Assignability of dichotomy spectrum for discrete time-varying linear control systems

Le Viet Cuong; Thai Son Doan

REVIEW 3 major objections 3 minor 18 references

Assignability of dichotomy spectrum for discrete time-varying linear control systems

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For discrete time-varying linear control systems, uniform complete controllability implies that any prescribed dichotomy spectrum—any finite union of disjoint closed intervals—can be realized by bounded linear feedback.

desk verdict A promising paper with a genuine strengthening of Lyapunov spectrum assignability, but the proof currently rests on an unproved two-sided triangularization theorem and a small zero-at-zero slip that breaks the construction's own hypotheses. read the letter →

arxiv 1908.04763 v1 pith:UXKTOX2B submitted 2019-08-12 math.OC

Le Viet Cuong , Thai Son Doan This is my paper · ORCID

classification math.OC MSC 34A3034D0934D0834H05

keywords time-varyinglinearsystemsdichotomyspectrumassignabilityuniformcompletecontrollabilityboundedfeedbackLyapunovupper-triangularkinematicalequivalence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

For a discrete time-varying linear control system $x_{n+1}=A_nx_n+B_nu_n$, this paper claims that uniform complete controllability is enough to assign, by a bounded linear feedback $u_n=U_nx_n$, any prescribed dichotomy spectrum made of finitely many disjoint closed intervals $[a_1,b_1],\dots,[a_\ell,b_\ell]$. This is a spectrum-level strengthening of the earlier pole-placement theorem that only assigned the Lyapunov spectrum, a finite set of growth rates; here the whole intervals between those rates are under control. If the claim is correct, one controllability hypothesis gives full freedom over the exponential growth and decay rates of all solutions of the closed-loop system, so the controlled dynamics can be shaped qualitatively at will.

What carries the argument

The machinery has three load-bearing pieces. Kinematical equivalence is the conjugacy relation that lets the paper replace the closed-loop system by an upper-triangular one without changing the dichotomy spectrum: two systems are equivalent when a bounded invertible transformation $T_n$ satisfies $A_nT_n=T_{n+1}C_n$. The two-sided upper-triangularization theorem (Theorem 10), adapted from [5, Theorem 4.6], is what turns uniform complete controllability into the existence of a bounded feedback placing the system in that triangular form with prescribed diagonal sequences. Finally, Proposition 9 computes the spectrum of an upper-triangular system whose diagonal entries are scalar Lyapunov sequences that are even under $n\mapsto -n$: the dichotomy spectrum is exactly the union of the scalar spectra, and a scalar sequence switching between $e^{a_i}$ and $e^{b_i}$ on alternating dyadic blocks has spectrum $[a_i,b_i]$.

What would settle it

Run the construction in dimension one, where no triangularization step is needed and the closed-loop coefficient is forced to be the sequence $p_n$ itself: with $p_0=0$, the closed-loop matrix at $n=0$ is not invertible, so the system is not in the Lyapunov class on which the definition of dichotomy spectrum rests. Computing the dichotomy spectrum of this scalar system—or replacing $p_0$ by a positive value and checking whether the spectrum remains $[a_i,b_i]$—would directly show whether the paper's claimed assignment is realized as written.

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Extended reading notes

Core claim

The paper's central claim is Theorem 5: if system (2) is uniformly completely controllable, then for any disjoint closed intervals $[a_1,b_1],\dots,[a_\ell,b_\ell]$ there exists a bounded feedback $U$ such that $A+BU$ is a Lyapunov sequence and $\Sigma_{\mathrm{ED}}(A+BU)=\bigcup_{i=1}^\ell[a_i,b_i]$. The proof works by reducing the feedback design to a triangularization problem: Theorem 10 asserts that a two-sided uniformly completely controllable system can be fed back so that it becomes kinematically equivalent to an upper-triangular system whose diagonal entries are prescribed positive scalar Lyapunov sequences, with the proof presented as a slight modification of the one-sided result in [5, Theorem 4.6] and deferred to that reference. Once the closed loop is in triangular form, Proposition 9 computes its dichotomy spectrum as the union of the scalar spectrum of each diagonal sequence, and the diagonal sequences are chosen to alternate between $e^{a_i}$ and $e^{b_i}$ on dyadic blocks, symmetric under $n\mapsto -n$, so that each scalar spectrum is exactly $[a_i,b_i]$.

