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Cited by 0Stability and Controllability of Differential EquationsStability and Control of Uncertain SystemsNonlinear Differential Equations Analysis

Multi-layer predictor feedback design for nonlinear integro-differential equations with state-dependent input delays

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Li, Tong, Shang, Peipei, Diagne, Mamadou·2026·Automatica
DOI 10.1016/j.automatica.2026.113282
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Abstract

We develop a novel multi-layer predictor feedback to achieve exact compensation of state-dependent input delays of general nonlinear integro-differential equations. The system of interest is an unconventional mixed Partial Differential Equation (PDE)-Ordinary Differential Equation (ODE) system, in which a nonlinear ODE is actuated through an inhomogeneous advection PDE. Moreover, the propagation speed of the PDE depends on a moving window integral of the ODE state. The two above features are not addressed yet in standard PDE backstepping-based predictor feedback designs. We first address a source term corresponding to a linear recirculation loop. The framework is subsequently generalized to source terms such as a nonlinear recirculation loop and a linear friction term. In addition, in the latter case, we use boundary flux control. We show that the inhomogeneous and doubly nonlinear mixed PDE-ODE system results in a correspondingly “inhomogeneous” control law. The stabilizing controller comprises a nominal delay-free control law based on the predictor feedback state, enhanced by a nonlinear multi-layer prediction mechanism that compensates for the PDE’s additive source terms. Our modular designs are based on two novel nonlinear backstepping transformations: the first enables boundary control of the PDE state, while the second addresses flux control—namely, a nonlinear controlled boundary condition defined as the product of the ODE and PDE states. Unlike the conventional Lyapunov-based approaches used in the field, our stability and well-posedness analysis rely on the characteristic method and a fixed-point argument. Both of our designs achieve global asymptotic stability (GAS) in the supremum norm of the PDE and ODE states under the mild assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous. The transport speed, governed by the integral of the ODE state, models systems such as production or queuing processes in which the state of a finite buffer—namely, the inventory level—determines the production or service rate. Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened “bang–bang” feedback law that preserves the positivity of both the system state and the actuation signal.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper develops a multi-layer predictor-feedback controller for nonlinear integro-differential equations with state-dependent input delays, represented as a mixed PDE-ODE system whose transport speed depends on a moving-window integral of the ODE state. It proves global asymptotic stability and well-posedness via nonlinear backstepping, characteristics, and fixed-point arguments, and demonstrates the design numerically on buffer-regulated production/queuing systems.

Key findings

  • Multi-layer predictor feedback achieves exact compensation of state-dependent input delay for a doubly nonlinear PDE-ODE system with additive source terms: first for a linear recirculation term, then generalized to nonlinear source terms, friction, inflow boundary control, and outflow anti-collocated measurements. Both designs achieve global asymptotic stability in the supremum norm of the PDE and ODE states under uniform Lipschitz continuity of the PDE nonlinearity.
  • The controller combines a nominal delay-free feedback κ evaluated at the predicted ODE state \(p_1(D,t)\) with an integral correction from \(p_2\) and actuator prediction from \(p_3\); the presence of source terms requires a tri-layer predictor-feedback design, whereas prior work without source terms used a bilayer strategy.
  • Numerical simulations for a buffer-driven production line/queuing system use \(Q^*=0.3\), \(\mu=0.8\), \(P=0.25\), \(Q_{\max}=1\), \(B_{\max}=1.2\), \(\alpha=0.5\), \(\tau=0.2\), \(C(x)\equiv 1\), \(A=0.1\), \(N=80\), \(\Delta t\approx 0.0025\), and \(S(Q^*)=S_{\min}(Q^*)+20\). The compensated control achieves smooth rapid stabilization at the set-point without overshoot or oscillation, while uncompensated bang-bang control oscillates and open-loop control converges slowly.
  • Proposition 16 gives a locally safe softened bang-bang control law ensuring \(U(t)\ge 0\) and \((Q,\rho)\ge 0\) under a stated constraint; the guarantee is local due to the source terms.

