_m^m=I, equation (2) gives

\begin{aligned} \sum_{k=0}^{m}\lVert A_m^k x\rVert_1 &=\lVert x\rVert_1+\sum_{k=0}^{m-1}|x_k|\\ &\le 2\lVert x\rVert_1. \end{aligned}

Thus (1) holds with B=2.

On the other hand, for every k<m,

A_m^ke_k=e_m,

so ‖A_m^k‖=1. If the system were uniformly exponentially stable, there would be constants N≥1 and α>0 satisfying

1=\lVert A_m^ke_k\rVert_1 \le Ne^{-\alpha(m-k)}

for all m>k. Letting m-k→∞ is a contradiction. Therefore (1) does not imply uniform exponential stability. ∎

Robustness

The proof skeleton records a weighted bilateral-shift variant on ℓ¹(ℤ) in which every generator is invertible, ‖A(n)‖=1, and the inverses are uniformly bounded. Hence noninvertibility is not the underlying obstruction.

The obstruction is instead the order of a supremum and a sum. Condition (1) bounds

\sup_{\lVert x\rVert=1}\sum_k\lVert A_m^kx\rVert,

whereas the known operator-norm criterion controls

\sum_k\sup_{\lVert x\rVert=1}\lVert A_m^kx\rVert.

On ℓ¹, different products can read disjoint coordinates of the same vector, so the first quantity stays bounded while every individual operator norm is one.

Verification record

Primary statement: https://arxiv.org/html/1305.2036#S7
Published source: https://doi.org/10.1016/j.camwa.2012.01.027
MathDB record: https://mathdb.com/p/381167
Exact finite-truncation checks: verify_counterexample.py
Detailed dependency and scope audit: proof-skeleton.md

No indexed published resolution of the exact conjecture was located in a forward-citation and erratum/correction audit through 2026-08-18. This is a mathematical proof of refutation, not yet a claim of external peer review or publication.

Lean: https://github.com/antoshashakov/Principia-Math-In-Progress/blob/main/mathdb-open-problems/problems/381167/Problem381167.lean

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