pith. sign in

arxiv: 1907.04580 · v1 · pith:LFG22NYZnew · submitted 2019-07-10 · 🧮 math.OC

Stabilization on periodic impulse control systems

Shulin Qin , Gengsheng Wang , Huaiqiang Yu This is my paper

Pith reviewed 2026-05-24 23:52 UTC · model grok-4.3

classification 🧮 math.OC
keywords stabilizationimpulse controlperiodic impulsesRiccati equationfeedback designdynamic programmingKalman decompositionlinear systems
0
0 comments X

The pith

Linear impulse control systems with periodic impulses can be stabilized by feedback laws designed from a discrete Riccati equation.

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

The paper examines stabilization of linear finite-dimensional systems controlled by impulses that occur at periodic times. It gives multiple ways to characterize when stabilization is possible. Methods are shown for designing the feedback laws that achieve it. Locations for the impulse instants are also provided to make stabilization work. A reader would care because this turns a continuous control problem into a discrete one that can be solved using standard optimization tools.

Core claim

This paper presents several characterizations on the stabilization of linear impulse control systems with periodic impulse instants in finite-dimensional spaces. It shows how to design feedback laws and provides locations for impulse instants to ensure the stabilization. In the proofs, a discrete LQ problem is set up, a discrete dynamic programming principle is derived, a variant of Riccati's equation is built, the Kalman controllability decomposition is applied repeatedly, and a controllability result from prior work is used.

What carries the argument

Variant of Riccati's equation derived via discrete dynamic programming on the reduced impulse system, which determines the stabilizing feedback.

Load-bearing premise

The system is linear and finite-dimensional, and the controllability result used in the proofs holds for the chosen impulse period and system matrices.

What would settle it

A concrete linear system and choice of periodic impulse period where the Riccati-derived feedback does not drive all states to zero over time.

read the original abstract

This paper studies the stabilization for a kind of linear and impulse control systems in finite-dimensional spaces, where impulse instants appear periodically. We present several characterizations on the stabilization; show how to design feedback laws; and provide locations for impulse instants to ensure the stabilization. In the proofs of these results, we set up a discrete LQ problem; derived a discrete dynamic programming principle, built up a variant of Riccati's equation; applied repeatedly the Kalman controllability decomposition; and used a controllability result built up in [17].

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

1 major / 0 minor

Summary. The manuscript studies stabilization for linear finite-dimensional impulse control systems with periodic impulse instants. It claims to present several characterizations of stabilization, methods for designing feedback laws, and selections of impulse locations. The proofs reduce the problem to a discrete LQ formulation, derive a dynamic programming principle, construct a variant of the Riccati equation, apply the Kalman controllability decomposition, and invoke a controllability result from reference [17].

Significance. If the claimed characterizations hold without hidden restrictions, the results would supply concrete tools for stabilizing periodic-impulse systems by linking them to discrete-time LQ theory. The explicit use of Kalman decomposition and Riccati methods is a methodological strength that could make the feedback-design and impulse-location statements falsifiable in principle.

major comments (1)
  1. [Abstract] Abstract: the stabilization characterizations, feedback design, and impulse-location results rest on repeated application of the controllability result from [17] to the discrete-time system obtained by periodic sampling. No verification is supplied that this result from [17] applies for arbitrary fixed period T and arbitrary (A,B) pairs; if [17] requires aperiodic sampling times, a minimum dwell time, or rank conditions that fail for some T, then discrete controllability (and hence the Riccati-based stabilizability) does not follow in general, undermining the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a potential gap in the justification of the controllability result from [17]. We address the single major comment below and indicate the revision that will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the stabilization characterizations, feedback design, and impulse-location results rest on repeated application of the controllability result from [17] to the discrete-time system obtained by periodic sampling. No verification is supplied that this result from [17] applies for arbitrary fixed period T and arbitrary (A,B) pairs; if [17] requires aperiodic sampling times, a minimum dwell time, or rank conditions that fail for some T, then discrete controllability (and hence the Riccati-based stabilizability) does not follow in general, undermining the central claims.

    Authors: We thank the referee for this observation. The controllability result cited from [17] is stated for general sequences of sampling instants and does not impose aperiodicity or a minimum dwell time; it requires only that the resulting discrete-time pair satisfies the standard rank condition for controllability. In our setting the periodic sampling with fixed T produces a discrete-time system to which the Kalman decomposition is applied, and the hypotheses of [17] hold for any T>0 under the paper's standing assumptions on (A,B). Nevertheless, the manuscript does not explicitly verify these hypotheses for arbitrary T, so we will add a short clarifying paragraph (or remark) in the revised version that recalls the precise conditions of [17] and confirms they are met by the periodic sampling. This addition improves transparency without altering any theorems or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard techniques plus external citation

full rationale

The paper's proof outline (discrete LQ from periodic sampling, dynamic programming principle, Riccati variant, repeated Kalman decomposition, controllability result from [17]) follows conventional control-theoretic steps without any exhibited reduction of a claimed prediction or characterization to a fitted parameter or self-defined quantity by construction. The citation to [17] is noted but does not trigger self-citation load-bearing circularity under the rules, as no evidence shows the central claims reduce solely to an unverified self-citation chain or that [17] itself is invoked to forbid alternatives in a definitional manner. The derivation remains self-contained against the provided excerpts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted from the provided text.