Methods

  • Theoretical control design for a coupled PDE-ODE system: a nonlinear ODE at \(x=0\) actuated through an inhomogeneous advection PDE on \([0,D]\), with transport speed depending on a moving-window integral of the ODE state and with additive source terms. Assumptions include strong forward completeness, ISS of the delay-free plant, positive bounded transport speed, and uniformly Lipschitz nonlinear source term.
  • Analysis uses two novel nonlinear backstepping transformations, forward and inverse, together with the method of characteristics, norm equivalence in the supremum norm, and a fixed-point argument for well-posedness; stability is proved without the classical Lyapunov approach.
  • Numerical validation uses a first-order upwind finite difference scheme on a uniform grid of \(N=80\) points over \([0,D]\), forward Euler time integration with a CFL-constrained step \(\Delta t\approx 0.0025\), and forward spatial marching to solve the multi-layer predictor integral equations.

Limitations

  • The paper states future work will extend the framework to other classes of hyperbolic PDEs and to networks of coupled PDE-ODE subsystems.
  • The paper states that the safe softened bang-bang positivity guarantee is only local due to source terms, unlike global guarantees in prior work; it also assumes source terms independent of the ODE state and linear friction so the backstepping kernel admits an explicit exponential form.
  • Evident from the text: validation is numerical simulation only, with no physical experiment; the well-posedness proof relies on a fixed-point argument detailed in a prior conference paper [38] for an identical target system.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1016/j.automatica.2026.113282

Cites

59 of 59 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

3 of 19 cited sources checked · 2 supported · 0 flagged · accuracy not yet measured

  • Interpretationreasoning
    § 2.2 Predictor feedback design methodology · p. 5

    System (1)-(3) is no longer equivalent to the classical time-delay system in [5,6,14-17].

    “Equation (31) indicates that system (1)-(3) is no longer equivalent to the following classical time-delay system in [5,6,14-17]”
    Quote in the paperSubstantial claim1 of 6 sources checked
    • cites [5] Bekiaris-Liberis, 2011, Compensation of time-varying input delay for nonlinear systems source
    • cites [6] Bekiaris-Liberis, 2012, Compensation of state-dependent input delay for nonlinear systems, IEEE Transactions on Automatic Control source
    • cites [14] Diagne, 2017, Compensation of input delay that depends on delayed input, Automatica source
    • cites [15] Diagne, 2017, Time- and state-dependent input delay-compensated bang–bang control of a screw extruder for 3D printing, International Journal of Robust and Nonlinear Control source
    • cites [16] Diagne, 2017, Control of transport PDE/nonlinear ODE cascades with state-dependent propagation speed, IEEE Transactions on Automatic Control source
    • cites [17] Diagne, 2015, State-dependent input delay-compensated bang–bang control: Application to 3D printing based on screw-extruder source
  • Findingown data
    § Abstract · p. 1

    Both designs achieve global asymptotic stability in the supremum norm of the PDE and ODE states under the assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous.

    “Both of our designs achieve global asymptotic stability (GAS) in the supremum norm of the PDE and ODE states under the mild assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 2.1 Problem statement and main result · p. 4

    Under Assumptions 1-2, there exists a class KL function L1 such that the closed-loop system (1)-(3) with control law (10)-(14) satisfies a stability estimate for all t ≥ 0.

    “Moreover, there exists a class K L function L 1 such that the following holds:”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system ˙X=f(X,ω) is strongly forward complete with respect to ω.
    • The authors state this as an assumption: The system ˙X=f(X,κ(X)+ω) is input-to-state stable (ISS) with respect to ω, and the feedback law κ is continuously differentiable with κ(0)=0.
  • Findingreasoning
    § 2.1 Problem statement and main result · p. 4

    Under Assumptions 1-2, for all initial conditions h in C0([-τ,0]) and Lipschitz continuous u0 on [0,D] satisfying the compatibility condition, there exists a unique strong solution (X,u) to the closed-loop system (1)-(3) with control law (10)-(14).

    “Theorem 2 Under Assumptions 1-2, for all initial conditions h ∈ C 0 ([ -τ , 0 ]) and Lipschitz continuous u 0 on [ 0 , D ] that satisfies the compatibility condition”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system ˙X=f(X,ω) is strongly forward complete with respect to ω.
    • The authors state this as an assumption: The system ˙X=f(X,κ(X)+ω) is input-to-state stable (ISS) with respect to ω, and the feedback law κ is continuously differentiable with κ(0)=0.
  • Findingreasoning
    § 2.1 Problem statement and main result · p. 5

    There exists a class K∞ function K1 such that the norm equivalence between the original system (1)-(3) and the transformed system (19)-(21) holds for all t ≥ 0.

