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Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems

Amadou Cissé

arXiv:2609.03136v1math.OCcs.SYeess.SYmath.AP

TL;DR

Temporally degenerate parabolic systems raise a prior question: whether feedback defines an admissible closed-loop evolution at all. The paper characterizes compatibility between the singular reaction and input–output operators, then reports that resolving this structural issue yields a regular evolution supporting stability certification, lifting, and feedback synthesis.

  • Problem

    For temporally degenerate systems, closed-loop well-posedness is not assured, and existing feedback approaches leave the structural admissibility of the interconnection unestablished.

    Source: §1 · p. 2

  • Method

    The paper derives operator compatibility linking the singular reaction with the actuator and observation operators, then uses the regularized evolution for stability analysis, lifting, and static output feedback synthesis.

    Source: §3.1 · p. 9, §8.2 · p. 29, §10 · pp. 41–42

  • Results

    Resolving compatibility restores a regular closed-loop evolution, from which stability certification, finite-to-infinite lifting, and feedback synthesis follow.

    Source: §10 · pp. 41–42

  • Takeaways & Limitations

    For the considered temporally degenerate linear parabolic systems, admissibility is a structural prerequisite to stability analysis and controller synthesis.

    Source: §10 · pp. 41–42

    Extending the theory beyond exact compatibility and to nonlinear degenerate systems remains open.

    Source: §10 · p. 42

Abstract

from arXiv · show

Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical--residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.

1 Introduction

For temporally degenerate parabolic systems, temporal degeneracy can undermine the admissibility of the closed-loop evolution itself, so feedback design must first address whether the interconnection is well posed. The paper characterizes this compatibility, regularizes the dynamics, and builds stability analysis and feedback synthesis on that formulation.

1 Introduction

  • Temporal degeneracy can make a closed-loop model fail to generate a regular evolution, challenging the well-posedness assumed in classical feedback design.Unlike spatially degenerate control problems, the degeneracy here acts on the time derivative and may affect admissibility before stability is considered.Source: §1 · p. 2
  • Existing feedback approaches share the premise that the closed-loop evolution is well posed, leaving structural admissibility of the interconnection unestablished.The cited directions include spectral feedback, finite-dimensional compensation, reduced-order methods, spillover-aware strategies, and static output feedback.Source: §1 · p. 2
  • The paper asks whether a feedback interconnection compatible with the degenerate dynamics can be defined before addressing stability, optimality, or robustness.This question is independent of any particular synthesis methodology.Source: §1 · p. 2
  • An operator compatibility condition links the dynamics with input–output operators, regularizes the evolution, and supports stability certification, finite-to-infinite lifting, and static output feedback synthesis.Numerical simulations validate the compatibility principle, regularization identity, and critical–residual stability mechanism.Source: §1 · p. 3

2 Mathematical Framework and Problem Formulation

The framework formulates an abstract temporally modulated parabolic system and a representative degenerate PDE with output feedback. Its central setup feature is an unweighted reaction operator that produces a singular term after desingularization, motivating the compatibility and stability questions.

2 Mathematical Framework and Problem Formulation

  • The abstract formulation is introduced as the foundation for the later analysis, with the next subsection specializing it to a representative temporally degenerate parabolic model.The analysis of the desingularized equation is carried out under stated assumptions.Source: §2 · p. 3, §2.1 · pp. 3–4, §2.3 · p. 6

2.1 Abstract Evolution Equation

  • The abstract model weights the time derivative by α(t) and the principal parabolic operator by β(t), alongside lower-order and reaction operators and input–output maps.The class includes additional regular lower-order dynamics through Ar.Source: §2.1 · pp. 3–4
  • A1 is not multiplied by α(t), unlike the remaining operators, making this structural asymmetry central to the subsequent analysis.The representative PDE sets Ar = 0, while the abstract model permits it.Source: §2.1 · p. 4, §2.2 · p. 5

2.2 Degenerate Parabolic Model and Closed-Loop Reformulation

  • The representative realization is a temporally degenerate parabolic PDE with positive temporal coefficients, homogeneous Dirichlet boundary conditions, and an initial condition.The measured output is supplied through the abstract output equation.Source: §2.2 · pp. 4–5
  • Substituting the stated output-feedback law yields a closed-loop equation with reaction term A1 − BK0C and regular term Ar − BK1C.After division by α(t), the desingularized equation contains the reaction mismatch scaled by 1/α(t).Source: §2.2 · p. 5

