Input-Power-to-State Stability of Time-Varying Systems

Antonio Russo; Hernan Haimovich; Jose L. Mancilla-Aguilar; Shenyu Liu

REVIEW 3 major objections 5 minor 19 references

Input-Power-to-State Stability of Time-Varying Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that a dissipation-form ISS-Lyapunov function certifies input-power-to-state stability (IPSS), and that ISS under mild Lipschitz assumptions implies IPSS.

desk verdict A useful new IPSS concept with a real but likely repairable gap in the ISS-to-IPSS theorem. read the letter →

arxiv 2505.24805 v1 pith:3KT3BBBX submitted 2025-05-30 eess.SY cs.SYmath.OC

Hernan Haimovich , Shenyu Liu , Antonio Russo , Jose L. Mancilla-Aguilar This is my paper · ORCID

classification eess.SYcs.SYmath.OC MSC 93D2593D3093C10

keywords input-power-to-statestabilitytime-varyingsystemsISSintegraldissipation-formISS-LyapunovfunctionconverseLyapunovtheoremaveragepowerdiscontinuousdynamics

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

Input-power-to-state stability (IPSS) bounds the state norm by a decaying term from the initial condition plus a gain in the maximum average input power over a fixed time window, making it useful exactly when input amplitude and total energy are unbounded. The paper's central claim is that, for time-varying systems, the existence of a dissipation-form ISS-Lyapunov function is a sufficient certificate for IPSS, while the more common implication-form ISS-Lyapunov function guarantees only ISS. A second main claim is that under uniform-in-time Lipschitz and near-continuity assumptions on the dynamics, ISS itself implies the existence of such a Lyapunov function and therefore IPSS. A third claim is that iISS with an exponential class-KL bound also implies IPSS, with explicit gains.

What carries the argument

The central object is the dissipation-form ISS-Lyapunov function, with the Dini derivative $D_f^+V$ taken along the right-hand side; its additively separated decay-and-gain structure is what lets the proof of Theorem 5 convert pointwise input bounds into a moving-average power bound via a transformation $W=\kappa\circ V$ and a comparison inequality of the form $\dot w\le -w+\rho(|u|)$. For the converse direction, the machinery is a scaled system $\dot x=f(t,x,\nu\varphi(|x|))$ with $\varphi\in C^1\cap K_\infty$, together with a converse Lyapunov theorem that builds $V=\sum_{k\ge1}2^{-k}W_k/(1+M_{k,k})$ from suprema over bounded disturbances, producing a function that is Lipschitz in time and state uniformly for all $t$.

What would settle it

Take a time-varying system satisfying Assumption 2 and check whether it is ISS but not IPSS, by constructing an input with finite maximum average power for which the state does not admit a bound of the form $\beta(|x(t_0)|,t-t_0)+\gamma(\|u\|_{\rho,T})$. Alternatively, test the time-varying extension of the scaled-system lemma directly: for a candidate ISS system, compute whether $\dot x=f(t,x,\nu\varphi(|x|))$ is globally asymptotically stable uniformly over all $\nu$ with $|\nu|\le 1$; a counterexample would break the first line of Theorem 9.

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Extended reading notes

Core claim

For the time-varying system $\dot x=f(t,x,u)$ satisfying Assumption 1, Theorem 5 proves that a locally Lipschitz $V(t,x)$ with $\alpha_1(|x|)\le V(t,x)\le\alpha_2(|x|)$ and Dini-derivative inequality $D_f^+ V(t,x,u)\le -\alpha_4(|x|)+\chi_4(|u|)$ for almost all $t$ yields the IPSS estimate $|x(t)|\le\beta(|x(t_0)|,t-t_0)+\gamma(\|u\|_{\rho,T})$. Theorem 9 proves that if $f$ is uniformly locally Lipschitz in $(x,u)$ and continuous in $t$ off a zero-measure set (Assumption 2), then ISS implies a dissipation-form ISS-Lyapunov function exists, constructed through a converse Lyapunov theorem for the scaled disturbance system $\dot x=f(t,x,\nu\varphi(|x|))$, and hence IPSS. Proposition 6 proves that an iISS system whose class-KL function is exponential is IPSS, with the averaging window $T$ chosen larger than $\log(K)/\lambda$.