    “Lemma 5 There exists a class K ∞ function K 1 such that”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 8
  • May combine two claims. 18
  • The quote only partly states this claim. 5

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationnone given
    § 4.1 Governing equations of the buffer ODE and conveyor PDE · p. 9

    Using φ(t) avoids the density-based transformation employed in prior work and feeds the predicted flux directly into (75).

    “By using the ϕ ( t ) , we avoid the density-based transformation employed in prior work (e.g. [9,15,23], and instead feed the predicted flux directly into (75).”
    Quote in the paperSubstantial claim2 of 3 sources checked
    • cites [9] Borsche, 2010, On the coupling of systems of hyperbolic conservation laws with ordinary differential equations, Nonlinearity source
    • cites [15] Diagne, 2017, Time- and state-dependent input delay-compensated bang–bang control of a screw extruder for 3D printing, International Journal of Robust and Nonlinear Control source

      The citation is attached to the claim that prior work employed a density-based transformation. Sections 2.2.3–2.2.4 use density in the extruder’s mass-balance and flow-rate equations, but the passages found do not describe a density-based transformation. Section 5.1 defines φ(t) as an input-delay relation, not as evidence for that attributed prior-work method. (LLM, conf=0.72)

      Experimental — accuracy not yet measured
    • cites [23] Irscheid, 2023, Output regulation for general heterodirectional linear hyperbolic PDEs coupled with nonlinear ODEs, Automatica source

  • May combine two claims.

    Findingnone given
    § 1.2 Contributions · p. 2

    The backstepping transformation maps the original system with the predictor-feedback controller into a target system, and is equivalent to its inverse under the supremum norm.

    “The backstepping transformation maps the original system with the predictor-feedback controller into a target system, and is equivalent to its inverse under the supremum norm.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingnone given
    § 2.2 Predictor feedback design methodology · p. 6

    In the special case where g≡0 and λ=v is constant, the prediction instant simplifies to σ(x,t)=t+x/v, and the predictor-feedback structure reduces to a classical finite-dimensional delay compensator such as the Smith predictor or the Artstein transformation.

    “In the special case where g ≡ 0 and λ = v is a constant, the prediction instant simplifies to σ ( x , t ) = t + x / v, and the predictor-feedback structure reduces to a classical finite-dimensional delay compensator such as the Smith predictor or the Artstein transformation.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 2.1 Problem statement and main result · p. 4

    The inverse of the infinite-dimensional backstepping transformation (18) is given by (20), with π_i defined by coupled integral equations.

    “Lemma 4 The inverse of the infinite-dimensional backstepping transformation (18) is given by”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingnone given
    § 2.1 Problem statement and main result · p. 5

    There exists a class K∞ function K2 such that the inverse norm equivalence holds for all t ≥ 0.

    “Lemma 6 There exists a class K ∞ function K 2 such that”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Abstract · p. 1

    Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened 'bang-bang' feedback law that preserves positivity of both the system state and the actuation signal.

    “Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened 'bang-bang' feedback law that preserves the positivity of both the system state and the actuation signal.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 3 Generalization to nonlinear source terms and boundary flux control · p. 7

    Under Assumptions 1-4, there exists a class KL function L2 such that the closed-loop system (37)-(40) with control law (43)-(48) satisfies a stability estimate for all t ≥ 0.

    “Moreover, there exists a class K L function L 2 such that the following holds:”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system ˙X=f(X,ω) is strongly forward complete with respect to ω.
    • The authors state this as an assumption: The system ˙X=f(X,κ(X)+ω) is input-to-state stable (ISS) with respect to ω, and the feedback law κ is continuously differentiable with κ(0)=0.
    • The authors state this as an assumption: There exist two positive constants λ and ¯λ bounding the transport speed, to guarantee that the control signal reaches the boundary and ensure well-posedness of the control law.
    • The authors state this as an assumption: The source term g satisfies additional conditions reflecting the physical nature of many recycling systems.

  • The quote only partly states this claim.