2.3 Analytical Setting and Problem Formulation

  • The temporal assumptions require α(t) and β(t) to be positive and d(t) = β(t)/α(t) to remain bounded above and bounded away from zero.The principal operator A0 is densely defined, self-adjoint, strictly positive, and has compact resolvent.Source: §2.3 · p. 6
  • The analysis frames three questions: whether temporal degeneracy imposes operator compatibility, how it can be characterized through A1, B, and C, and when the regularized evolution is uniformly exponentially stable.The singular operator has no analogue in classical nondegenerate parabolic equations and is the work’s central focus.Source: §2.3 · p. 6

3 Compatibility Principle Induced by Temporal Degeneracy

Strong temporal degeneracy imposes a structural compatibility condition on the system operators before stabilization is considered. The paper derives this condition from regular solvability and identifies its dependence on the singular reaction and input–output structure.

3.1 Structural Obstruction and Compatibility Principle

  • The trace obstruction forces the singular reaction mismatch to vanish on admissible initial traces, yielding the compatibility principle independently of stabilization.The theorem’s consequence is determined by the interaction of A1 with the input and observation operators B and C.Source: §3 · p. 7, §3.1 · p. 9
  • After desingularization, the principal parabolic contribution cannot be treated as a bounded perturbation; its influence depends on the temporal coefficient near the degenerate time.The analysis introduces a strong temporal degeneracy assumption to identify this structural constraint.Source: §3.1 · p. 7
  • Regular solvability for a dense set of initial states under strong temporal degeneracy requires the operator identity A1 = BK0C.The identity is a necessary consequence of admissible regular solutions, not a controller design requirement.Source: §3.1 · pp. 8–9
  • The compatibility identity is independent of the dissipative properties of A0 and does not rely on a stability argument.Its necessity follows from the singular reaction and input–output structure.Source: §3.1 · p. 9
  • The principle identifies the identity required for regular solvability but does not by itself establish whether a given system can realize it.The following analysis characterizes solvability of the operator equation and gives necessary and sufficient conditions for structural compatibility.Source: §3.1 · p. 10

4 Operator-Theoretic Characterization of the Compatibility Principle

The section completely characterizes when the compatibility equation can be solved, reducing admissibility to geometric conditions on the reaction, observation, and actuator operators. This characterization supports elimination of the singular reaction term and reduction to a regular evolution equation.

4.1 Operator Factorization

  • When these conditions hold, A1 factors through C and the resulting operator can be lifted through B to construct a bounded compatibility gain K0.The construction defines the induced operator on Ran(C), lifts it through the actuator, and extends the gain using a projection.Source: §4.1 · p. 11
  • Every compatible reaction operator has finite rank, so an infinite-rank reaction operator cannot yield an admissible regular closed loop.This is an intrinsic limitation of finite-dimensional sensing and actuation.Source: §4.1 · p. 12

4 Operator-Theoretic Characterization of the Compatibility Principle

  • Compatibility is solvable exactly when ker(C) ⊆ ker(A1) and Ran(A1) ⊆ Ran(B).The first condition makes the reaction factor through the observation operator; the second permits lifting through the actuator.Source: §4.1 · pp. 10–12
  • Once compatibility is established, the singular reaction term is eliminated and the closed-loop problem reduces to a regular evolution equation for subsequent stability analysis.The characterization depends on operator geometry, not on dissipativity of the principal operator or a stability argument.Source: §4.1 · p. 12

5 Structural Analysis of the Regular Evolution

The section identifies geometric and spectral obstructions to uniform stabilization of the regularized evolution. It shows that feedback cannot stabilize unstable dynamics hidden from observation, and that necessary spectral tests must hold across every admissible frozen parameter value.

