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arxiv: 2605.01344 · v1 · submitted 2026-05-02 · 🧮 math.OC · cs.SY· eess.SY

Unified Lyapunov Method for ISS of PDEs: A Tutorial on Constructing Generalized Lyapunov Functionals for Parabolic and Hyperbolic Equations

Jun Zheng , Guchuan Zhu This is my paper

Pith reviewed 2026-05-09 14:57 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords input-to-state stabilitygeneralized Lyapunov functionalspartial differential equationsparabolic equationshyperbolic equationsboundary disturbancesLyapunov methodinfinite-dimensional systems
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The pith

Generalized Lyapunov functionals that depend explicitly on external inputs establish input-to-state stability estimates in L^q spaces for parabolic and hyperbolic PDEs.

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

This tutorial shows that the generalized Lyapunov method constructs functionals incorporating the input directly to overcome limitations of classical Lyapunov approaches when boundary disturbances are present. It provides explicit step-by-step constructions for an N-dimensional nonlinear parabolic equation with mixed nonlinear boundary disturbances, a first-order nonlinear hyperbolic equation with boundary disturbances, and a second-order linear hyperbolic wave equation with boundary damping and disturbances. Each construction yields ISS estimates valid in any L^q space for q from 2 to infinity. A sympathetic reader cares because distributed-parameter systems in control often involve complex boundary inputs where standard methods become cumbersome or inapplicable. If the constructions work as described, they supply a systematic procedure for stability analysis that extends more readily to controller synthesis than input-independent functionals.

Core claim

The generalized Lyapunov method relies on generalized Lyapunov functionals that depend on the external input to derive explicit input-to-state stability estimates in L^q spaces for the three PDE classes through direct accounting for boundary disturbances, providing greater flexibility than classical input-independent Lyapunov functionals especially for Dirichlet-type conditions.

What carries the argument

Generalized Lyapunov functionals (GLFs), input-dependent functionals that serve as the core tool in the generalized Lyapunov method to establish ISS bounds by incorporating disturbances directly into the functional.

If this is right

  • ISS estimates hold in every L^q space with q in [2, infinity] for each of the three PDE classes once the corresponding GLF is built.
  • The constructions handle mixed nonlinear boundary disturbances and Dirichlet-type inputs more directly than classical Lyapunov theorems.
  • Step-by-step GLF construction applies uniformly across the parabolic, first-order hyperbolic, and second-order hyperbolic cases.
  • The method supports explicit derivation of decay rates and gain functions for the ISS property in these systems.
  • Remaining open problems include extending the constructions to wider PDE families and integrating them into controller design.

Where Pith is reading between the lines

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

  • The same input-dependent construction pattern may simplify boundary controller synthesis for fluid or structural systems governed by these PDEs.
  • Numerical approximation schemes for evaluating the constructed functionals could turn the analytic ISS bounds into practical verification tools.
  • Links to integral input-to-state stability or other robustness notions might emerge by modifying the same GLF forms.
  • The tutorial's approach could reduce reliance on semigroup or operator-theoretic methods when only ISS rather than full asymptotic stability is required.

Load-bearing premise

Suitable generalized Lyapunov functionals can be systematically constructed for the nonlinear parabolic, first-order hyperbolic, and wave equations to produce the stated ISS estimates in L^q spaces.

What would settle it

A concrete PDE instance from one of the three classes for which no input-dependent functional yields a valid ISS estimate, or a simulation showing the system fails to satisfy the derived ISS bound despite the constructed functional.

Figures

Figures reproduced from arXiv: 2605.01344 by Guchuan Zhu, Jun Zheng.

Figure 1
Figure 1. Figure 1: Relationship between x, u, hi , V, Vb, and R for I = {1, 2, ..., N}. (ii) Coercivity w.r.t. x. By virtue of condition (12), the functional Vb(x, u) is coercive w.r.t. x in the sense that for any fixed u, ∥x∥X → +∞ ⇒ Vb(x, u) → +∞. (iii) Non-coercivity w.r.t. (x, u). Note that Vb(x, u) is non-coercive w.r.t. (x, u) in the sense that ∥x∥X + ∥u∥U → +∞ ̸⇒ Vb(x, u) → +∞. In the sequel, we always assume that con… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of a domain in R 2 . We apply the technique of Stampacchia’s truncation to construct a GISS-LF. For any p ∈ (1, ∞), let g(s) := ( s p , s ∈ R≥0, 0, s ∈ R<0, (22a) G(s) := Z s 0 g(τ ) dτ =    1 p + 1 s p+1, s ∈ R≥0, 0, s ∈ R<0. (22b) Note that g and G have the following properties, which will be extensively used in this paper: (G1) G(s) = g(s) = 0 for all s ∈ R≤0; (G2) G(s) = 1 p+1 g(s)s for… view at source ↗
read the original abstract

This tutorial provides an overview of the generalized Lyapunov method (GLM) for analyzing input-to-state stability (ISS) of partial differential equations (PDEs). We begin by revisiting the classical Lyapunov method and the standard ISS-Lyapunov theorem, highlighting their limitations when applied to systems with complex boundary disturbances. In contrast, the GLM, based on the concept of generalized Lyapunov functionals (GLFs) that explicitly depend on the external input, offers greater flexibility and efficiency, particularly for PDEs with Dirichlet-type disturbances. The main objective of this tutorial is to demonstrate how to systematically construct GLFs to establish ISS estimates in $L^q$ spaces with any $q\in[2,\infty]$ for different PDEs. Specifically, we consider three representative classes of PDEs: (i) an $N$-dimensional nonlinear parabolic equation with mixed nonlinear boundary disturbances, (ii) a first order nonlinear hyperbolic equation with boundary disturbances, and (iii) a second order linear hyperbolic equation, i.e., a wave equation, with boundary damping and disturbances. For each case, we provide step-by-step constructions of appropriate GLFs and derive explicit ISS estimates, illustrating the general applicability of the GLM. Finally, we discuss open challenges and future directions, including the systematic construction of GLFs for broader classes of PDEs and their applications in controller design.