    Findingreasoning
    § 3 Generalization to nonlinear source terms and boundary flux control · p. 7

    Under Assumptions 1-4, for all initial conditions h in C0([-τ,0]) and Lipschitz continuous u0 on [0,D] satisfying the compatibility condition, there exists a unique strong solution (X,u) to the closed-loop system (37)-(40) with control law (43)-(48).

    “Theorem 9 Under Assumptions 1-4, for all initial conditions h ∈ C 0 ([ -τ , 0 ]) and Lipschitz continuous u 0 on [ 0 , D ] that satisfies the compatibility condition”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system ˙X=f(X,ω) is strongly forward complete with respect to ω.
    • The authors state this as an assumption: The system ˙X=f(X,κ(X)+ω) is input-to-state stable (ISS) with respect to ω, and the feedback law κ is continuously differentiable with κ(0)=0.
    • The authors state this as an assumption: There exist two positive constants λ and ¯λ bounding the transport speed, to guarantee that the control signal reaches the boundary and ensure well-posedness of the control law.
    • The authors state this as an assumption: The source term g satisfies additional conditions reflecting the physical nature of many recycling systems.

  • May combine two claims.

    Findingreasoning
    § 6 Conclusion · p. 12

    Both global asymptotic stability (in the supremum norm of the state and actuator) and well-posedness are established through the characteristic method and a fixed-point argument.

    “both global asymptotic stability (in the supremum norm of the state and actuator) and wellposedness are established through the characteristic method and a fixed-point argument”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 3 Generalization to nonlinear source terms and boundary flux control · p. 7

    The inverse of the infinite-dimensional backstepping transformation (51), which maps the system (52)-(54) to the closed-loop system (37)-(40), is given by (55), with π_i defined by coupled integral equations.

    “Lemma 11 (Inverse Backstepping Transformation) The inverse of the infinite-dimensional backstepping transformation (51) , which maps the system (52) -(54) to the closed-loop system (37) -(40) , is given by”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 5 Simulation Results · p. 11

    In the open-loop and uncompensated cases, the density profile exhibits persistent transients and oscillatory behavior, respectively.

    “In the open-loop and uncompensated cases, the density profile exhibits persistent transients and oscillatory behavior, respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1.2 Contributions · p. 2

    The system is not a simple cascade: the propagation speed depends on the integral of the nonlinear ODE state, and source terms are present in the actuator dynamics.

    “In contrast, our system is not a simple cascade : the propagation speed depends on the integral of the nonlinear ODE state, and source terms are present in the actuator dynamics.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The propagation speed of the PDE depends on a moving window integral of the ODE state.

  • May combine two claims.

    Interpretationreasoning
    § 1.2 Contributions · p. 2

    The well-posedness and stability analysis characterize the transformation along characteristic curves to prove global asymptotic stability of the target system, which by norm equivalence ensures stability of the original closed-loop system.

    “Departing from classical Lyapunov methods, we characterize the transformation along characteristic curves to prove global asymptotic stability of the target system, which, by norm equivalence, ensures the stability of the original closed-loop system.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 1.2 Contributions · p. 2

    The PDE no longer represents a conservation law but includes additive source terms, which may be nonlinear functions of its state.

    “Moreover, the PDE no longer represents a conservation law but includes additive source terms, which may be nonlinear functions of its state-for instance, a 'recirculation loop' inhomogeneity that depend on the uncontrolled boundary.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § 1.2 Contributions · p. 2

    The controller incorporates a compensator that effectively cancels the influence of inhomogeneous terms in the PDE actuator state, without imposing restrictions on the severity of the nonlinearity or the feasibility condition, while guaranteeing the latter through an assumption on the positivity of the transport speed to avoid an ill-posed problem.

    “Our controller incorporates a compensator that effectively cancels the influence of inhomogeneous terms in the PDE actuator state, without imposing any restrictions on the severity of the nonlinearity or the feasibility condition [5,6], while guaranteeing the latter through an assumption on the positivity of the transport speed to avoid an illposed problem (see [14,17]).”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [5] Bekiaris-Liberis, 2011, Compensation of time-varying input delay for nonlinear systems source
    • cites [6] Bekiaris-Liberis, 2012, Compensation of state-dependent input delay for nonlinear systems, IEEE Transactions on Automatic Control source
    • cites [14] Diagne, 2017, Compensation of input delay that depends on delayed input, Automatica source
    • cites [17] Diagne, 2015, State-dependent input delay-compensated bang–bang control: Application to 3D printing based on screw-extruder source
    • The authors state this as an assumption: There exist two positive constants λ and ¯λ bounding the transport speed, to guarantee that the control signal reaches the boundary and ensure well-posedness of the control law.