pith-pipeline@v0.9.0 · 5603 in / 1018 out tokens · 19764 ms · 2026-05-24T23:52:33.008309+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Ai, Stabilization and optimization of linear systems via pathw ise state-feedback impulsive control, Journal of the Franklin Institute, 354(3) (2017) 1637-1657

    Z. Ai, Stabilization and optimization of linear systems via pathw ise state-feedback impulsive control, Journal of the Franklin Institute, 354(3) (2017) 1637-1657

    work page 2017

  2. [2]

    Bensoussan, J.-L

    A. Bensoussan, J.-L. Lions, Impulse Control and Quasi-Variational Inequalities , Bordas, Paris, 1984

    work page 1984

  3. [3]

    H. Chen, X. Li, J. Sun, Stabilization, controllability and optimal control of Boo lean net- works with impulsive effects and state constraints , IEEE Transactions on Automatic Control, 60(3)(2015) 806-811

    work page 2015

  4. [4]

    Z. Guan, T. Qian, X. Yu, Controllability and observability of linear time-varying impulsive systems, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 49 (2002) 119 8-1208

    work page 2002

  5. [5]

    Z. Guan, T. Qian, X. Yu, On controllability and observability for a class of impulsi ve systems , Systems Control Lett., 47 (2002) 247-257

    work page 2002

  6. [6]

    Kurepa, On The Quadratic Functional , Publications de l’Institut Mathmatique, 13(19) (1959) 57-72

    S. Kurepa, On The Quadratic Functional , Publications de l’Institut Mathmatique, 13(19) (1959) 57-72

    work page 1959

  7. [7]

    Lakshmikantham, D.D

    V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations , World Scientific Publishing Co. Pte. Ltd., 1989

    work page 1989

  8. [8]

    Lawrence, Stabilizability of Linear Impulsive Systems , 2011 American Control Confer- ence, (2011) 4328-4333

    D.A. Lawrence, Stabilizability of Linear Impulsive Systems , 2011 American Control Confer- ence, (2011) 4328-4333

    work page 2011

  9. [9]

    Lawrence, On Output Feedback Stabilization for Linear Impulsive Syst ems, 2012 Amer- ican Control Conference, (2012) 5936-5941

    D.A. Lawrence, On Output Feedback Stabilization for Linear Impulsive Syst ems, 2012 Amer- ican Control Conference, (2012) 5936-5941

    work page 2012

  10. [10]

    Y. Li, X.Z. Liu, H. Zhang, Dynamical analysis and impulsive control of a new hyperchao tic system, Math. Comput. Modelling, 42 (2005) 1359-1374

    work page 2005

  11. [11]

    X. Z. Liu, Impulsive control and optimization , Appl. Math. Comput. 73 (1995) 77-98. 22

    work page 1995

  12. [12]

    Medina, D.A

    E.A. Medina, D.A. Lawrence, State feedback stabilization of linear impulsive systems , Auto- matica, 45 (6) (2009) 1476-1480

    work page 2009

  13. [13]

    Medina, D.A

    E.A. Medina, D.A. Lawrence, Feedback-reversibility and reachability of linear impuls ive sys- tems, Automatica, 46(6) (2010) 1101-1106

    work page 2010

  14. [14]

    Medina, D.A

    E.A. Medina, D.A. Lawrence, Output Feedback Stabilization for Linear Impulsive System s, 2010 American Control Conference, (2010) 4211-4216

    work page 2010

  15. [15]

    Y. Peng, X. Xiang, A class of nonlinear impulsive differential equation and opt imal controls on time scales , Discrete Contin. Dyn. Syst. Ser. B, 16 (2011) 1137-1155

    work page 2011

  16. [16]

    Phung, G

    K.D. Phung, G. Wang, Y. Xu, Impulse output rapid stabilization for heat equations , J. Dif- ferential Equations, 263(8) (2017) 5012-5041

    work page 2017

  17. [17]

    S. Qin, G. Wang, Controllability of impulse controlled systems of heat equa tions coupled by constant matrices, J. Differential Equations, 263 (2017) 6456-6493

    work page 2017

  18. [18]

    Sontag, Mathematical control theory: deterministic finite dimensi onal systems, Vol

    E.D. Sontag, Mathematical control theory: deterministic finite dimensi onal systems, Vol. 6, Springer Science & Business Media, 2013

    work page 2013

  19. [19]

    Tr´ elat, L

    E. Tr´ elat, L. Wang, Y. Zhang, Impulse and sampled-data optimal control of heat equations , and error estimates , SIAM J. Control Optim., 54 (2016) 2787-2819

    work page 2016

  20. [20]

    G. Xie, L. Wang, Controllability and observability of a class of linear impu lsive systems , J. Math. Anal. Appl., 304 (2005) 336-355

    work page 2005

  21. [21]

    Yang, Impulse Control Theory , Springer-Verlag, Berlin, Heidelberg, 2001

    T. Yang, Impulse Control Theory , Springer-Verlag, Berlin, Heidelberg, 2001

    work page 2001

  22. [22]

    J. Yong, P. Zhang, Necessary conditions of optimal impulse controls for distr ibuted parameter systems, Bull. Aust. Math. Soc., 45 (1992) 305-326

    work page 1992

  23. [23]

    S. Zhao, J. Sun, Controllability and observability for a class of time-vary ing impulsive systems , Nonlinear Anal. Real World Appl., 10 (2009) 1370-1380. 23

    work page 2009