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arxiv: 1906.09109 · v1 · pith:C2BFYW37new · submitted 2019-06-19 · 📡 eess.SY · cs.SY· math.OC

Finite-Time Stability of Hybrid Systems: A Multiple Generalized Lyapunov Functions Approach

Kunal Garg , Dimitra Panagou This is my paper

Pith reviewed 2026-05-25 19:50 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords finite-time stabilityhybrid systemsmultiple Lyapunov functionsgeneralized Lyapunov functionsdwell timeswitching systemsstability analysis
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The pith

Finite-time stability of hybrid systems holds even when generalized Lyapunov functions increase during flows and jumps, if a convergent mode stays active long enough.

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

The paper develops sufficient conditions for finite-time stability in a class of hybrid systems by introducing multiple generalized Lyapunov functions. It demonstrates that stability is possible despite growth in these functions between switches, as long as the finite-time convergent mode operates for a sufficient duration to dominate any increases. This relaxes prior requirements that functions must strictly decrease during flows and remain non-increasing at jumps, yielding less conservative results. A numerical example confirms the conditions work in practice.

Core claim

The origin of the hybrid system is finite-time stable under the existence of multiple generalized Lyapunov functions where the functions may increase during continuous flows and discrete jumps, provided the finite-time convergent mode is active for a dwell time long enough to ensure overall finite-time convergence.

What carries the argument

Multiple generalized Lyapunov functions paired with a minimum dwell-time requirement on the finite-time convergent mode.

If this is right

  • The stability criteria apply to hybrid systems whose Lyapunov functions are permitted to increase temporarily.
  • Prior results requiring monotonic decrease become special cases of the new conditions.
  • Controller design can prioritize sufficient activation time for convergent modes rather than global decrease.
  • The approach certifies stability for a wider set of switching sequences that include growth phases.

Where Pith is reading between the lines

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

  • The dwell-time condition could be used to schedule modes in applications such as switched power systems or legged robots to guarantee finite-time settling.
  • Extensions might relax the single convergent mode assumption to multiple modes with varying convergence rates.
  • Similar multiple-function techniques could be explored for asymptotic stability or input-to-state stability of hybrid systems.

Load-bearing premise

There exist multiple generalized Lyapunov functions and at least one finite-time convergent mode whose activation time is long enough to overcome growth occurring in other modes.

What would settle it

Construct or simulate a hybrid system satisfying the multiple-function and dwell-time conditions yet whose trajectories fail to reach the origin in finite time.

Figures

Figures reproduced from arXiv: 1906.09109 by Dimitra Panagou, Kunal Garg.

Figure 1
Figure 1. Figure 1: Conditions (i), (ii) and (iii) of Theorem [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Increment in the value of VF during the switched-off period (condition (v) of Theorem 2). Condition (i) means that at switching instants, the cumulative value of the differences between the consecutive Lyapunov functions is bounded by a class-GK function; condition (ii) means that the cumulative increment in the value of the individual Lyapunov functions when the respective modes are active, is bounded by … view at source ↗
Figure 3
Figure 3. Figure 3: Switching signal σf (t) for the considered hybrid system. 0 2 4 6 8 10 Time [sec] -3 -2 -1 0 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: System states x1(t) and x2(t) for t ∈ [0, 10] sec. 0 10 20 30 40 50 Time [sec] 10-6 10-4 10-2 100 102 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The norm of the state vector x(t) for the considered hybrid system. IV. CONCLUSIONS AND FUTURE WORK In this paper, we studied FTS of hybrid systems. We showed that under some mild conditions on the bounds on the difference of the values of generalized Lyapunov functions, if the FTS mode is active for a minimum required time, then FTS can be achieved. Our proposed method allows the individual generalized Ly… view at source ↗
Figure 6
Figure 6. Figure 6: Generalized Lyapunov functions Vi(t) for t ∈ [0, 10] sec for the considered hybrid system. both spatial and temporal requirements. More specifically, we are investigating how to incorporate prescribed-time, rather than merely finite-time, convergence for the system modes, so that the overall framework can be used in the synthesis and analysis of controllers with spatiotemporal specifications. APPENDIX A PR… view at source ↗
read the original abstract

This paper studies finite-time stability of a class of hybrid systems. We present sufficient conditions in terms of multiple generalized Lyapunov functions for the origin of the hybrid system to be finite-time stable. More specifically, we show that even if the value of the generalized Lyapunov functions increase between consecutive switches, finite-time stability can be guaranteed if the finite-time convergent mode is active long enough. In contrast to earlier work where the Lyapunov functions are required to be decreasing during the continuous flows and non-increasing at the discrete jumps, we allow the generalized Lyapunov functions to increase \emph{both} during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the related literature. Numerical example demonstrates the efficacy of the proposed methods.

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

3 major / 2 minor

Summary. The paper claims to provide sufficient conditions, in terms of multiple generalized Lyapunov functions, for finite-time stability of a class of hybrid systems. The central result asserts that finite-time stability holds even when the generalized Lyapunov functions increase during both continuous flows and discrete jumps, provided a designated finite-time convergent mode remains active for a dwell time long enough to dominate any growth in the other modes. This is presented as less conservative than prior literature that requires the Lyapunov functions to be non-increasing.