Load-bearing premise

The load-bearing step is a lemma imported from the time-invariant setting: that ISS of the original system makes the scaled system $\dot x=f(t,x,\nu\varphi(|x|))$ decay globally, uniformly in disturbances and initial time. The paper invokes the proof of that lemma without supplying the time-varying version, and the entire converse-Lyapunov route to IPSS rests on it.

Editorial extensions

If this is right

For time-varying systems, practitioners can certify IPSS, and hence ISS and iISS, by finding a dissipation-form ISS-Lyapunov function rather than an implication-form one.
Under Assumption 2, ISS and IPSS coincide, so an ISS certificate automatically delivers a power-based state bound.
When an iISS estimate is known with an exponential class-KL function, IPSS follows with explicit formulas for the new gain and decay rate.
The IPSS property is independent of the window length $T$ up to gain redefinition, so the user can choose the averaging window that matches the application.

Reading between the lines

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

A natural conjecture left open by the paper is that IPSS is equivalent to the existence of a dissipation-form ISS-Lyapunov function under assumptions weaker than Assumption 2; the first test would be a time-varying ISS system that is not iISS but still has bounded solutions for all bounded-average-power inputs.
The explicit tradeoff in Proposition 6, namely that the averaging window must exceed $\log(K)/\lambda$, suggests that IPSS-based design favors systems with fast exponential decay; this could be tested by applying Theorem 5 to adaptive or event-triggered controller designs.
Because IPSS bounds depend only on the maximum energy over a sliding window, it may provide a natural Lyapunov-level specification for disturbance models in networked control, where bursts of high amplitude are separated by quiet periods.

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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

Quick take: this is a genuinely useful paper. It introduces IPSS, a stability property that bounds the state by the maximum moving-average power of the input, and gives three sufficient conditions. The first two are solid. Theorem 5, that a dissipation-form ISS-Lyapunov function implies IPSS, is the real contribution; the kappa-construction and the sliding-interval argument are self-contained and convincing. Proposition 6 (iISS with an exponential KL bound implies IPSS) is a simple but nice corollary, and Lemma 7's proof is complete. The converse Lyapunov theorem for D-URGAS time-varying systems (Theorem 11) is developed in full detail and appears sound.

The soft spot is Theorem 9, the headline 'ISS under Assumption 2 implies IPSS'. The proof reduces to showing the scaled system x_dot=f(t,x,nu phi(|x|)) is D-URGAS, and this is justified by 'following the proof of [19, Lemma 2.12]'. That lemma is time-invariant, and the time-varying version is neither stated nor proved. Since the paper itself notes that time-varying ISS systems may lack dissipation-form Lyapunov functions, you can't automatically port a time-invariant proof that relies on a converse Lyapunov step. Assumption 2 gives uniform Lipschitzness and a.e. continuity in t, but the paper doesn't show that rescues the argument. The stress-test note is right: this is a load-bearing gap, not a stylistic one. Without this step, Theorem 9 and the claim that ISS implies IPSS under Assumption 2 are unsupported. I suspect the gap is fillable—ISS estimates can often be turned into a state-dependent small-gain argument—but it needs to be written out or cited precisely.

The citation practice is fine: self-citations appear for motivating examples and moving-average norm equivalence, not to assume the central claim. No fitting; this is theorem-based.

Who this is for: control theorists working on ISS/Lyapunov theory for time-varying systems. Not a paradigm shift, but IPSS is a natural missing piece and Theorem 5 is likely to be cited. I'd send it to peer review and ask the authors to substantiate the D-URGAS step in Theorem 9, either by proving the time-varying extension of the Sontag-Wang lemma or by giving a precise reference. If they do, the paper is rigorous. As it stands, it's conditional.

Referee Report

3 major / 5 minor

Summary. The paper introduces input-power-to-state stability (IPSS) for time-varying systems whose dynamics are not necessarily continuous in time. IPSS bounds the state norm by a KL term in the initial condition and a K-infinity function of the maximum moving-average of a weighted input norm, so it remains informative for inputs with unbounded amplitude and unbounded energy. The main results are: (a) Theorem 5, a dissipation-form ISS-Lyapunov function implies IPSS; (b) Proposition 6, an iISS estimate with a linear exponential KL bound implies IPSS via a new comparison lemma; and (c) Theorem 9, under a uniform-in-time Lipschitz assumption (Assumption 2), ISS implies the existence of a dissipation-form ISS-Lyapunov function and hence IPSS. The proof of (c) relies on a converse Lyapunov theorem for D-URGAS systems with time-discontinuous right-hand sides (Theorem 11), proved in Section 4.4 along the lines of a known time-invariant construction.

