{"id":"d5d14eb6-2fd1-4734-b692-63eafdf2fead","arxiv_id":"2501.08709","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"kEDMD-MPC: practical asymptotic stability of the MPC closed loop follows from cost controllability of the true system and pointwise proportional error bounds, without invariance assumptions.","lead":"Using kernel EDMD as a data-driven surrogate, the authors prove that model predictive control stabilizes nonlinear control-affine systems to a small neighborhood of the origin, without terminal conditions or invariance assumptions on the dictionary. The result matters because it extends certified data-driven MPC to kernel surrogates, where basis functions are chosen by the kernel rather than hand-crafted.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's perturbed cost-controllability bound grows exponentially in the horizon when 2(L_F+Cε)^2>1, so the claimed bounded sequence B^ε_k is not established; Theorem 3's α condition is therefore unquantified.","rationale":"I read the paper as aiming to transfer robust MPC stability to kernel EDMD surrogates via cost controllability, avoiding the invariance assumptions of earlier EDMD-MPC work. The strongest positive element is Theorem 2's pointwise proportional error bound, which genuinely gives a state-dependent model error. The load-bearing step is Proposition 1: without a bounded sequence B^ε_k, the standard relaxed Lyapunov machinery invoked through [9, Thm 11.10] does not apply, and the α condition in Theorem 3 is not justified. The proof as written cannot yield boundedness because the constants c^ε_1 and c^ε_2 involve powers of d = 2(L_F + C ε_hX)^2, which exceed 1 for typical Lipschitz constants; this is a distinct and more fundamental issue than the reader's observation that the α formula mixes nominal and perturbed constants. The εc = 0 restriction is an additional limitation of the proven guarantee, but it is not the central gap. The approach is plausible and likely repairable with a direct controllability argument for the surrogate or an explicit h_X-dependent horizon condition, so I do not recommend rejection. My verdict remains CONDITIONAL, matching the reader's overall assessment but with a sharper reason: the main theorem's key hypothesis is not established as stated.","tokens_in":11909,"tokens_out":16073,"duration_ms":178301,"concrete_test":"For the scalar linear benchmark x^+ = a x + u with a = 1.5 and stage cost x²+u² (nominal B_N ≡ 1), compute the true cost-controllability ratio sup_{|x|≤1} V^ε_N(x)/x² for the perturbed surrogate x^+ = a x + u + δ x, for δ ∈ {10^-3, 10^-6} and N = 2, 4, 8, 16, 32. If this ratio stays bounded as N grows, Proposition 1's conclusion is true and the exponential constants in its proof are merely loose, so Theorem 3 can be repaired by replacing the same-control argument; if it grows, bounded cost controllability is genuinely lost and the main theorem's assumption fails in the simplest possible case.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central stability theorem rests on Proposition 1, which asserts that the kEDMD surrogate inherits cost controllability with a monotonically increasing and bounded sequence (B^ε_k). In the proof, comparing surrogate and nominal trajectories under the same control yields J^ε_N ≤ (B_N + c^ε_1 C ε_hX λ̄/λ + c^ε_2 C² ε_hX² λ̄/λ) ℓ*(x), with c^ε_1 = Σ_{k=0}^{N-1} B_k (Σ_{i=0}^k d^i)(Σ_{i=0}^{N-1-k} d^i), c^ε_2 = Σ_{k=0}^{N-2} B_k Σ_{i=0}^{N-2-k} d^i, and d = 2(L_F + C ε_hX)^2. For any fixed h_X with 2L_F² > 1, d > 1, and these coefficients grow like d^N. Hence B^ε_N is not bounded in N, contrary to the proposition's claim. The van der Pol example has L_F > 1 (Δt = 0.05), so this regime is not exotic. Because Theorem 3 then states α_N using the nominal B_k and defers the proof to [3, Thm 10], there is no demonstrated choice of h_X making the perturbed α^ε_N ∈ (0,1) for the horizon used. The PAS result is therefore conditional on an unproven finite-horizon controllability margin.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes kEDMD-MPC, a model predictive control scheme that uses a kernel-EDMD surrogate of a control-affine nonlinear system. Building on pointwise error bounds proportional to the state distance, the authors aim to prove practical asymptotic stability (PAS) of the origin for the closed loop without terminal ingredients, using cost controllability of the original system. The scheme is stated in Algorithm 1, theoretical results are given in Section 4 (Lemma 1, Proposition 1, Theorem 3), and numerical experiments on a van der Pol oscillator are reported in Section 5.","tokens_in":12249,"tokens_out":9141,"duration_ms":85353,"significance":"If the proof were complete, the paper would provide a useful advance: it would remove the restrictive invariance assumptions often needed in EDMD-based MPC stability proofs and would give deterministic, data-dependent stability guarantees for kernel-EDMD surrogates. The central idea of leveraging pointwise proportional error bounds and cost controllability is promising, and the numerical examples support the qualitative behavior. However, the main stability theorem depends on a cost-controllability