5.1 Common Invisible Dynamics

  • An invariant subspace invisible to measurements evolves independently of feedback, so uniform stabilization requires its open-loop dynamics to be uniformly exponentially stable.The result applies to subspaces that remain invariant under the regular dynamics and lie in ker(C).Source: §5.1 · pp. 13–14

5.2 Spectral Necessary Conditions

  • Uniform stabilization requires every frozen operator in the parameter interval to satisfy both Hautus conditions for eigenvalues with Re λ ≥ 0.No such eigenmode may remain invisible to observation or inaccessible through actuation.Source: §5.2 · pp. 16–17
  • Failure of either spectral condition at a single admissible frozen parameter value rules out uniform exponential stabilization of the non-autonomous family.Uniform stabilization also implies exponential stability of each corresponding frozen closed-loop system.Source: §5.2 · pp. 15, 17
  • The geometric and spectral conditions delimit when stabilization can be pursued but are necessary conditions, not a stability certificate.The subsequent critical–residual analysis supplies sufficient conditions for uniform exponential stability.Source: §5 · p. 13, §5.2 · p. 17

6 Uniform Stability Certification

The section develops a uniform stability certificate by separating critical and residual dynamics, estimating residual dissipation and spillover, and combining these with a quadratic estimate for the critical subsystem. The resulting criterion applies uniformly across admissible parameter trajectories.

6 Uniform Stability Certification

  • The certificate combines a critical-subsystem quadratic estimate with residual dissipation and bounds on the spillover couplings.Stability of the two subsystems separately is insufficient unless their interaction is also controlled.Source: §6 · p. 17, §6.3 · p. 24

6.1 Critical–Residual Decomposition

  • A spectral decomposition splits each state into finite-dimensional critical and infinite-dimensional residual components for separate stability analysis.The components are defined by orthogonal projections onto spectral subspaces of A0.Source: §6.1 · pp. 17–18

6.2 Residual Dissipation and Spillover Analysis

  • For sufficiently large truncation orders, the residual dynamics are uniformly dissipative when d−λN+1 > ℓs(K1).The spectral coercivity of A0 supplies dissipation against the bounded contribution of LK1.Source: §6.2 · pp. 19–20

6.3 Uniform Stability Certificate

  • A common quadratic certificate for critical modes, positive residual dissipation, and sufficiently weak spillover jointly yield trajectory-uniform exponential stability.The stability estimate is uniform over the entire admissible parameter trajectory, not just individual frozen systems.Source: §6.3 · pp. 22, 24

6.4 Explicit Stability Criterion

  • The explicit small-gain criterion expresses the conditions under which the regular evolution family is uniformly exponentially stable.It packages the critical certificate, residual dissipation margin, and spillover bounds.Source: §6.4 · p. 24

7 Structural Finite-to-Infinite Lifting

Finite-rank coupling makes spillover vanish as the spectral truncation grows, allowing the stability certificate to be verified at sufficiently large finite truncations. The lifting result then transfers this certificate to the regular infinite-dimensional evolution.

7 Structural Finite-to-Infinite Lifting

  • The lifting principle reduces verification of infinite-dimensional stability conditions to analysis of a sufficiently large finite-dimensional critical subsystem.The section establishes the structural link between the regularized dynamics and the certificate developed earlier.Source: §7 · p. 25, §7.3 · p. 28

7.1 Structural Decay of the Spillover Operators

  • Because LK has finite rank, the spillover operators vanish asymptotically as the spectral truncation order increases.This decay is the structural ingredient used by the lifting procedure.Source: §7.1 · pp. 25–26

7.2 Asymptotic Verification of the Stability Certificate

  • Asymptotic spillover decay makes the uniform stability certificate's hypotheses hold for sufficiently large truncation orders.This verification supplies the condition needed for finite-to-infinite lifting.Source: §7.2 · pp. 26–27

7.3 Structural Finite-to-Infinite Lifting

  • Under the compatibility condition and the stated hypotheses, a sufficiently large truncation satisfies the stability certificate, yielding uniform exponential stability of the regular evolution.The stability constants are independent of the admissible parameter trajectory.Source: §7.3 · pp. 27–28

8 Constructive Static Output Feedback Synthesis

The synthesis first constructs a compatibility gain that eliminates the singular reaction term, then designs a dynamic gain on the critical subsystem. Finite-to-infinite lifting transfers the resulting stability to the regularized evolution.