Significance. If the results hold, the paper gives a useful ISS-type notion for time-varying systems with moving-average input bounds and clarifies the gap between ISS and iISS in the time-varying setting. The paper is theorem-based with no fitted parameters; Theorem 5 and Lemma 7 are self-contained, and the converse Lyapunov theorem for D-URGAS systems with merely measurable time dependence is potentially valuable in its own right. The main caveat is that the central implication ISS implies IPSS depends on an unproved time-varying extension of a known time-invariant lemma, as detailed below.

major comments (3)

[Section 4.2, proof of Theorem 9, Eq. (25)] The proof asserts that, because the original system (1) is ISS, the scaled time-varying system x_dot = f(t, x, nu phi(|x|)) is D-URGAS by 'following the proof of [19, Lemma 2.12]'. That lemma is a time-invariant result, and its proof in [19] uses the existence of a smooth dissipation-form ISS-Lyapunov function. The present paper itself recalls (Section 1 and [3]) that time-varying ISS systems may lack dissipation-form ISS-Lyapunov functions, and Assumption 2 is not shown to restore the converse. Since Theorem 9 and the whole route from ISS to IPSS depend on this step, the D-URGAS claim for (25) is a load-bearing gap that must be closed by a proof, not by an appeal to following a time-invariant lemma.
[Section 4.4, proof of Theorem 11, Eqs. (33)-(37)] The construction of V from the sequence W_k follows a known time-invariant converse Lyapunov theorem, but the paper does not verify the time-varying hypotheses in detail. In particular, the estimate (33) from Sontag's Lemma is cited without statement, and the passage from D-URGAS to the time-uniform Lipschitz bounds on W_k is only sketched. Since the uniform Lipschitz property of V is essential in the proof of Theorem 9, the authors should either state and prove the time-varying version of the auxiliary lemma or give a precise reference with a verification that all hypotheses are satisfied.
[Proposition 6 and Lemma 7] Proposition 6 assumes the specific linear-in-r KL bound beta(r,t) = K r e^{-lambda t}. The abstract and the conclusion state more broadly that 'iISS with exponential class-KL function implies IPSS'. If 'exponential class-KL' is intended to mean beta(r,t) = c(r) e^{-lambda t} for arbitrary c in K-infinity, then Lemma 7 does not prove that case, because its iteration argument uses the linear dependence on g(t0) with a fixed constant K. The wording should be aligned with the actual assumption of Proposition 6.

minor comments (5)

[Abstract and Section 1] The abstract says 'necessary and sufficient conditions for IPSS are developed', but the paper's main results are sufficient conditions (Theorems 5 and 9, Proposition 6) plus the necessary condition in Lemma 2 that IPSS implies ISS and iISS. The phrasing should be softened to avoid overclaiming.
[Section 4.2, footnote 2] The footnote states that a C-infinity function phi actually exists but C1 suffices. If this is not used later, it can be omitted; if it is used, the regularity requirement should be stated explicitly in the theorem.
[Section 2.3, Definition 2] The sentence 'Note that ||u||_{rho,T} is not necessarily a norm' could be expanded to note that for a given rho in K-infinity, ||u||_{rho,T} is finite for all locally essentially bounded inputs, which is what makes the definition applicable to the stated input class.
[Section 4.4, Eq. (33)] The notation 'Sontag's Lemma [18, Proposition 7]' is not self-contained; the lemma should be stated in the preliminaries or quoted in full, since the estimate is crucial for the boundedness of W_k.
[References] Reference [15] is cited as the source for Theorem B.31, but several other results in the proof of Theorem 11 are attributed to the same book with different notation; a more precise citation map (lemma or theorem numbers) would help the reader verify the time-varying modifications.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main implications are derived from Lyapunov inequalities and external theorems, with no fitted parameters or self-referential definitions.

full rationale

The paper's derivation chain is theorem-based and self-contained against external benchmarks. Theorem 5 derives IPSS from a dissipation-form ISS-Lyapunov inequality via an explicit kappa-circumflex-V transformation and average-power estimate; this is a genuine sufficient-condition proof, not a restatement of the definition. Theorem 9 uses Assumption 2 and ISS to build a scaled system and then invokes the converse Lyapunov Theorem 11, whose proof is given in Section 4.4 following Mironchenko's time-invariant construction; the result does not assume IPSS or a dissipation-form Lyapunov function as input. The only load-bearing external citation is [19, Lemma 2.12], used to assert D-URGAS of the scaled time-varying system; this is a time-invariant lemma whose time-varying extension is asserted but not proved in the manuscript, which is a rigor/completeness gap rather than a circular reduction. Self-citations [6,12] appear only for background counterexamples and equivalence of moving-average norms, not for the central claim. No fitted parameters are renamed as predictions and no quantity is defined in terms of the target result, so no circular step can be exhibited.