Load-bearing premise

The argument depends on the two-sided upper-triangularization theorem (Theorem 10), which is asserted as a 'slight modification' of the one-sided result in [5, Theorem 4.6] with its proof omitted, and on the diagonal sequences in (13) being positive Lyapunov sequences although those sequences set $p_i^0=0$, so if either the unproved extension fails or the zero cannot be repaired, the construction collapses.

Editorial extensions

If this is right

Any finite union of disjoint closed intervals can be realized as the dichotomy spectrum of a closed-loop system, provided the open-loop system is uniformly completely controllable.
By taking $a_i=b_i=\lambda_i$, the dichotomy spectrum and the Lyapunov spectrum of the closed-loop system both equal $\{\lambda_1,\dots,\lambda_\ell\}$; the dichotomy-spectrum result therefore recovers the Lyapunov-spectrum pole placement theorem of [5] as a corollary.
The upper-triangular spectral formula (Theorem 8) shows that in the triangular normal form, off-diagonal couplings do not enlarge the dichotomy spectrum beyond the union of the diagonal spectra.
The construction isolates a one-dimensional phenomenon: a scalar coefficient that switches between $e^{a}$ and $e^{b}$ on long dyadic blocks has dichotomy spectrum exactly $[a,b]$, which is the atomic building block for assigning arbitrary intervals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension the authors do not state: the same dyadic construction should assign the one-sided dichotomy spectra $\Sigma^+_{\mathrm{ED}}$ and $\Sigma^-_{\mathrm{ED}}$ on the positive and negative half-lines, since each half of the symmetric sequence (13) independently yields the interval $[a_i,b_i]$.
Because Theorem 10 requires positive Lyapunov diagonal sequences while (13) sets $p_i^0=0$, a minimal repair would be to replace that single zero by a positive value; checking whether the scalar dichotomy spectrum remains $[a_i,b_i]$ would test whether the construction is coherent or needs modification.
Through the 1-time discretization of Lemma 12, the discrete assignment theorem suggests a route to assigning the dichotomy spectrum of sampled continuous-time control systems, a problem the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this paper has a genuinely interesting claim and a plausible route, but as written the main theorem is not proved. The proof depends on a two-sided upper-triangularization result (Theorem 10) that is stated with no proof, just a reference to a one-sided theorem in [5]; and the scalar sequences used in the construction set the n=0 term to zero, which violates the positivity and Lyapunov assumptions the argument requires. Both issues look fixable, but they are real.

What's new: the claim that uniform complete controllability can assign the whole dichotomy spectrum, not just the Lyapunov spectrum, is a natural strengthening of the 2017 pole-placement result. The symmetric dyadic construction for scalar sequences with prescribed interval spectrum is a neat idea, and I think it works once the n=0 slip is corrected. The discrete version of the Battelli-Palmer triangular spectral inclusion (Proposition 9) is also a solid piece of work, and the appendix lemma linking continuous and sampled spectra is clean.

The soft spots: Theorem 10 is the load-bearing wall. Extending [5, Theorem 4.6] from one-sided to two-sided time is not a 'slight modification' you can hand-wave; you need to construct the feedback and the Lyapunov transformation on all of Z, with a boundary condition at 0 and uniform bounds in both directions. The manuscript doesn't supply that. If the two-sided version is false, the main theorem has no support. This isn't a small gap; it's the central step. The zero at n=0 in (13) is smaller—likely a typo—but as written it breaks the LLya requirement and the positivity condition, and it makes the closed-loop matrix singular at one time. Fix it and that part goes away.

