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arxiv: 2606.08798 · v1 · pith:T3HHONSPnew · submitted 2026-06-07 · 🧮 math.DS

State-Feedback Control of Logistic-Based Gene Regulatory Networks: Closed-Form Lyapunov Certificates, Monostabilization, and Delay-Uniform Stability

Ismail Belgacem This is my paper

Pith reviewed 2026-06-27 17:36 UTC · model grok-4.3

classification 🧮 math.DS
keywords gene regulatory networksstate-feedback controlLyapunov stabilitylogistic functionsmonostabilizationdelay stabilityglobal exponential stability
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The pith

A feedforward-plus-proportional law turns any positive setpoint into a globally exponentially stable equilibrium of a logistic gene network under an explicit gain condition.

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

The paper constructs an additive state-feedback controller for gene regulatory networks modeled by logistic production terms. It proves that a feedforward term plus proportional gains can enforce any chosen positive concentration vector as a closed-loop equilibrium, independent of the open-loop vector field. Global exponential stability then follows from a common quadratic Lyapunov function whose negative definiteness is guaranteed by the inequality (γ1+K1)(γ2+K2) > κ1 κ2 λ²/64 together with a closed-form decay rate. The same framework supplies a uniform monostabilization gain budget for bistable self-activation switches and a Halanay-type theorem that preserves global exponential stability for arbitrary constant delays once the gains exceed κλ/4.

Core claim

A feedforward-plus-proportional control law renders any positive setpoint a closed-loop equilibrium of a logistic-based GRN and guarantees global exponential stability when (γ1+K1)(γ2+K2) > κ1 κ2 λ²/64, with the proof relying on a common quadratic Lyapunov function constructed from the logistic sector bound; additionally, a parameter-uniform monostabilization budget K* = κλ/4 - γ is established for bistable switches, and Halanay-type delay-uniform GES holds under γ + K > κλ/4.

What carries the argument

The common quadratic Lyapunov function built from the logistic sector bound, whose derivative supplies the explicit global exponential stability inequality for the closed-loop error system.

If this is right

  • Any positive concentration vector can be made an equilibrium by choice of the feedforward term alone.
  • Global exponential stability with an explicit rate holds once the proportional gains satisfy the quadratic inequality.
  • Bistable self-activation switches are monostabilized by any gain exceeding the uniform budget K* = κλ/4 - γ.
  • Global exponential stability remains uniform in delay size whenever γ + K > κλ/4, with two-sided rate bounds.
  • The logistic coupling term remains strictly positive near the origin, unlike the vanishing Hill coupling.

Where Pith is reading between the lines

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

  • The same Lyapunov construction may extend directly to networks with more than two nodes if a suitable diagonal or block-diagonal P matrix can be found.
  • The explicit settling-time bound within 1 percent of simulation suggests the certificates are tight enough for quantitative synthetic-biology design.
  • Because the logistic model preserves network coupling at low concentrations, the controller may remain effective in regimes where Hill-based designs lose controllability.

Load-bearing premise

The logistic production function admits a sector bound that permits a quadratic Lyapunov function whose time derivative is negative definite under the stated gain condition.

What would settle it

A simulation or experiment on a two-node logistic GRN in which the product (γ1+K1)(γ2+K2) is set below κ1 κ2 λ²/64 and the closed-loop trajectories are observed to diverge from the setpoint or converge only locally.

Figures

Figures reproduced from arXiv: 2606.08798 by Ismail Belgacem.

Figure 1
Figure 1. Figure 1: Feedback control of a bistable T-cell switch ( [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Closed-loop response of the logistic two-gene oscillator under the state-feedback [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop response of the Hill two-gene oscillator ( [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Side-by-side trajectories of logistic (solid) and Hill (dashed) state-feedback under [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Setpoint scan exposing the structural advantage of logistic over Hill state [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
read the original abstract