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

2 major / 0 minor

Summary. The paper is a tutorial on the generalized Lyapunov method (GLM) for input-to-state stability (ISS) of PDEs. It reviews limitations of the classical Lyapunov method and standard ISS-Lyapunov theorem for systems with complex boundary disturbances, then introduces generalized Lyapunov functionals (GLFs) that explicitly depend on external inputs. The main contribution consists of step-by-step constructions of such GLFs for three representative classes—N-dimensional nonlinear parabolic equations with mixed nonlinear boundary disturbances, first-order nonlinear hyperbolic equations with boundary disturbances, and second-order linear hyperbolic (wave) equations with boundary damping and disturbances—yielding explicit ISS estimates in L^q spaces for arbitrary q ∈ [2, ∞]. The manuscript concludes with a discussion of open challenges and future directions.

Significance. If the constructions are rigorous and the estimates tight, the tutorial could provide a useful resource for researchers in infinite-dimensional control theory by offering concrete guidance on handling Dirichlet-type disturbances and obtaining ISS in multiple L^q norms. The explicit, step-by-step format and coverage of both parabolic and hyperbolic cases are strengths that may facilitate applications in controller design. However, the significance is tempered by whether the GLM constitutes a genuine unified framework or primarily a collection of tailored examples.

major comments (2)
  1. Abstract: The central claim that the GLM enables 'systematic construction' of GLFs with 'general applicability' across PDE classes is load-bearing for the title and abstract but rests on three distinct, case-specific constructions rather than a single overarching theorem or algorithm that derives the GLF form directly from the PDE structure and disturbance type. Without such a general procedure, the constructions risk appearing illustrative rather than unified, weakening the assertion of a 'unified' method.
  2. Sections on the three PDE classes (parabolic, first-order hyperbolic, wave equation): The explicit ISS estimates are asserted to hold in L^q for any q ∈ [2, ∞], yet the manuscript supplies no general error bounds, tightness analysis, or cross-validation against standard energy methods. This is load-bearing because the tutorial's value hinges on the estimates being both explicit and verifiable; case-by-case derivations alone do not automatically confirm they are sharper or more flexible than existing approaches.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed report on our tutorial manuscript. The comments help clarify how to better present the unified aspects of the generalized Lyapunov method. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: Abstract: The central claim that the GLM enables 'systematic construction' of GLFs with 'general applicability' across PDE classes is load-bearing for the title and abstract but rests on three distinct, case-specific constructions rather than a single overarching theorem or algorithm that derives the GLF form directly from the PDE structure and disturbance type. Without such a general procedure, the constructions risk appearing illustrative rather than unified, weakening the assertion of a 'unified' method.

    Authors: The GLM is unified at the methodological level by the consistent use of generalized Lyapunov functionals that explicitly incorporate external inputs to overcome limitations of classical Lyapunov functions when handling boundary disturbances. The tutorial illustrates this unified principle through systematic, step-by-step constructions for three representative classes (parabolic, first-order hyperbolic, and wave equations), each following the same core strategy of designing input-dependent functionals suited to the system structure and disturbance type. We do not claim or provide a single algorithmic procedure that automatically generates the GLF for arbitrary PDEs, as the manuscript explicitly lists the development of such a general procedure as an open challenge. To address the concern, we will revise the abstract and introduction to more precisely state that the GLM provides a unified framework demonstrated via systematic constructions for key PDE classes, rather than implying a fully general derivation algorithm. revision: partial

  2. Referee: Sections on the three PDE classes (parabolic, first-order hyperbolic, wave equation): The explicit ISS estimates are asserted to hold in L^q for any q ∈ [2, ∞], yet the manuscript supplies no general error bounds, tightness analysis, or cross-validation against standard energy methods. This is load-bearing because the tutorial's value hinges on the estimates being both explicit and verifiable; case-by-case derivations alone do not automatically confirm they are sharper or more flexible than existing approaches.

    Authors: The explicit ISS estimates in L^q spaces (q ∈ [2, ∞]) are obtained rigorously from the differential inequalities satisfied by the constructed GLFs in each case, and the derivations themselves serve as the verification. The manuscript does not include general error bounds, a dedicated tightness analysis, or systematic cross-validation against classical energy methods, as its primary aim is to demonstrate the construction process and resulting explicit estimates rather than comparative sharpness. We will add a concise discussion in the concluding section outlining how the obtained estimates compare to standard energy approaches for the specific systems considered and note that a comprehensive tightness study lies beyond the tutorial's scope. revision: partial

standing simulated objections not resolved
  • A single overarching theorem or algorithm that derives the precise form of the GLF directly from arbitrary PDE structure and disturbance type, which the manuscript identifies as an open challenge for future work.

Circularity Check

0 steps flagged

No circularity; tutorial presents explicit constructions building on standard ISS-Lyapunov theorem without self-referential reductions.

full rationale

The paper revisits the classical Lyapunov method and standard ISS-Lyapunov theorem as external foundations, then provides step-by-step constructions of input-dependent generalized Lyapunov functionals for three specific PDE classes to derive ISS estimates in L^q spaces. No equations or steps in the abstract or description reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The claim of systematic construction is illustrated via case-specific examples rather than derived tautologically from prior results by the same authors. The derivation chain remains self-contained against external benchmarks like the standard ISS theorem, with no evidence of renaming known results or smuggling ansatzes via self-citation.

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. The GLM is presented as an extension of the classical Lyapunov method without detailing new postulates.

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