preservation result whose proof is incomplete, and the theorem itself uses nominal rather than perturbed constants, so the claimed guarantees are not established as stated.","major_comments":[{"comment":"The boundedness of the sequence (B^ε_k) is not proven and is in fact false under the stated assumptions. In the proof, the coefficients c^ε_1 and c^ε_2 contain sums of powers of d = 2(L_F + C ε_hX)^2. When d > 1, these coefficients grow like d^N, so the expression for B^ε_N grows exponentially in N. The proposition asserts the existence of a monotonically increasing and bounded sequence (B^ε_k) but imposes no condition implying d < 1, and for the van der Pol example in Section 5 the Lipschitz constant L_F is greater than 1, so d ≈ 2L_F^2 > 1. Thus the proof does not establish the claimed bounded sequence, and the cost-controllability preservation result is not available in the form needed by Theorem 3.","section":"Section 4, Proposition 1"},{"comment":"The horizon condition in Theorem 3 uses the nominal constants B_k in the formula for α_N, whereas the perturbed constants B^ε_k from Proposition 1 are what enter the stability analysis of the surrogate-based MPC loop. The argument needs to show that α^ε_N ∈ (0,1) for the chosen horizon N, using the perturbed constants, and to state a quantitative smallness condition on h_X that guarantees this. The sentence 'The proof resembles the proof of Theorem 10 in [3]' is not a substitute for this step, since the perturbed constants change the relevant inequalities. Because Proposition 1 only claims pointwise convergence B^ε_k → B_k as h_X → 0 for each fixed k, the theorem requires an additional uniformity or margin argument that is not provided.","section":"Section 4, Theorem 3"},{"comment":"All the theoretical stability guarantees are proven only for εc = 0, i.e., for exact data at the virtual cluster points. The numerical section uses εc = √2/d > 0 and states that PAS 'is also obtained' in that case, but the theory does not cover this regime. This is a significant scope limitation: the advertised data-driven stability guarantee does not apply to the sampling noise or data corruption represented by nonzero cluster radius. The manuscript should either extend the theorem to εc > 0 or explicitly frame the εc > 0 behavior as a conjecture supported by simulations.","section":"Section 4, Lemma 1 / Proposition 1 / Theorem 3"}],"minor_comments":[{"comment":"The symbol ε is used both for the total error bound in (8) and as a perturbation parameter in definitions such as the surrogate F^ε and the admissibility set U^ε_N. This overloading makes statements like 'for all ε ∈ (0, ε0]' ambiguous, since ε in (8) is determined by h_X and εc. Please introduce a separate symbol for the error level (for example, δ or η) and use it consistently.","section":"Section 2, Theorem 2"},{"comment":"The quantifier order in Definition 2 is confusing: 'for each fill distance h_X and cluster radius ε_c such that (8) holds with ε ∈ (0, ε0]' suggests that h_X and ε_c are chosen after ε0, but the error bound ε depends on these quantities. Please rephrase to clarify that the existence of ε0 quantifies over sufficiently small h_X and εc.","section":"Section 3, Definition 2"},{"comment":"The sentence 'Note that the first summands in J^ε_N (ˆx, u) and J_N (ˆx, u) coincide' should say 'the first summand (k=0) coincides', since the two sums are over the same k-range and the initial state is the same. This is a minor wording issue.","section":"Section 4, Proposition 1 proof"},{"comment":"The Lipschitz constant L_F of the van der Pol map (18) is not reported. Since the validity of the bound in Proposition 1 depends on whether d = 2(L_F + C ε_hX)^2 is less than or greater than 1, reporting L_F (or the relevant bound) would help the reader assess the example against the assumptions.","section":"Section 5, paragraph after Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the authors' own prior work: the error bound in Theorem 2 is adapted from [4] (a preprint), and the proof of Theorem 3 is delegated to [3] (a published IEEE TAC paper). This is not by itself disqualifying, but the reviewer should be aware that the novelty increment over [3] and [4] is concentrated in Proposition 1 and its application, and those are exactly the parts with the unproven boundedness claim. If the authors can fix Proposition 1 and supply a fully detailed proof of Theorem 3 with the perturbed constants, the paper would be suitable; otherwise the claimed stability guarantee is not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the main stability theorem is not proven in the text, and Proposition 1, which is supposed to do the heavy lifting, has a real gap. The stress-test note is right: the coefficients c_1^ε and c_2^ε grow like d^N with d = 2(L_F + Cε_hX)^2. For d > 1, which holds for the van der Pol example and for any system with L_F > 1/√2, the sequence B_k^ε is not bounded as claimed. The proof of Proposition 1 only shows a finite-horizon bound that grows exponentially in N; it does not give the uniform bound that cost controllability requires and that Theorem 3's α formula assumes. On top of that, Theorem 3 states the α condition using the nominal B_k, not the perturbed B_k^ε, and defers the proof to [3]. So as written, the claimed ε-PAS is conditional on an unquantified controllability margin.