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

The paper's central claim rests on standard assumptions on the dynamics plus one unproved extension of a time-invariant lemma to the time-varying setting. There are no fitted free parameters or new physical entities.

assumptions (4)

domain assumption Assumption 1: f(t,0,0)=0, measurable in t, continuous in (x,u), bounded on bounded sets
Ensures local existence, continuation, and well-posedness of solutions for all inputs. This is the standing assumption throughout.
domain assumption Assumption 2: f is uniformly Lipschitz in (x,u) and continuous in (x,u) outside a zero-measure set in t
Needed for Theorem 9, which derives IPSS from ISS; it is a strengthening of Assumption 1.
ad hoc to paper The Sontag-Wang Lemma 2.12 on input scaling, originally for time-invariant systems, extends to time-varying systems under Assumption 2
Theorem 9's proof asserts without proof that ISS implies D-URGAS of the scaled system (22) via 'following the proof of [19, Lemma 2.12]', which is a time-invariant statement.
standard math Sontag's Lemma on KL functions (existence of θ1,θ2) and the converse Lyapunov construction of Mironchenko [15, Theorem B.31]
Used in the proof of Theorem 11 to construct the Lyapunov function V and to bound W_k. These are cited results with no proof included.

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Pith. "Pith review of Input-Power-to-State Stability of Time-Varying Systems." pith.science (2026). https://pith.science/paper/3KT3BBBX

@misc{pith2026250524805,
  author       = {Pith},
  title        = {Pith review of: Input-Power-to-State Stability of Time-Varying Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KT3BBBX}},
  note         = {Machine review of arXiv:2505.24805}
}

abstract

When the state of a system may remain bounded even if both the input amplitude and energy are unbounded, then the state bounds given by the standard input-to-state stability (ISS) and integral-ISS (iISS) properties may provide no useful information. This paper considers an ISS-related concept suitable in such a case: input-power-to-state stability (IPSS). Necessary and sufficient conditions for IPSS are developed for time-varying systems under very mild assumptions on the dynamics. More precisely, it is shown that (a) the existence of a dissipation-form ISS-Lyapunov function implies IPSS, but not necessarily that of an implication-form one, (b) iISS with exponential class-$\KL$ function implies IPSS, and (c) ISS and stronger assumptions on the dynamics imply the existence of a dissipation-form ISS-Lyapunov function and hence IPSS. The latter result is based on a converse Lyapunov theorem for time-varying systems whose dynamics (i.e. state derivative) is not necessarily continuous with respect to time.

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Works this paper leans on

19 extracted references · 19 canonical work pages

[19]

E. D. Sontag and Y. Wang. On characterizations of the input- to-state stability property. Systems and Control Letters, 24:351–359, 1995. 11

work page 1995
[3]

Edwards, Y

H.A. Edwards, Y. Lin, and Y. Wang. On input-to-state stability for time varying nonlinear systems. InProceedings of the 39th IEEE Conference on Decision and Control, volume 4, pages 3501–3506, 2000

work page 2000
[1]

Angeli and D

D. Angeli and D. Neˇ si´ c. Power characterizations of input- to-state stability and integral input-to-state stability. IEEE Trans. on Automatic Control, 46(8):1298–1303, 2001

work page 2001
[2]

Angeli, E

D. Angeli, E. D. Sontag, and Y. Wang. A characterization of integral input-to-state stability. IEEE Trans. on Automatic Control, 45(6):1082–1097, 2000

work page 2000
[4]

Efimov and E

D. Efimov and E. Fridman. On ISS with respect to average value of disturbances: A time-delay approach. IEEE Trans. on Automatic Control, 69(5):3434–3440, 2024

work page 2024
[5]