The citation pattern is fine; the single self-citation is in a supporting discretization lemma, not in the main line. This is honest work, not a disguised triviality.

Who should read it: anyone working on nonautonomous linear control and spectral assignability. It deserves a serious referee—the problem is worth solving and the approach is promising. My recommendation: send it to review, but ask the referee to insist on a full proof of Theorem 10 and the corrected definition of p_i^0, and to check whether Proposition 9's symmetry assumption is really enough for the two-sided dichotomy spectrum.

Referee Report

3 major / 3 minor

Summary. The paper claims that for a discrete time-varying linear control system x_{n+1}=A_n x_n+B_n u_n on Z, uniform complete controllability implies assignability of the dichotomy spectrum: for any prescribed union of ℓ disjoint closed intervals there is a bounded feedback U such that the closed-loop system lies in LLya and its dichotomy spectrum equals that union. The proof strategy is to triangularize the closed-loop system via feedback, then compute the spectrum of the triangular system from its diagonal entries. Section 3.1 proves an inclusion for upper-triangular systems (Theorem 8), Section 3.2 states a two-sided upper-triangularization result (Theorem 10) as a 'slight modification' of [5, Theorem 4.6], and Section 3.3 constructs diagonal sequences p_i^n with prescribed dichotomy spectra and applies these two results.

Significance. If fully proved, the main theorem would substantially strengthen the Lyapunov spectrum assignment result of Babiarz, Czornik, Makarov, Niezabitowski, and Popova, because the dichotomy spectrum is a coarser object that contains the Lyapunov spectrum, and Remark 6 correctly explains the implication. The paper's high-level mechanism is natural: use uniform complete controllability to put the closed-loop system into upper-triangular form with prescribed diagonal sequences, then compute the dichotomy spectrum of the triangular system. The manuscript is clearly organized and the intended reduction is transparent. However, as written the proof rests on three load-bearing gaps, described in the major comments; the central claim is plausible but is not established by the present text.

major comments (3)

[Section 3.2, Theorem 10] Theorem 10 is stated for two-sided systems, but its proof is the single sentence 'See [5, Theorem 4.6]'. The cited result is explicitly described in the text as a one-sided theorem. The two-sided extension is not a routine corollary: the one-sided construction iterates forward from 0 and does not address the additional requirement that the Lyapunov transformation T_n be bounded and invertible for all n∈Z, including negative times, nor the consistency condition at n=0 for the feedback and the transformation. Since Theorem 10 is the only mechanism in the proof of Theorem 5 that produces the upper-triangular closed-loop system, the main theorem is not proved as written. The authors need to supply a complete proof of the two-sided version or cite a reference that actually states it.
[Section 3.3, Eq. (13)] The sequences p_i^n are defined by p_i^n=e^{a_i}, p_i^n=e^{b_i}, and '0, for n=0'. This makes p_i^0=0, so the sequences are not positive and do not belong to LLya. Theorem 10 requires arbitrary scalar positive Lyapunov sequences, and Proposition 9 requires Lyapunov sequences satisfying the symmetry condition (9). Moreover, the scalar system (14) is not invertible at n=0, so its two-sided evolution operator is not defined in the sense used throughout the paper. Thus the construction in the proof of Theorem 5 violates the hypotheses of the results it invokes. This is likely a typo, but as written it is internally inconsistent and must be fixed, for example by assigning a positive value at n=0 and checking that the dichotomy spectrum remains [a_i,b_i].
[Section 3.1, proof of Theorem 8] The proof of Theorem 8 identifies the discrete system (6) with the 1-time discretization of the piecewise-constant continuous system dot{x}=W(t)x with W(t)=D_n on [n,n+1). By the definition in the Appendix, the coefficient of the associated 1-time discrete system is A_n=Φ_W(n+1,n), which in this piecewise-constant case equals e^{D_n}, not D_n in general. For a diagonal scalar example, the diagonal entries of the discretized system are e^{a_n} and e^{b_n}, not a_n and b_n. Consequently the equalities in (8) are not justified by the given argument. Theorem 8 may be true, but the proof as written needs to be replaced by a direct discrete-time argument or by a correct continuous embedding that actually produces the ordered evolution over [n,n+1).

