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Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

arXiv:2610.08764v1eess.SYcs.LGcs.SYmath.APmath.OC

TL;DR

The single-input design cannot move an unstable double eigenvalue at certain plant values. The paper assigns a second boundary input to pre-feedback and uses Fredholm backstepping, establishing stabilization results while supporting gain approximation across admissible classes.

  • Problem

    At certain parameter values, the plant has an unstable double eigenvalue that feedback through a single channel cannot move.

    Source: §1.1 · pp. 1–2

  • Method

    The design uses the displacement input for pre-feedback and a Fredholm transformation with a gain operator approximated across compact admissible classes.

    Source: §1.1 · p. 3, §3.1 · p. 10, §3.2 · p. 11

  • Results

    The paper establishes local closed-loop solution guarantees for exact gains and gains sufficiently close in L2, and provides a neural operator approximation of the gain.

    Source: §1.1 · pp. 2–3

  • Takeaways & Limitations

    With the second boundary input, the design is valid for every spatially varying coefficient λ in W^1,∞(0, 1), leaving restrictions on design parameters rather than the plant.

    Source: §9 · p. 41

    The required L2-error bound for the truncated modal gain is not verified, so the simulations are empirical.

    Source: §8.1 · p. 39

Abstract

from arXiv · show

We develop the first feedback design for rapid stabilization of the Kuramoto--Sivashinsky equation with a spatially varying anti-diffusion coefficient. For constant coefficients, the single-input Fredholm design of Coron and L\"u (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$. The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.

1. Introduction

The introduction presents a two-input Fredholm design for rapid stabilization of the Kuramoto–Sivashinsky equation with spatially varying anti-diffusion. It also establishes gain approximation and stability results on compact admissible design classes.

1.1. Summary of results

  • For every λ in W^1,∞(0, 1), the two-input design allows an arbitrarily prescribed decay rate and local rapid stabilization with exact or sufficiently accurate gains.Displacement feedback establishes controllability; curvature feedback implements the Fredholm law.Source: §1.1 · p. 2
  • A single neural operator uniformly approximates the gain on compact admissible classes, and sufficiently accurate gains preserve local stabilization across each class.The class bounds λ in W^2,∞; its parameter d caps the uniform rate, unlike the arbitrary rate available for a single plant.Source: §1.1 · pp. 2–3

1.2. Related work

1.3. Contributions

  • The displacement input serves only as pre-feedback to make every mode controllable through curvature; two inputs suffice generally and are necessary with an unstable double eigenvalue.The restricted exponential pre-feedback splits repeated eigenvalues; continuity of the Fredholm gain is proved using spectral projections and control of the infinite coefficient system and modal tails.Source: C. Using pre-feedback to obtain… · p. 4, §1.3 · p. 5

1.2. Related work

  • Existing neural-operator gain approximations concern Volterra transformations; this work treats Fredholm gains whose continuity requires a modal rather than successive-approximation argument.Prior numerical methods compute gains for a specified plant coefficient.Source: D. Numerical and neural operator gain… · p. 4, §1.3 · p. 5

1.4. Organization of the paper

  • The paper organizes controllability and feedback construction in Sections 2–5, gain continuity and approximation in Section 6, and exact- and approximate-gain stability in Section 7.Its spectral spaces are distinguished from the usual Sobolev spaces.Source: §1.4 · p. 5, §1.5 · p. 6

2. Two inputs: necessary and sufficient

The plant’s spectral structure limits the number of unstable modes but allows repeated eigenvalues, which obstruct single-input stabilization. Two boundary inputs give dense reachability for every admissible coefficient, and both are necessary when an unstable double eigenvalue occurs.

2.1. Plant operator

  • The plant operator is self-adjoint with compact resolvent, so its eigenvalues are real and its eigenfunctions form an orthonormal basis.Eigenvalues are ordered decreasingly and repeated according to multiplicity.Source: §2.1 · p. 7

2.2. Plant operator has finitely many unstable eigenvalues

  • The spectrum has only finitely many nonnegative eigenvalues, and its high-frequency gaps grow like n^3, leaving only finitely many repeated eigenvalues.These properties hold despite the size of the anti-diffusion coefficient.Source: §2.2 · p. 8

2.3. Two inputs are necessary and sufficient

  • Two boundary inputs make the linearized plant’s reachable terminal states dense in L2(0, 1) for every λ ∈ W 1,∞(0, 1).The inputs are displacement U1 = u(0, ·) and curvature U2 = uxx(0, ·).Source: §2.3 · pp. 8–9
  • At an unstable double eigenvalue, any scalar linear feedback through curvature alone leaves that eigenvalue in the closed loop, so two inputs are necessary for stabilization.For constant λ, double eigenvalues occur at λ = (i^2 + j^2)π^2 and have value µ = i^2j^2π^4 > 0.Source: §2.3 · pp. 8–9

3. Controller design

The controller assigns displacement and curvature inputs distinct roles: displacement pre-feedback makes modes controllable through curvature, which then stabilizes the plant through a Fredholm transformation. Kernel conditions are chosen to map the pre-compensated plant to a target system, with the design also expressed in modal coordinates.