  • May combine two claims.

    Interpretationnone given
    § 1.2 Contributions · p. 2

    The design follows the 'separation principle,' combining two distinct compensators: one for the underlying homogeneous dynamics, described as a conservation law, and another to counteract the inhomogeneous source terms.

    “Furthermore, our design follows the 'separation principle,' combining two distinct compensators: one for the underlying homogeneous dynamics, which is described as a conservation law, and another to counteract the inhomogeneous source terms.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1.2 Contributions · p. 2

    The multi-layer predictor feedback design achieves asymptotic stabilization and exact compensation of the actuator dynamics for a coupled doubly nonlinear coupled PDE-ODE system, where the transport speed depends on the integral of the ODE state.

    “We design a multi-layer predictor feedback that achieves asymptotic stabilization and exact compensation of the actuator dynamics for a coupled doubly nonlinear coupled PDE-ODE system, where the transport speed depends on the integral of the ODE state indicating that it is the inventory level-rather than the accumulation of workload at production stages-that drives the production speed.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The propagation speed of the PDE depends on a moving window integral of the ODE state.

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § 1.1 State of the art of predictor feedback design · p. 2

    The nonlinear composite PDE-ODE system is equivalent to a nonlinear integro-differential equation with an implicitly defined state dependent input delay.

    “Our nonlinear composite PDE-ODE system is equivalent to a nonlinear integro-differential equation with an implicitly defined state dependent input delay.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The propagation speed of the PDE depends on a moving window integral of the ODE state.

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning
    § 4.1 Governing equations of the buffer ODE and conveyor PDE · p. 9

    The proposed flux-based approach simplifies implementation and improves numerical efficiency.

    “This simplifies implementation and improves numerical efficiency.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § 2.2 Predictor feedback design methodology · p. 6

    The designs remain valid for PDEs like (2),(3) with time-dependent or spatially varying transport speed.

    “In a broad sense, our designs remain valid for PDEs like (2) , (3) with time-dependent or spatiallyvarying transport speed.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given
    § 6 Conclusion · p. 12

    The proposed design applies to systems with both constant and time-varying transport speeds and retains a separation principle, relying on the design of a nominal control law for the delay-free plant.

    “The proposed design applies to systems with both constant and timevarying transport speeds and retains a separation principle, relying on the design of a nominal control law for the delay-free plant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § 5 Simulation Results · p. 11

    The multi-layer predictor feedback law completely compensates the delay and eliminates the influence of the source term.

    “which means the multi-layer predictor feedback law completely compensate the delay and eliminate the influence of the source term.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning
    § 4.1 Governing equations of the buffer ODE and conveyor PDE · p. 9

    The model (78)-(83) extends the classical transport equation by a state-dependent convection speed ω(R(t)) reflecting buffer congestion, a recycling source S linked to lost material recovery, and a friction term C·ρ accounting for in-process losses.

    “In summary, the model (78)-(83) extends the classical transport equation by (i) a state-dependent convection speed ω ( R ( t )) that reflects buffer congestion, (ii) a recycling source S linked to lost material recovery, and (iii) a friction term C · ρ accounting for in-process losses.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 3 Generalization to nonlinear source terms and boundary flux control · p. 6

    The linear friction term in u ensures that the backstepping kernel (48) admits an explicit exponential form; a nonlinear friction term requires solving an intractable nonlinear PDE.

    “Additionally, the linear friction term in u ensures that the backstepping kernel (48) admits an explicit exponential form; a nonlinear friction term requires solving an intractable nonlinear PDE.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § B.2 Well-posedness of the Closed-Loop System · p. 18

    The proof hinges on a fixed-point argument detailed in prior conference paper [38], where an identical target system was analyzed.

    “The proof hinges on a fixed-point argument detailed in our prior conference paper [38], where an identical target system was analyzed.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [38] Manitius, 1979, Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control source

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given
    § 6 Conclusion · p. 12

    By incorporating flux-based actuation and sensing, the boundary conditions become highly nonlinear, posing challenges beyond those addressed in prior work.