Gene regulatory networks (GRNs) are high-value targets for therapeutic and synthetic-biology control. Classical Hill models carry a structural defect: the production term vanishes when the activator is absent, causing loss of controllability under multiplicative actuation and a collapse of network coupling under additive actuation -- precisely where biological operation is most common. Building on logistic functions as robust Hill alternatives, we develop an additive state-feedback framework for logistic-based GRNs, with two companion scalar results. A feedforward-plus-proportional law turns any positive setpoint into a closed-loop equilibrium, regardless of the uncontrolled dynamics. We prove local exponential stability under a Gershgorin gain bound and, via a common quadratic Lyapunov function built on the logistic sector bound, global exponential stability under the explicit condition $(\gamma_1+K_1)(\gamma_2+K_2)>\kappa_1\kappa_2\lambda^2/64$, with a closed-form rate. A diagonal Lyapunov certificate $P=\mathrm{diag}(B,A)$ yields an explicit settling-time bound (within $1\%$ of simulation) and an ISS ultimate-bound estimate. We further establish a parameter-uniform monostabilization budget $K^{*}=\kappa\lambda/4-\gamma$ for bistable self-activation switches, and a Halanay-type delay-uniform global exponential stability theorem under $\gamma+K>\kappa\lambda/4$, with two-sided closed-form rate bounds. Numerical comparison with the Hill counterpart confirms robust tracking in the nominal range while exposing the structural divergence near the boundary of the positive orthant: the Hill coupling $|[J_f]_{21}|=\Theta(x_{d,1}^{n-1})\to0$ as $x_{d,1}\to0$, whereas the logistic coupling stays strictly positive.

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

0 major / 2 minor

Summary. The paper develops an additive state-feedback control framework for logistic-based gene regulatory networks as an alternative to Hill models. It shows that a feedforward-plus-proportional law renders any positive setpoint a closed-loop equilibrium, proves local exponential stability via Gershgorin and global exponential stability via a common quadratic Lyapunov function on the logistic sector bound under the explicit gain condition (γ1+K1)(γ2+K2) > κ1 κ2 λ^{2}/64 with closed-form rate, provides a parameter-uniform monostabilization budget K* = κλ/4 - γ for bistable switches, and establishes Halanay-type delay-uniform global exponential stability under γ+K > κλ/4, with numerical comparisons highlighting structural advantages over Hill models near the positive orthant boundary.

Significance. If the derivations hold, the explicit closed-form Lyapunov certificates, parameter-uniform monostabilization budget, and delay-uniform stability bounds constitute a practical contribution to control of GRNs, directly addressing controllability loss in Hill models under additive actuation. The reproducible numerical comparisons and use of standard tools (Gershgorin, sector-bound quadratic Lyapunov, Halanay) add value for synthetic-biology applications.

minor comments (2)
  1. [Abstract] Abstract: the claim of an 'explicit settling-time bound (within 1% of simulation)' should be tied to a specific theorem or corollary number and the corresponding simulation figure or table in the main text for reproducibility.
  2. [Global stability paragraph] The logistic sector bound |σ(λe) - σ(0)| ≤ (λ/4)|e| is invoked for the common quadratic Lyapunov function; the main text should explicitly state the logistic function definition and confirm the bound holds with equality only at the origin for the chosen scaling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the thorough and positive assessment of our manuscript on additive state-feedback control for logistic-based GRNs. The recommendation for minor revision is noted. No specific major comments or points requiring clarification were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations follow from standard Lyapunov analysis

full rationale

The central results (global exponential stability under the gain condition, monostabilization budget K*=κλ/4−γ, Halanay delay theorem) are obtained by substituting the logistic sector bound |σ(λe)−σ(0)|≤(λ/4)|e| into the derivative of the diagonal quadratic Lyapunov function V=(1/2)B e1²+(1/2)A e2², then applying the elementary estimate |e1 e2|≤(e1²+e2²)/2 to obtain the explicit threshold (γ1+K1)(γ2+K2)>κ1κ2λ²/64. All steps are direct algebraic consequences of the closed-loop vector field and standard comparison lemmas; no parameter is fitted to data and then re-labeled as a prediction, no self-citation supplies a uniqueness theorem, and no ansatz is smuggled in. The feedforward-plus-proportional law is constructed explicitly to enforce the setpoint equilibrium and does not presuppose the stability result. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard Lyapunov and comparison lemmas plus the modeling choice that logistic functions admit a usable sector bound; no new entities are postulated and no parameters are fitted to data.

axioms (2)
  • standard math Lyapunov stability theorems and comparison lemmas apply to the closed-loop system
    Invoked to obtain global exponential stability and settling-time bounds from the quadratic form.
  • domain assumption Logistic functions satisfy a sector bound permitting a common quadratic Lyapunov function
    Central to the global stability proof described in the abstract.

pith-pipeline@v0.9.1-grok · 5860 in / 1488 out tokens · 24708 ms · 2026-06-27T17:36:00.964088+00:00 · methodology

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Reference graph

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