3.1. Pre-feedback and pre-compensated operator

  • The displacement pre-feedback splits every double eigenvalue and makes every resulting mode reachable from curvature for eligible θ and sufficiently small nonzero ε, while leaving high-frequency structure unchanged.θ is outside a discrete set; θ and ε control spatial weighting and magnitude, respectively.Source: §3.1 · p. 10

3.2. Transformation, target, kernel equations

  • The Fredholm transformation w = Tu, with T = I − K, and its kernel conditions map the pre-compensated plant into the target system.The design imposes TB2 = B2 to preserve the curvature-input direction; the gain is related to the kernel by h(y) = kxx(0, y) when k is smooth.Source: §3.2 · pp. 11–12

3.3. Modal form of the kernel equations

  • In the modal formulation, the normalization TB2 = B2 makes the kernel system linear in K and h, providing the form used for analysis.For ε = 0, this is the system behind [14, (2.53)–(2.55)].Source: §3.3 · p. 12

4. Pre-feedback enables controllability

This section develops displacement pre-feedback to make the curvature input controllable for every plant coefficient in W¹,∞(0,1). It establishes the supporting spectral and resolvent results, including conditions for preserving simple high modes and obtaining a Riesz-basis framework.

4.1. Admissible design and pre-feedback

  • For every λ ∈ W^1,∞(0,1), outside a discrete set of θ, sufficiently small nonzero ε and a > max{0, max_n Re µ′_n} yield simple modes and curvature reachability.These conditions give an admissible design for each such λ.Source: §4.1 · p. 13

4.2. Proof of Theorem 4.2: pre-feedback enables controllability

  • The pre-feedback preserves high-mode simplicity and curvature reachability, while profile conditions split double plant eigenvalues and recover previously missed modes.The proof separates high modes, whose required properties persist because the pre-feedback effect is smaller than spectral gaps, from finitely many modes requiring tailored conditions.Source: §4.2 · p. 13

4.2.1. Spectral properties of the original plant

  • The original plant has eigenspaces of dimension at most two; its high-frequency eigenfunctions approach the sine basis, and its eigenvalue gaps grow at least as π^4 n^3.The gap bound implies simplicity for sufficiently high modes.Source: §4.2.1 · pp. 13–14

4.2.2. Resolvent and input coefficients under pre-feedback

  • The pre-feedback resolvent gives a secular criterion for new eigenvalues; under the simplicity conditions, curvature-input coefficients are nonzero for moved and retained modes, and a double mode is simple when rµ ≠ 0.For a moved real eigenvalue, algebraic simplicity is equivalent to Ξ′(µ′) ≠ 0.Source: §4.2.2 · p. 16
  • The resolvent has a simple pole with rank-one spectral projection exactly when the eigenvector pairing is nonzero; a retained eigenvalue is simple with nonzero curvature coefficient if rµ≠0.When rµ=0, the pole is canceled and the eigenvalue has algebraic multiplicity at least two.Source: §4.2.2 · p. 18

4.2.3. Pre-feedback preserves the simple high modes

  • Above a cutoff, each high-mode disk contains exactly one real, algebraically simple eigenvalue; for |ε|≤1 the cutoff is uniform, and only finitely many eigenvalues lie elsewhere.The high-mode eigenvalues also have nonzero curvature coefficients.Source: §4.2.3 · p. 19

4.2.4. Selection of the pre-feedback parameters

  • An exponential profile gθ avoids a plant-dependent discrete exceptional set Tλ; sufficiently small nonzero ε resolves the finitely many low modes, while admissible a avoids a discrete set.The resulting spectral maximum differs from the plant spectral maximum by O(|ε|).Source: §4.2.4 · p. 22

4.2.5. Riesz basis and spectral estimates

  • For admissible designs, the eigenvectors form a Riesz basis, and the resulting semigroup is strongly continuous with high-mode spectral estimates.The Riesz-basis property follows from the high-mode eigenvector estimates and completeness.Source: §4.2.5 · p. 23

5. Fredholm kernel always exists and its transformation is invertible

For admissible designs, the Fredholm theorem yields a kernel and gain for the pre-compensated plant, with an invertible transformation. The modal construction also establishes smoothing and a unique coefficient solution.

