Predictor-Based Stabilization of a Delayed Hetero-Dimensional Shoreline--Bathymetry System

• Prior partial-actuation stabilization results concern common spatial domains, and existing mixed-dimensional and predictor-based studies do not address this combination.Source: §1 · pp. 1–2
• The problem combines hetero-dimensional coupling, partial actuation, unstable shoreline modes, and constant input delay, while the coupling need not dissipate or cancel in the energy balance.Control acts on the one-dimensional shoreline, while the two-dimensional bathymetry is influenced indirectly.Source: §1 · p. 2
• Predictor compensation restores decay of critical shoreline dynamics and prevents delay-induced amplification from propagating to the unactuated bathymetry.Without compensation, delayed feedback can strongly amplify both shoreline and bathymetric responses.Source: §1 · p. 2
• The design separates critical shoreline modes from stable residual dynamics, applies feedback on the critical subspace, and uses a transport equation and predictor to compensate for input delay.The stability analysis combines shoreline stabilization, bathymetric coercivity, and weighted input–output estimates under a small-gain condition.Source: §1 · p. 2

2.1 Mathematical Model

• The model couples a one-dimensional shoreline subsystem to a two-dimensional bathymetric subsystem, with constant-delay distributed actuation applied only to the shoreline.The bathymetry has homogeneous Neumann conditions, while shoreline perturbations have pinned-endpoint Dirichlet conditions.Source: §2.1 · pp. 2–3

2.2 Structural Properties

• The coupling operators are uniformly bounded, but this does not imply dissipativity or energy cancellation; the interaction can take both positive and negative values.The paper gives an admissible coefficient configuration for which the coupling contribution takes both signs as the state varies.Source: §2.2 · p. 4
• The control objective is to stabilize the full system through delayed shoreline actuation alone despite the sign-indefinite interaction with the unactuated bathymetry.Source: §2.2 · p. 4

3.1 Predictor-Based Spectral Feedback Design

• The critical shoreline subsystem is controllable, so a feedback gain can make its closed-loop operator Hurwitz.The critical subspace is finite-dimensional; the result establishes controllability of the critical pair and the existence of the stabilizing gain.Source: §3.1 · p. 5
• Lifting the critical-mode feedback to the full shoreline space yields an analytic exponentially stable semigroup, while accounting for feedback-induced components in the stable residual space.The feedback action may generate a stable-residual component because multiplication by g need not preserve the critical subspace.Source: §3.1 · p. 6
• The predictor transformation removes the explicit input delay from nominal shoreline dynamics while retaining the residual term d in the closed-loop predictor equation.With the lifted feedback U(t) = KcrPcrY(t), the predictor dynamics are Ẏ = AFY + eAnDd.Source: §3.1 · pp. 6–7
• Under the stated coefficient, actuator, and coupling conditions, predictor feedback gives a unique global energy solution on every finite time interval.The theorem also provides finite-time bounds and a continuous transport-state representation.Source: §3.1 · pp. 7–8

3.2 Closed-Loop Stability Analysis

• The stability analysis estimates the mappings d to u_x and u_x to d separately, and uses bathymetric coercivity to obtain a weighted bound on d in terms of the shoreline gradient.The bathymetric coercivity result assumes the existence of µv > 0.Source: §3.2 · pp. 8–10
• The interconnection closes when the product of weighted gains satisfies Γ(θ) := Gud(θ)Gdu(θ) < 1, making the shoreline-gradient estimate uniform in T.The gains correspond to the bathymetric-to-shoreline and shoreline-to-bathymetric mappings.Source: §3.2 · pp. 10–11
• Uniform weighted bounds, combined with finite-time control of u, v, and z on 0 ≤ t < D, establish pointwise exponential decay of the full delayed closed-loop state.The finite-time bound covers the initial delay interval, while the proof combines estimates for the predictor, bathymetric component, and transport state.Source: §3.2 · pp. 11–12
• Under condition (34), with Kcr from Proposition 3 and AF exponentially stable as in Theorem 4, predictor feedback exponentially stabilizes the delayed coupled system when condition (42) holds.The theorem asserts an exponential bound with some M ≥ 1.Source: §3.2 · p. 11

4.1 Numerical Setting and Critical-Mode Design

• The shoreline uses 180 interior finite-difference points, while bathymetry uses a 90-by-50 tensor grid; the discrete coercivity check does not verify the continuous small-gain condition.The numerical energies are unweighted discrete L2 diagnostics, not the weighted functional used in the stability proof.Source: §4.1 · p. 12

4.2 Delay Robustness and Representative Delay

• Increasing uncompensated delay progressively weakens terminal total-energy decay, and the empirical decay-loss threshold is not a stability margin for the continuous system.The transition is attributed to feedback using a shoreline state increasingly out of phase with the state at actuation time.Source: §4.2 · p. 13, Figure 1 · p. 13

4.3 Closed-Loop Shoreline–Bathymetry Response

• 99.29% lower final total energy than uncompensated delayed feedback and 44.06% lower than open loop are reported with predictor compensation.The comparison concerns terminal energy in the full discretized system.Source: §4.3 · p. 13
• Uncompensated delayed feedback produces pronounced oscillatory shoreline amplification, whereas predictor compensation maintains a substantially lower-amplitude space–time response.The three shoreline panels use identical scales for open loop, uncompensated delayed feedback, and predictor compensation.Source: §4.3 · p. 13, Figure 3 · p. 14
• 97.99% lower bathymetric energy than under uncompensated delayed feedback is reported, with the result close to its open-loop level.The paper attributes this to limiting shoreline amplification and its propagation through coupling, not to direct bathymetric regulation.Source: §4.3 · p. 14

5 Conclusion

• The feedback is designed on the finite-dimensional critical shoreline subspace, lifted to the full shoreline dynamics, and paired with predictor compensation that removes explicit input delay from nominal predictor dynamics.Source: §5 · p. 14
• Weighted input–output estimates and a small-gain condition combine predictor-based shoreline stabilization with bathymetric coercivity to establish exponential stability of the full delayed closed-loop system.Source: §5 · p. 14
• Predictor compensation restores decay of the critical shoreline component and prevents delay-induced amplification from propagating to the unactuated bathymetric subsystem.The bathymetric centerline is shown for open loop, uncompensated delayed feedback, and predictor compensation at D = D⋆, with identical scales across panels.Source: §5 · p. 14, Figure 4 · p. 14