minor comments (3)

[Section 3.2, Theorem 10] In the statement of Theorem 10, the feedback is written as U=(U_n)_{n∈N}; the index set should be Z to be consistent with the surrounding definitions and with the requirement U∈L∞(Z,R^{s×d}).
[Introduction] The sentence 'The structure of the paper is follows' should read 'The structure of the paper is as follows'.
[Appendix, Definition 11] Definition 11 refers to 'the dichotomy spectrum of (1)' when defining the continuous-time spectrum; the equation number should refer to the continuous system (16), not the discrete system (1).

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the target proof reduces to external triangularization theorems, not to its own conclusion; the only self-citation is minor and not load-bearing.

full rationale

The derivation chain is not circular. Theorem 5 is proved by constructing scalar sequences (13), computing their dichotomy spectra via Proposition 9, and invoking Theorem 10 to find a feedback U such that A+BU is kinematically equivalent to an upper-triangular system C with prescribed diagonal sequences. The final identity Sigma_ED(A+BU)=Sigma_ED(C)=union of intervals follows from Proposition 9 and the definition of the pi sequences. Each load-bearing ingredient is either proved in this paper (Proposition 9, using Theorem 8 and Lemma 12, with external Theorem 7 from Battelli-Palmer) or is an external result, namely [5, Theorem 4.6], which is a one-sided triangularization theorem and is not equivalent to the target assignability statement. No parameter is fitted to data, and no 'prediction' is defined in terms of the quantity it is supposed to predict. The only self-citation is [8] in the Appendix, used only to recall the notion of a 1-time discretization; it is not used in the proof of the main theorem. The two-sided extension stated as Theorem 10 is asserted with the proof line 'See [5, Theorem 4.6]' and the paper does not supply the modification argument; this is a completeness/rigor risk, not circularity. Likewise, the construction in (13) sets p_i^0=0, which conflicts with the hypotheses that pi are positive Lyapunov sequences and that C is in LLya; this is an internal inconsistency, not a circular reduction. Accordingly, the circularity score is 1, reflecting only a minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof imports the spectral theorem for dichotomies, the triangular spectrum inclusion, and the upper-triangularization theorem from prior literature. No numbers are fitted to data and no new physical or abstract entities are introduced; the only ad hoc additions are the unproved two-sided extension and the inconsistent zero value at time zero.

assumptions (4)

domain assumption The dichotomy spectrum of a bounded discrete linear system is a union of at most d disjoint intervals.
Invoked implicitly throughout and supported by references [3,13]; the paper does not reprove the spectral theorem.
domain assumption For upper-triangular differential systems, the dichotomy spectrum lies between the union of the diagonal spectra and the union of their one-sided spectra.
Used as Theorem 7, quoted from [7, Section 4] with no proof.
ad hoc to paper The two-sided upper-triangularization theorem (Theorem 10) is a valid extension of [5, Theorem 4.6].
The proof is omitted and merely cites [5, Theorem 4.6], so the extension is an unverified assumption on which the main theorem rests.
ad hoc to paper The scalar diagonal sequences p_i^n in (13) can be simultaneously positive, symmetric, and Lyapunov while realizing the prescribed intervals.
The actual definition sets p_i^0 = 0, contradicting positivity; the proof implicitly assumes this inconsistency can be repaired without changing the spectrum.