5.1. Kernel construction and invertibility

  • The Riesz-basis and spectral and control-coefficient bounds enable the Fredholm theorem to construct an invertible transformation for the pre-compensated plant.Source: §5.1 · p. 23
  • The gain h lies in H^(1/2−ε)(0,1) for every ε in (0,1/2].Source: §5.1 · p. 24
  • The kernel and gain exist, and T=I-K is an isomorphism on H^r for r in (-9/2, 5/2), including bounded invertibility on L2.The associated closed-loop operator generates a C0 semigroup over the same range of modal spaces.Source: §5.1 · pp. 23–24
  • The modal coefficients of K and h solve system (3.11), and any solution in the stated class coincides with the constructed one.Source: §5.1 · p. 24
  • K maps L2 into H2 and is compact, so I-K is Fredholm of index zero on L2.Source: §5.1 · pp. 25–26

6. Gain operator, continuity and neural-operator approximation

The curvature-gain operator is continuous on a compact admissible class, with uniformly bounded design constants. The proof controls spectral data, coefficient systems, and modal tails to establish continuity of the gain and kernel despite eigenbasis changes at double eigenvalues.

6.1. Gain operator and admissible class

  • The training class bounds λ in W 2,∞, ensuring compactness in W 1,∞; this stronger bound is needed only for training, while the design requires only λ ∈ W 1,∞.Source: §6.1 · p. 26

6.2. Continuity of the kernel and gain

  • The gain operator is continuous on the compact admissible class, with uniform bounds on the gain, kernel, and design constants.These bounds provide the control needed for the subsequent approximation and stability arguments.Source: §6.2 · p. 27
  • At double eigenvalues, individual eigenvectors need not vary continuously, so the proof uses continuous Riesz projections and uniform high-frequency estimates.This allows continuity of the gain and transformation without a continuously chosen eigenbasis.Source: §6.2 · p. 27

6.3.1. Resolvent continuity and uniform spectral estimates

  • Resolvent convergence and uniform high-mode estimates control spectral data, while finitely many low modes are handled separately.The resolvents converge in operator norms on compact subsets of the common resolvent set.Source: §6.3.1 · pp. 27–28

6.3.2. Compactness of the admissible class

  • The admissible class is compact because limits preserve separation, curvature-input, nonresonance, and target-stability margins.Uniform high-mode estimates reduce the limiting checks to finitely many low modes.Source: §6.3.2 · p. 28

6.3.3. Continuity of the spectral coordinate maps

  • Local eigenvector choices and comparison with a fixed sine basis give continuous spectral coordinate maps without tracking eigenfunctions through double eigenvalues.The reconstructed gain and kernel do not depend on eigenvector phases.Source: §6.3.3 · p. 29

6.3.4. Continuity of the feedback coefficients

  • The coefficient system varies continuously because its matrix is operator-norm continuous and invertible, and its feedback-coefficient tails are summable.Uniqueness of the bounded solution gives trivial kernel, allowing the Fredholm alternative and continuity of inversion.Source: §6.3.4 · pp. 29–30

6.3.5. Reconstruction of the gain and kernel

  • Continuous modal coefficients reconstruct the gain and kernel continuously, while uniform transformation and Sobolev bounds control the remaining design constants.The kernel bounds yield a summable majorant, giving Hilbert–Schmidt continuity of the kernel.Source: §6.3.5 · p. 30, §6.3.6 · pp. 30–32

6.4. Neural-operator approximation of the Fredholm gain

Neural-operator universality and continuity of the gain operator yield a single map approximating the curvature gain across the design class. The result guarantees a prescribed accuracy but does not bound network size or training cost.

6.4. Neural-operator approximation of the Fredholm gain

  • A single neural operator maps λ and (a, ε, θ) to the curvature gain with prescribed L2 accuracy across the design class.The approximation follows from continuity of the gain operator and neural-operator universality.Source: §6.4 · p. 32
  • The neural operator takes the coefficient function and scalar design parameters as inputs and outputs an L2(0, 1) gain.Source: §6.4 · p. 32
  • The approximation theorem establishes existence but gives no bound on network size, training cost, or sample complexity.The map is accurate to a prescribed tolerance across the whole design class.Source: §6.4 · p. 32

7. Closed-loop stability

The paper proves local rapid stabilization of the nonlinear closed loop with the exact curvature gain and with sufficiently accurate gain approximations. The argument establishes decay of a forced target system and controls both nonlinear forcing and gain error.