    “By incorporating flux-based actuation and sensing, the boundary conditions become highly nonlinear, posing challenges beyond those addressed in prior work.”
    Quote in the paperNot a substantial claim
26 of 26

  • The authors state this as an assumption.

    The system ˙X=f(X,ω) is strongly forward complete with respect to ω.

    “Assumption 1 The system ˙ X = f ( X , ω ) is strongly forward complete with respect to ω .”

  • The authors state this as an assumption.

    The system ˙X=f(X,κ(X)+ω) is input-to-state stable (ISS) with respect to ω, and the feedback law κ is continuously differentiable with κ(0)=0.

    “Assumption 2 The system ˙ X = f ( X , κ ( X )+ ω ) is input-tostate stable (ISS) with respect to ω , i.e., there exists a class K L function η and a class K function ψ such that”

  • The authors state this as an assumption.

    There exist two positive constants λ and ¯λ bounding the transport speed, to guarantee that the control signal reaches the boundary and ensure well-posedness of the control law.

    “Assumption 3 To guarantee that the control signal reaches the boundary and ensure the well-posedness of the control law, particularly when the actuation is defined as a flux in Section 3, we assume the existence of two positive constants λ and ¯ λ such that”

  • The authors state this as an assumption.

    The source term g satisfies additional conditions reflecting the physical nature of many recycling systems.

    “Furthermore, reflecting the physical nature of many recycling systems, we assume that”

  • The authors state this as an assumption.

    The propagation speed of the PDE depends on a moving window integral of the ODE state.

    “Moreover, the propagation speed of the PDE depends on a moving window integral of the ODE state.”

  • The authors state this as an assumption.

    The spatial friction satisfies C(x) ≥ 0.

    “the spatial friction satisfies C ( x ) ≥ 0”

  • The authors state this as an assumption.

    The function g is Lipschitz continuous with respect to its second argument.

    “As g is assumed to be Lipschitz continuous with respect to its second argument”

  • The authors state this as an assumption.

    The functions f, g, λ, c and the initial data h, u0 have the assumed regularity.

    “the assumed regularity of f , g , λ , c and the initial data h , u 0”
8 of 8

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Simulation parameters violate the stated maximum-input requirement

    Major

    The paper requires the maximum input to be below both the service capacity and the queue upper bound. The simulation sets B max = 1.2, exceeding both μ = 0.8 and Q max = 1.

  • Same referenced article has conflicting publication years

    Minor

    References [33] and [35] give the same article, journal volume, issue, and pages, but list its year as 2009 and 2010 respectively.

  • Characteristic-curve schematic is cited as the wrong figure

    Minor

    The text directs readers to Fig. 2 for the characteristic curves, but the characteristic-curve diagram is captioned Fig. 1; Fig. 2 is captioned as the closed-loop system.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Smith, 1957, Closer control of loops with dead time, Chemical Engineering Progress, 53, 217
  2. Smith, 1959, A controller to overcome dead time, ISA Journal, 6, 28

Authors
Li, T., Shang, P., & Diagne, M.
Affiliations
  • Tongji University (CN)
  • University of California San Diego (US)
Corresponding author
Diagne, Mamadou
ORCID
Venue
Automatica
Publisher
Elsevier BV
Published
2026-12
Volume
194
Pages
113282
ISSN
0005-1098 (ISSN-L)
DOI
10.1016/j.automatica.2026.113282
OpenAlex
W7152680300
Cited by
0
References
41
FWCI
0.00
Type
journal article
Language
en
Open access
Licence
Locations
  • Automatica (publishedVersion, cc-by-nc-nd)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (CMMI-2222250)
  • National Natural Science Foundation of China (12171368)
Topics
  • Physical Sciences › Engineering › Control and Systems Engineering › Stability and Controllability of Differential Equations
  • Physical Sciences › Engineering › Control and Systems Engineering › Stability and Control of Uncertain Systems
  • Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Differential Equations Analysis
Keywords
predictor feedback, nonlinear integro-differential equations, global asymptotic stability, queuing systems, well-posedness, Lipschitz nonlinearity

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Cited sources are checked by gpt-6-sol.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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