8.1 Compatibility Feedback Construction

  • The compatibility gain K0 eliminates the singular reaction term and reduces the closed-loop system to the regularized evolution.Its construction relies on the compatibility conditions.Source: §8.1 · p. 29

8.2 Critical Static Output Feedback Synthesis

  • The remaining gain K1 is designed on the finite-dimensional critical subsystem to satisfy the common quadratic certificate.The lifting result then transfers this construction to the original infinite-dimensional evolution.Source: §8.2 · pp. 29–30

8.3 Constructive Synthesis Procedure

  • The constructive procedure combines compatibility verification, regularization, finite-dimensional critical-subsystem design, and finite-to-infinite lifting to obtain a stabilizing static output feedback law.The procedure chooses a sufficiently large truncation order before designing the dynamic gain.Source: §8.3 · p. 30

8 Constructive Static Output Feedback Synthesis

  • The synthesis separates structural regularization from stabilization: compatibility removes the singular term, while dynamic feedback stabilizes the critical subsystem.The finite-to-infinite lifting guarantees uniform exponential stability of the regularized evolution.Source: §8.3 · p. 30

9 Numerical Validation

The numerical experiments illustrate the compatibility obstruction and regularization, then assess stability certification, finite-to-infinite lifting, and controller implementation. Across these tests, the regularized and degenerate trajectories agree closely, and the same critical controller gives similar trajectories across truncation orders.

9.1 Numerical Setting

  • The benchmark uses a 60-mode Galerkin approximation over t ∈ [10−3, 20], with ode15s tolerances RelTol = 10−8 and AbsTol = 10−10.The numerical configuration is held fixed throughout the validation study.Source: §9.1 · pp. 31–32

9.2 Compatibility Principle and Regularization

  • The compatible degenerate and regularized trajectories are visually indistinguishable, with maximum absolute and relative discrepancies of 3.15 × 10−9 and 2.12 × 10−8.The reported discrepancies remain at the prescribed integration tolerances.Source: §9.1 · p. 32, Figure 2 · p. 35
  • Compatibility defects amplify the initial transient and preserve a singular contribution near the degenerate time, while the compatible realization has RN = 0 and no singular term.Incompatible trajectories also become more sensitive as the lower integration threshold approaches the degenerate time, unlike the compatible trajectory.Source: §9.2 · p. 33, Figure 1 · p. 34

9.3 Certified Critical–Residual Stabilization

  • With K1, the complete state decays exponentially and the certified envelope remains above the computed trajectory; without K1, critical modes grow rapidly.The feedback gain is synthesized from two unstable modes, while the smallest cutoff satisfying the full small-gain condition is n⋆ = 3.Source: §9.3 · pp. 35–36, Figure 3 · p. 37
  • As the spectral cutoff increases, the spillover quantities decrease markedly, including a more-than-seven-order decline in γss(n); the certified decay rate eventually plateaus.The normalized small-gain margin first becomes positive at the smallest truncation order satisfying the theoretical certificate.Source: §9.3 · pp. 36–37, Figure 3 · p. 37, Figure 4 · p. 38

9.4 Finite-to-Infinite Lifting and Control Implementation

  • Using the same controller, trajectories for Galerkin orders 20, 40, 60, and 80 remain visually indistinguishable, with discrepancies already at integration accuracy for N = 20.The authors describe this as a numerical illustration of lifting, not an independent proof of infinite-dimensional convergence.Source: §9.4 · pp. 38–39, Figure 5 · p. 39
  • Near the degenerate initial time, the compatibility component dominates control effort; later the stabilizing component becomes dominant, and both contributions decay with the state.The distributed actuator commands remain smooth and converge asymptotically to zero without singular control effort near t = 0.Source: §9.4 · p. 40, Figure 6 · p. 41

10 Conclusion

The paper identifies structural admissibility as a prerequisite for feedback design in temporally degenerate parabolic systems. Its compatibility principle restores regular closed-loop evolution, from which stability certification, finite-to-infinite lifting, and feedback synthesis follow.

10 Conclusion

  • For temporally degenerate parabolic systems, temporal degeneracy can make existence of an admissible closed-loop evolution a prerequisite to later control objectives.The compatibility principle removes this structural obstruction and restores regular closed-loop evolution.Source: §10 · pp. 41–42
  • The theory leaves extensions beyond exact compatibility, nonlinear degenerate systems, and actuator–sensor co-design as open questions.These are described as directions for further development rather than shortcomings of the analysis.Source: §10 · p. 42
  • The results support treating structural admissibility as fundamental to singular infinite-dimensional control, alongside stability.For such systems, establishing admissible evolution may be as fundamental as establishing stability.Source: §10 · p. 42

Funding

The paper reports no funding.

Funding

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