{"id":"330b585e-2c01-4635-9ec4-4021fb63418c","arxiv_id":"2607.01861","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Preservation of fractional Sobolev regularity under saturated feedback enables semi-uniform input-to-state stability for impedance passive systems.","lead":"The paper shows that fractional Sobolev regularity of the free output is preserved under saturated nonlinear feedback for impedance passive systems, which then yields semi-uniform input-to-state stability with disturbances. A smart generalist might read it to see how regularity arguments can simplify stability proofs for nonlinear infinite-dimensional systems like vibrating structures.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Linear exp. stability assumption is load-bearing for regularity transfer; unclear if it suffices for nonlinear observability without domain ID.","rationale":"The reader's weakest_assumption correctly isolates the linear exponential stability hypothesis as the point where regularity and observability are transferred; the concern above is a direct elaboration of that transfer step rather than a new objection. Because the abstract-only review left the details of the transfer unexamined, the verdict moves from UNVERDICTED to CONDITIONAL pending verification that the shared regularity is strong enough for the observability step.","tokens_in":1656,"tokens_out":326,"duration_ms":25523,"concrete_test":"Take the multidimensional wave example of §5; pick initial data in the interpolation space D(A^{1/2+ε}) and numerically integrate the saturated closed-loop system for two different saturation levels; check whether the observed decay rate of the energy matches the linear observability constant up to a uniform factor independent of saturation amplitude.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (semi-uniform ISS) rests on showing fractional Sobolev regularity of the free output is preserved under saturated feedback, so that linear observability estimates apply directly to nonlinear trajectories (avoiding nonlinear generator domain identification). This transfer is justified by invoking exponential stability of the unsaturated linear closed-loop. However, saturation is a bounded nonlinear map; the argument does not explicitly verify that the interpolation-space trajectories remain compatible with the linear observability inequality once the saturated input and disturbances are present, leaving open whether the constants or the semi-uniform character survive the nonlinearity.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates the long-time behavior of impedance passive (possibly infinite-dimensional) systems under saturated output feedback in the presence of external disturbances. Assuming exponential stability of the linear closed-loop without saturation or disturbances, it proves that fractional Sobolev regularity of the free output is preserved by the nonlinear saturated feedback when initial data lie in suitable interpolation spaces associated with the linear closed-loop generator. This shared regularity allows direct application of linear observability estimates to nonlinear trajectories, yielding a characterization of asymptotic behavior and, under impedance passivity plus exact observability and regularity assumptions, a semi-uniform input-to-state stability property. The results are illustrated by a multidimensional wave equation with nonlinear boundary damping.","tokens_in":1761,"tokens_out":345,"duration_ms":19188,"significance":"If the regularity-transfer argument is complete, the work offers a route to semi-uniform ISS estimates for nonlinear infinite-dimensional systems that avoids explicit identification of the nonlinear generator domain, a common technical obstacle in PDE control with nonlinear boundary conditions. The wave-equation example supplies a concrete test case.","major_comments":[{"comment":"The central regularity-transfer step (the argument that linear exponential stability of the unsaturated closed-loop implies preservation of interpolation-space regularity and direct applicability of the linear observability inequality once the bounded saturation map and disturbances are present) is load-bearing for the semi-uniform ISS claim. The manuscript must explicitly confirm that the constants and semi-uniform character survive the insertion of the saturated input; without this verification the transfer from linear to nonlinear trajectories remains incomplete.","section":"regularity preservation argument (main theorem on fractional Sobolev regularity)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the importance of the regularity-transfer argument. We respond to the single major comment below.","responses":[{"response":"We agree that an explicit verification paragraph improves clarity. The boundedness of the saturation map (which maps into a fixed ball) together with the given external disturbance ensures that the effective input to the linear observability inequality remains controlled in the same function space as in the linear case. Because the main regularity theorem already places both the state and output of the nonlinear closed-loop system in the identical interpolation spaces used for the linear observability estimate, the constants and the semi-uniform dependence on the initial-data norm carry over verbatim. In the revised manuscript we will add, immediately after Theorem 3.2, a short remark that records this fact and cites the precise linear observability statement (Assumption 4.1) to make the transfer fully explicit.","revision_made":"yes","referee_comment":"[regularity preservation argument (main theorem on fractional Sobolev regularity)] The central regularity-transfer step (the argument that linear exponential stability of the unsaturated closed-loop implies preservation of interpolation-space regularity and direct applicability of the linear observability inequality once the bounded saturation map and disturbances are present) is load-bearing for the semi-uniform ISS claim. The manuscript must explicitly confirm that the constants and semi-uniform character survive the insertion of the saturated input; without this verification the transfer from linear to nonlinear trajectories remains incomplete."}],"tokens_in":1299,"tokens_out":320,"duration_ms":20476,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main contribution is the claim that if the linear unsaturated feedback exponentially stabilizes the system, then fractional regularity of the output is preserved under saturation. This shared regularity lets them apply the linear observability estimates straight to the nonlinear trajectories and disturbances, avoiding any need to identify the domain of the nonlinear generator.