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Cite this review

Pith. "Pith review of Assignability of dichotomy spectrum for discrete time-varying linear control systems." pith.science (2026). https://pith.science/paper/UXKTOX2B

@misc{pith2026190804763,
  author       = {Pith},
  title        = {Pith review of: Assignability of dichotomy spectrum for discrete time-varying linear control systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXKTOX2B}},
  note         = {Machine review of arXiv:1908.04763}
}

read the original abstract

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in A. Babiarz et. al. Pole placement theorem for discrete time-varying linear systems.SIAM J. Control Optim. 55 (2017), no. 2, 671-692 about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[5]

Babiarz, A

A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova. Pole placement theorem for discrete time-varying linear sy stems. SIAM J. Control Optim. 55 (2017), no. 2, 671-692

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Aulbach and S

B. Aulbach and S. Siegmund. A spectral theory for nonauto nomous dif- ference equations. Proc. of 5th Int. Conference on Diﬀerence Equations and Applications, Temuco/Chile, 45–55, 2000

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Babiarz, I

A. Babiarz, I. Banshchikova, A. Czornik, E.K. Makarov, M . Niez- abitowski, S. Popova. Necessary and suﬃcient conditions fo r assignabil- ity of the Lyapunov spectrum of discrete linear time-varyin g systems. IEEE Trans. Automat. Control 63 (2018), no. 11, 3825-3837

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Barreira and C, Valls

L. Barreira and C, Valls. Stability of Nonautonomous Diﬀerential Equa- tions. Lecture Notes in Mathematics, 1926. Springer, Berlin, 2008

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Battelli and K.J

F. Battelli and K.J. Palmer. Criteria for exponential di chotomy for triangular systems. J. Math. Anal. Appl. 428 (2015), 525–543

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Cuong, T.S

L.V. Cuong, T.S. Doan and S. Siegmund. A Sternberg theore m for nonautonomous diﬀerential equations. To appear in J. Dynam. Diﬀer- ential Equations , https://doi.org/10.1007/s10884-017-9629-8

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Babiarz, A

A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova. Pole placement theorem for discrete time-varying linear sy stems. SIAM J. Control Optim. 55 (2017), no. 2, 671-692

work page 2017

[2] [1]

L. Ya. Adrianova. Introduction to Linear Systems of Diﬀerential Equa- tions. Translated from the Russian by Peter Zhevandrov. Translati ons of Mathematical Monographs, 146. American Mathematical Society, Prov- idence, RI, 1995

work page 1995

[3] [2]

Aulbach, C

B. Aulbach, C. P¨ otzsche and S. Siegmund. A smoothness th eorem for invariant ﬁber bundles. J. Dynam. Diﬀerential Equations 14 (2002), 519-547. 13

work page 2002

[4] [3]

Aulbach and S

B. Aulbach and S. Siegmund. A spectral theory for nonauto nomous dif- ference equations. Proc. of 5th Int. Conference on Diﬀerence Equations and Applications, Temuco/Chile, 45–55, 2000

work page 2000

[5] [4]

Babiarz, I

A. Babiarz, I. Banshchikova, A. Czornik, E.K. Makarov, M . Niez- abitowski, S. Popova. Necessary and suﬃcient conditions fo r assignabil- ity of the Lyapunov spectrum of discrete linear time-varyin g systems. IEEE Trans. Automat. Control 63 (2018), no. 11, 3825-3837

work page 2018

[6] [6]

Barreira and C, Valls

L. Barreira and C, Valls. Stability of Nonautonomous Diﬀerential Equa- tions. Lecture Notes in Mathematics, 1926. Springer, Berlin, 2008

work page 1926

[7] [7]

Battelli and K.J

F. Battelli and K.J. Palmer. Criteria for exponential di chotomy for triangular systems. J. Math. Anal. Appl. 428 (2015), 525–543

work page 2015

[8] [8]

Cuong, T.S

L.V. Cuong, T.S. Doan and S. Siegmund. A Sternberg theore m for nonautonomous diﬀerential equations. To appear in J. Dynam. Diﬀer- ential Equations , https://doi.org/10.1007/s10884-017-9629-8

work page doi:10.1007/s10884-017-9629-8