E. I. Grøtli, E. Panteley, A. Chaillet, and J. T. Gravdahl. Robustness of ISS systems to inputs with limited moving average: Application to spacecraft formations. Int. Journal of Robust and Nonlinear Control, 26:816–833, 2016

work page 2016
[6]

Haimovich and J

H. Haimovich and J. L. Mancilla-Aguilar. ISS implies iISS even for switched and time-varying systems (if you are careful enough). Automatica, 104:154–164, 2019

work page 2019
[7]

J. P. Hespanha, D. Liberzon, and A. Teel. Lyapunov conditions for input-to-state stability of impulsive systems. Automatica, 44(11):2735–2744, 2008

work page 2008

Show all 19 references

[8]

Karafyllis and Z.-P

I. Karafyllis and Z.-P. Jiang. Stability and stabilization of nonlinear systems. Springer London, 2011

2011
[9]

Karafyllis and Z.-P

I. Karafyllis and Z.-P. Jiang. A vector small-gain theorem for general non-linear control systems. IMA Journal of Mathematical Control and Information, 28(3):309–344, 2011

2011
[10]

H. Khalil. Nonlinear Systems. Prentice-Hall, New Jersey, 3rd edition, 2002

2002
[11]

S. Liu, A. Tanwani, and D. Liberzon. ISS and integral- ISS of switched systems with nonlinear supply functions. Mathematics of Control, Signals, and Systems, 34:297–327, 2022

2022
[12]

J. L. Mancilla-Aguilar and H. Haimovich. On zero-input stability inheritance for time-varying systems with decaying- to-zero input power. Systems and Control Letters, 104:31–37, 2017

2017
[13]

J. L. Mancilla-Aguilar, H. Haimovich, and R. A. Garc ´ ıa. Global stability results for switched systems based on weak Lyapunov functions. IEEE Trans. on Automatic Control, 62(6):2764–2777, 2017

2017
[14]

J. L. Mancilla-Aguilar, J. E. Rojas-Ruiz, and H. Haimovich. Characterization of integral input-to-state stability for nonlinear time-varying systems of infinite dimension. SIAM J. Control and Optimization, 61(4):1979–2003, 2023

1979
[15]

Mironchenko

A. Mironchenko. Input-to-State Stability: Theory and Applications. Springer Cham, 01 2023

2023
[16]

Praly and Y

L. Praly and Y. Wang. Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to- state stability. Mathematics of Control, Signals and Systems, 9:1–33, 1996

1996
[17]

E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. on Automatic Control, 34:435– 443, 1989

1989
[18]

E. D. Sontag. Comments on integral variants of ISS. Systems & Control Letters, 34(1):93 – 100, 1998

1998

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

E. D. Sontag and Y. Wang. On characterizations of the input- to-state stability property. Systems and Control Letters, 24:351–359, 1995. 11

work page 1995

[2] [3]

Edwards, Y

H.A. Edwards, Y. Lin, and Y. Wang. On input-to-state stability for time varying nonlinear systems. InProceedings of the 39th IEEE Conference on Decision and Control, volume 4, pages 3501–3506, 2000

work page 2000

[3] [1]

Angeli and D

D. Angeli and D. Neˇ si´ c. Power characterizations of input- to-state stability and integral input-to-state stability. IEEE Trans. on Automatic Control, 46(8):1298–1303, 2001

work page 2001

[4] [2]

Angeli, E

D. Angeli, E. D. Sontag, and Y. Wang. A characterization of integral input-to-state stability. IEEE Trans. on Automatic Control, 45(6):1082–1097, 2000

work page 2000

[5] [4]

Efimov and E

D. Efimov and E. Fridman. On ISS with respect to average value of disturbances: A time-delay approach. IEEE Trans. on Automatic Control, 69(5):3434–3440, 2024

work page 2024

[6] [5]

E. I. Grøtli, E. Panteley, A. Chaillet, and J. T. Gravdahl. Robustness of ISS systems to inputs with limited moving average: Application to spacecraft formations. Int. Journal of Robust and Nonlinear Control, 26:816–833, 2016

work page 2016

[7] [6]

Haimovich and J

H. Haimovich and J. L. Mancilla-Aguilar. ISS implies iISS even for switched and time-varying systems (if you are careful enough). Automatica, 104:154–164, 2019

work page 2019

[8] [7]

J. P. Hespanha, D. Liberzon, and A. Teel. Lyapunov conditions for input-to-state stability of impulsive systems. Automatica, 44(11):2735–2744, 2008

work page 2008