\n\nThe argument is set up cleanly inside the impedance passivity framework. The wave equation example with nonlinear boundary damping makes the setting concrete and shows where the interpolation spaces come from. The assumption of linear exponential stability is used explicitly to carry both regularity and observability over.\n\nThe soft spot is exactly the transfer step flagged in the stress test. Saturation is a bounded nonlinearity, but it is not obvious that the semi-uniform constants in the observability inequality survive once the saturated input and external disturbances act on the system. The abstract states that the estimates carry over, yet without the detailed estimates in the proofs it is hard to see whether the semi-uniform decay rate remains intact or degrades. If the constants do hold, the result is useful; if they do not, the semi-uniform ISS claim weakens.\n\nThis is for people already working on infinite-dimensional passive systems and boundary control of PDEs. A reader who needs a way around nonlinear domain identification will find the technique worth checking. It is worth sending to peer review because the regularity idea is targeted and the example is standard enough for referees to test the transfer argument directly.","headline":"The paper shows fractional Sobolev regularity transfers from the linear closed-loop to the saturated nonlinear case, letting linear observability give semi-uniform ISS without nonlinear domain work.","tokens_in":2210,"tokens_out":362,"would_cite":false,"duration_ms":17493,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Saturated output feedback preserves fractional Sobolev regularity in impedance passive systems and establishes semi-uniform input-to-state stability.","keywords":["impedance passive systems","saturated feedback","semi-uniform stability","input-to-state stability","fractional Sobolev regularity","wave equation","boundary damping","observability estimates"],"falsifier":"A simulation or calculation for the wave equation example that checks whether the state norm satisfies the semi-uniform decay bound under saturated feedback and a nonzero constant disturbance.","tokens_in":2565,"feed_emoji":"","tokens_out":621,"duration_ms":21208,"temperature":0.7,"pith_summary":"The paper shows that for impedance passive systems, saturated output feedback preserves the fractional Sobolev regularity of both the output and the state whenever the initial condition lies in a suitable interpolation space tied to the linear closed-loop generator. This shared regularity between linear and nonlinear cases allows direct application of linear observability estimates to characterize long-time behavior even with external disturbances. The approach avoids explicit identification of the nonlinear generator domain. Under the impedance passivity assumption together with exact observability and regularity conditions, the result yields a semi-uniform input-to-state stability property, illustrated on a multidimensional wave equation with nonlinear boundary damping.","feed_headline":"Saturated feedback yields semi-uniform stability for passive systems","feed_subtitle":"Fractional regularity is preserved in interpolation spaces, enabling observability-based ISS estimates with disturbances.","key_machinery":"The interpolation space associated with the linear closed-loop generator, which transfers fractional regularity to the nonlinear saturated feedback via the impedance passivity framework.","core_discovery":"Under the assumption that the linear output feedback exponentially stabilizes the system in the absence of saturation and disturbances, the nonlinear closed-loop system preserves fractional Sobolev regularity of the free output in the corresponding interpolation spaces. Combined with exact observability estimates for the linear system, this regularity yields a characterization of the asymptotic behavior and establishes semi-uniform input-to-state stability for the nonlinear system in the presence of disturbances.","pith_inferences":["The regularity-transfer technique could simplify analysis for other classes of nonlinear boundary feedback in distributed systems.","The framework may extend to time-dependent or state-dependent saturation levels while retaining the same stability conclusion.","Similar interpolation-space arguments might apply to other passive structures such as port-Hamiltonian systems."],"forward_implications":["The output and state inherit the fractional regularity of the linear system when initial data are in the interpolation space.","Linear observability estimates apply directly to bound the nonlinear closed-loop trajectories.","Semi-uniform input-to-state stability holds under impedance passivity and exact observability.","The result covers infinite-dimensional examples such as the wave equation with nonlinear boundary damping."],"fun_headline_variants":["Semi-uniform ISS for impedance passive systems with saturated feedback","Fractional regularity preserved by saturated nonlinear feedback","Semi-uniform input-to-state stability via exact observability","Regularity yields asymptotic behavior for saturated passive systems","Saturated feedback maintains Sobolev regularity in passive systems"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The linear output feedback exponentially stabilizes the system in the absence of saturation and disturbances, allowing transfer of regularity and observability to the nonlinear case.","fun_headline_variants_meta":{"raw":{"variants":["Semi-uniform ISS for impedance passive systems with saturated feedback","Fractional regularity preserved by saturated nonlinear feedback","Semi-uniform input-to-state stability via exact observability","Regularity yields asymptotic behavior for saturated passive systems","Saturated feedback maintains Sobolev regularity in passive systems"]},"model":"grok-4.3","cost_usd":0.004009,"raw_usage":{"total_tokens":2025,"prompt_tokens":628,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":40087000,"prompt_tokens_details":{"text_tokens":628,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1326,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":628,"tokens_out":71,"duration_ms":10690,"temperature":1.0,"reasoning_tokens":1326,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T09:50:51.398723+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A simulation or calculation for the wave equation example that checks whether the state norm satisfies the semi-uniform decay bound under saturated feedback and a nonzero constant disturbance.","supporting_citations":[],"review_version":1}