Significance. If the stated conditions are shown to be sufficient and to exclude pathological executions, the result would extend finite-time stability analysis to switched hybrid systems in which temporary growth of the Lyapunov functions is permitted. The numerical example is noted as demonstrating the approach, but the absence of reproducible code, machine-checked proofs, or explicit falsifiable predictions limits the immediate impact.

major comments (3)
  1. [Main stability theorem (likely Theorem 3.1 or 4.1)] The central claim requires that the finite-time convergent mode accumulate sufficient continuous-time activation to satisfy a differential inequality of the form dot V <= -c V^alpha (alpha < 1). No section or theorem statement supplies an explicit lower bound on the dwell time of this mode that is uniform across all executions and independent of the switching signal.
  2. [Hybrid system definition and stability conditions] Hybrid systems can admit Zeno executions in which inter-event times approach zero. The manuscript provides no argument that the proposed dwell-time condition on the convergent mode, or the hybrid time-domain restrictions, precludes Zeno behavior; without such an argument the 'long enough' activation requirement cannot be guaranteed on every solution.
  3. [Statement of main result] The abstract states that the functions may increase 'both during the continuous flows and the discrete jumps,' yet the precise growth bounds (e.g., the constants or functions quantifying the increase at jumps versus the decay rate in the convergent mode) are not visible in the provided abstract and must be checked against the full derivation for consistency.
minor comments (2)
  1. [Abstract] The abstract refers to 'generalized Lyapunov functions' without an immediate definition or reference to the precise class of functions employed; a short clarifying sentence would improve readability.
  2. [Abstract] The numerical example is mentioned but its specific system equations, switching signal, and computed dwell times are not summarized in the abstract; adding one sentence would help readers assess the example's relevance.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below, indicating revisions where appropriate to strengthen the presentation while preserving the core contributions.

read point-by-point responses

  1. Referee: [Main stability theorem (likely Theorem 3.1 or 4.1)] The central claim requires that the finite-time convergent mode accumulate sufficient continuous-time activation to satisfy a differential inequality of the form dot V <= -c V^alpha (alpha < 1). No section or theorem statement supplies an explicit lower bound on the dwell time of this mode that is uniform across all executions and independent of the switching signal.

    Authors: The theorem states sufficient conditions that must hold for a given switching signal: the total activation time of the convergent mode must be long enough for its decay to dominate the possible growth during flows and jumps in other modes. The required duration is derived from the specific growth constants and is therefore signal-dependent rather than a single uniform lower bound independent of the signal. This is standard in multiple-Lyapunov analyses of switched systems. In revision we will add an explicit formula for the minimal required activation time expressed in terms of the growth bounds, decay rate, and number of switches. revision: partial

  2. Referee: [Hybrid system definition and stability conditions] Hybrid systems can admit Zeno executions in which inter-event times approach zero. The manuscript provides no argument that the proposed dwell-time condition on the convergent mode, or the hybrid time-domain restrictions, precludes Zeno behavior; without such an argument the 'long enough' activation requirement cannot be guaranteed on every solution.

    Authors: We agree that an explicit argument ruling out Zeno executions is needed to guarantee the activation-time condition on every solution. The manuscript implicitly relies on the standard hybrid time-domain definition (Goebel et al.) but does not prove that the dwell-time requirement precludes Zeno behavior. We will add a short lemma or remark showing that the positive lower bound on activation time of the convergent mode, combined with the hybrid time-domain axioms, implies that only finitely many switches can occur in finite time, thereby excluding Zeno executions. revision: yes

  3. Referee: [Statement of main result] The abstract states that the functions may increase 'both during the continuous flows and the discrete jumps,' yet the precise growth bounds (e.g., the constants or functions quantifying the increase at jumps versus the decay rate in the convergent mode) are not visible in the provided abstract and must be checked against the full derivation for consistency.

    Authors: The abstract is intentionally a high-level summary of the main idea. The precise growth bounds (linear or exponential growth during flows, multiplicative jumps, and the finite-time decay inequality in the convergent mode) are stated explicitly in the assumptions preceding Theorem 3.1 and are used throughout the proof. The abstract does not claim specific numerical values and is therefore consistent with the detailed statements in the body. No revision to the abstract is required. revision: no

Circularity Check

0 steps flagged

No circularity; derivation self-contained

full rationale

The paper derives sufficient conditions for finite-time stability of hybrid systems via multiple generalized Lyapunov functions, permitting increases during flows/jumps provided a designated convergent mode has sufficient dwell time. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are visible in the abstract or description that reduce the central claim to its inputs by construction. The approach extends standard Lyapunov analysis to hybrid settings without self-referential loops. External concerns such as Zeno executions affect correctness but do not constitute circularity under the defined criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated or derivable.

pith-pipeline@v0.9.0 · 5656 in / 906 out tokens · 23688 ms · 2026-05-25T19:50:35.694491+00:00 · methodology

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Reference graph

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