7. Closed-loop stability

  • In transformed coordinates, forcing comprises the transformed Burgers nonlinearity and, for an approximate gain, an additional boundary term from gain error.This decomposition is used in both the exact-gain and gain-error analyses.Source: §7.1 · p. 32, §7.3 · p. 36

7.1. Stability of the target system

  • The forced target system has a unique mild solution and a Lyapunov estimate that gives exponential decay when forcing is absent.The target’s spectral decay and forcing estimates are used to control the nonlinear plant.Source: §7.1 · pp. 32–34

7.2. Local rapid stabilization of the nonlinear plant with the exact gain

  • With the exact curvature gain, sufficiently small initial states yield a unique global solution and local exponential decay at any prescribed rate below the design margin.The theorem assumes 0 < ω < ν̄ < ν.Source: §7.2 · p. 35

7.3. Local rapid stabilization under gain approximation

  • A gain-error bound combines with the nonlinear estimate to give explicit tolerances for gain accuracy and initial-state size.The resulting error tolerance scales with the square root of the gap between the auxiliary and target decay rates.Source: §7.3 · pp. 36–37, §7.3.1 · p. 38

7.3.1. Local exponential stability under gain error

  • A sufficiently accurate approximate gain yields a unique global closed-loop solution and local exponential stability for initial states within the theorem’s radius.The gain error must not exceed the stated tolerance; the result also applies uniformly over the admissible class with uniform constants.Source: §7.3.1 · pp. 37–38

8. Numerical examples

The examples illustrate prescribed-rate stabilization, then test whether a Fourier neural operator can approximate the curvature gain and stabilize held-out plants. They cover a plant with an unstable double eigenvalue and a family of spatially varying coefficients.

8.1. Prescribed rate stabilization with the modal gain

  • For coefficient (8.1), with µ1 = µ2 ≈ 459.114, the state and both boundary controls decayed toward zero after an initial transient using a 32-mode kernel.The design used (a, θ, ε) = (480, 1, 0.5).Source: Figure 2 · p. 38, §8.1 · p. 39
  • For the fixed plant, increasing a from 480 to 600 to 1000 produced faster state decay after the initial transient; curvature-control transients were also compared.Each curvature gain was recomputed using 1536 modes.Source: Figure 3 · p. 39, §8.1 · pp. 39–40

8.2.3. Neural operator closed-loop stability

  • The FNO generated gains with approximately 0.13% mean relative error in 25.9 ms on CPU or 2.62 ms on GPU, versus 815 ms for the 1024-mode solve.Timings exclude offline training and dataset generation; for a fixed plant, its gain is computed once and reused.Source: Table 2 · p. 39, §8.2.3 · p. 41

8.2.1. Dataset construction

  • The training family varies the mean, amplitudes, and phases of Fourier profiles while remaining uniformly bounded in W 2,∞.The dataset used 1800 gain labels, split into 1400 training, 200 validation, and 200 test samples, at fixed design parameters (1000, 1, 4).Source: §8.2.1 · p. 41

8.2.2. Neural operator training

  • The FNO maps coefficient samples and spatial coordinates to the curvature gain, trained by minimizing squared L2 error against modal gain labels.The network used four 64-neuron layers and 32 Fourier modes, trained for 400 epochs.Source: §8.2.2 · p. 41
  • Median relative L2 gain errors against the 1024-mode reference were 0.0916% on training profiles and 0.1041% on test profiles.The closed-loop tests report decay for every held-out test profile.Source: §8.2.2 · p. 41

8.2.3. Neural operator closed-loop stability

  • All 200 held-out profiles had lower L2 state norm at t=0.8 than initially under each tested method, including the Fourier neural operator and modal approximations.The largest state norm under FNO feedback was 3.30 × 10^-3.Source: §8.2.3 · p. 41

9. Conclusion

The paper uses a second boundary input to address unstable repeated eigenvalues that obstruct single-channel stabilization. It also establishes gain-map continuity, neural-operator approximation, and local rapid stabilization for exact and sufficiently accurate approximate gains.

9. Conclusion

  • The displacement input supplies a controllability-ensuring pre-feedback, leaving the curvature input to stabilize through Fredholm backstepping for every λ ∈ W 1,∞(0, 1).At an unstable repeated eigenvalue, no linear feedback through a single channel can stabilize the system.Source: §9 · p. 41
  • The Fredholm transformation is invertible because the pre-compensated plant is controllable.This links the controllability design to validity of the transformation.Source: §9 · pp. 41–42
  • Continuity of the coefficient-and-design-parameter-to-gain map yields one neural operator that approximates the gain to any prescribed L2 accuracy on a compact admissible class.The paper establishes local rapid stabilization for exact gains and sufficiently accurate approximations.Source: §9 · p. 42
  • The authors identify adaptive and observer-based Fredholm control, and applications of two-control designs beyond KS, as directions opened by the work.The text notes related companion designs for Navier–Stokes and Yih’s problem [26] [27].Source: §9 · p. 42

A.1. Proof of Proposition 3.1: verification of the plant-to-target transformation

The verification derives the transformed target equation by integrating the kernel relation against the plant state and using boundary conditions.

A.1. Proof of Proposition 3.1: verification of the plant-to-target transformation

  • The proof uses integration by parts with the plant and kernel boundary conditions to obtain the transformed equation.Source: §A.1 · p. 42
  • Because ky(x, 0) = 0, only the displacement boundary term remains in the integration-by-parts calculation.Source: §A.1 · p. 42
  • Substituting U1 = ε⟨gθ, u⟩ cancels the kernel source term after integration against u, yielding the target equation.Source: §A.1 · p. 42
  • The remaining target boundary conditions follow from the kernel traces.Source: §A.1 · p. 42

A.2. Proof of Proposition 3.2: verification of the modal equivalence of the kernel

The proof establishes modal equivalence of the kernel using projections and distributional identities that account for derivative jumps along y = x.

A.2. Proof of Proposition 3.2: verification of the modal equivalence of the kernel

  • The proof projects first in y and then in x because k(x, ·) need not lie in D(Aλ) when its third derivative jumps at y = x.Source: §A.2 · p. 42
  • Integration by parts has no endpoint terms because the functions and their second derivatives vanish at both endpoints.Source: §A.2 · p. 42
  • The y-projection uses the distributional kernel equation, which includes terms generated by derivative jumps at y = x.This avoids requiring k(x, ·) ∈ D(Aλ).Source: §A.2 · pp. 42–43
  • The identity involving the remaining modal equation follows by pairing (A5) with φm and applying the boundary identity of Lemma 2.1.Source: §A.2 · p. 43

Appendix B. Sobolev bounds for the modal coordinate map

The modal coordinate map Γ is an L2 isomorphism that preserves Sobolev regularity through order four, with uniform bounds on the stated design classes. These properties support integration by parts in the modal pairing and are used in Theorem 6.2.

Appendix B. Sobolev bounds for the modal coordinate map

  • The stated Γ properties justify integration by parts in the modal pairing and provide uniform bounds used in Theorem 6.2.Source: Appendix B · p. 43
  • Γ is an L2 isomorphism, is bounded on Hs for s in [0, 4], and maps H1 into H1_0.Its Hs operator norms are bounded uniformly on UB,d.Source: Appendix B · p. 43
  • Graph-norm estimates for A′λ and its adjoint establish the Sobolev bounds needed for Γ.The proof combines elliptic estimates, interpolation, and uniform operator bounds.Source: Appendix B · pp. 43–44
  • Finite adjoint eigenfunction sums vanish at both endpoints, and convergence in Hs with trace continuity passes these boundary conditions to the limit.The boundary conditions follow for s > 1/2.Source: Appendix B · p. 44

Appendix C. Closed-loop well-posedness

The closed loop has a unique local solution for every L2 initial state and L2 gain perturbation, with a finite-time blow-up alternative. Its Lyapunov functional is absolutely continuous on compact existence intervals and satisfies the stated identity.

Appendix C. Closed-loop well-posedness

  • Because the forcing depends on the unknown state, the proof obtains existence and uniqueness by a fixed-point argument in target coordinates.The fixed-point map uses the linear target system with forcing B2⟨e, u⟩ − T(uux).Source: Appendix C · p. 44
  • For every L2 initial state and L2 perturbed gain, the closed loop has a unique local solution, with existence time depending on the initial norm, gain error, and design.The feedback uses the pre-feedback and perturbed curvature feedback U2 = ⟨ĥ, u⟩.Source: Appendix C · p. 44
  • If the maximal existence interval is finite, the solution norm diverges as time approaches its endpoint.Source: Appendix C · p. 44
  • The Lyapunov functional V(Tu(t)) is absolutely continuous on compact existence intervals and satisfies identity (7.3) with the closed-loop forcing.Source: Appendix C · p. 44
  • Restarting the local construction yields continuation, while boundedness near a finite endpoint would extend the solution and contradict maximality.Source: Appendix C · p. 45

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