\n\nThat is a pity, because the paper has a lot going for it. The combination of kernel EDMD error bounds from [4] with the stability framework from [3] is a natural and worthwhile step, and the pointwise proportional error bound is the right tool. The idea of removing the dictionary invariance assumption is genuinely new and important for the Koopman-MPC subfield. The numerical section is honest and illustrates the expected behavior, including the offset for εc > 0.\n\nThe other limitations are less severe: the theory requires εc = 0, so exact data at cluster points; sampling noise is not covered. The proof of Lemma 1 depends on derivative error bounds that are plausible but not fully detailed. And the paper leans on several self-authored preprints, but those are independent results with their own proofs, so that is not a circularity problem.\n\nNet: the conceptual contribution is solid and the errors are in the right place, but the central perturbation estimate is not correct as stated. The paper needs a substantial revision of Proposition 1 and Theorem 3—either by deriving a finite-horizon controllability margin that is compatible with the growth rates, or by adding a condition on L_F and C. Until then, the stability guarantee is not established.\n\nWho should read this: anyone working on data-driven MPC with Koopman surrogates will benefit from the problem setup and the error framework, but they should treat the main theorem as open. I would send it out for review because the question matters and the authors have a credible path to fixing it, but I would not accept it as is.","headline":"Clear idea and promising error framework, but Proposition 1's perturbed cost-controllability bound is not bounded as claimed, so the main stability theorem is unproven.","tokens_in":12785,"tokens_out":3845,"would_cite":false,"duration_ms":36452,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D15","93C10","93C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Kernel EDMD-based model predictive control can stabilize nonlinear systems from data without terminal conditions, with practical asymptotic stability guarantees.","keywords":["kernel extended dynamic mode decomposition","Koopman operator","model predictive control","practical asymptotic stability","cost controllability","Wendland kernels","data-driven control","error bounds"],"falsifier":"For the van der Pol example with $\\varepsilon_c=0$, refine the Chebyshev grid (increase $d$) and record the asymptotic offset of the closed-loop trajectory: the theorem predicts the offset goes to zero as $h_X\\to 0$. If the offset does not shrink toward zero with increasing $d$, or if, for a fixed small $h_X$, the value-function decrease $\\alpha$ computed from the surrogate violates $\\alpha\\in(0,1)$, the claimed guarantee fails.","tokens_in":11698,"feed_emoji":"📉","tokens_out":5428,"duration_ms":56030,"temperature":0.7,"pith_summary":"The paper's aim is a stability certificate for model predictive control (MPC) when the prediction model is a data-driven kernel EDMD surrogate of a nonlinear control-affine system. It claims that, if the true system is cost controllable and the surrogate error is pointwise proportional to the distance to the data grid, then the MPC closed loop is practically asymptotically stable, meaning it converges to a neighborhood of the origin whose size shrinks with the data density, without any stabilizing terminal cost or terminal constraint. This matters because terminal ingredients are often the hard part of nonlinear MPC design, and EDMD-based surrogates have previously needed restrictive invariance assumptions to get such guarantees. The paper proves the claim through Theorem 3 and supports it with simulations on a van der Pol oscillator.","feed_headline":"Data-driven Koopman MPC gets proven stability without terminal costs","feed_subtitle":"Kernel EDMD surrogates inherit cost controllability, so closed-loop error shrinks as data density grows.","key_machinery":"The load-bearing object is the two-level kernel EDMD surrogate: first, local linear regression at virtual cluster points $x_i$ approximates the system matrices $[g_0(x_i)\\mid G(x_i)]$ from sample triples; second, Wendland-kernel interpolation lifts these approximations into a control-affine surrogate $x^+=g^\\varepsilon_0(x)+G^\\varepsilon(x)u$. Its key property is the pointwise error bound $\\|F(x,u)-F^\\varepsilon(x,u)\\|_\\infty\\le C_1 \\varepsilon_{h_X}\\,\\mathrm{dist}(x,X)+C_2 c\\|K_X^{-1}\\|\\varepsilon_c$, which with $\\varepsilon_c=0$ becomes proportional to $\\|x\\|$ because $0\\in X$. Proposition 1 converts that proportional error into perturbed cost controllability, and Theorem 3 feeds the perturbed constants into the relaxed-Lyapunov framework of the nonlinear-MPC PAS framework used in [9] to obtain practical asymptotic stability.","core_discovery":"The central discovery is that kernel EDMD surrogates inherit cost controllability with controllable accuracy: under exact data at cluster points ($\\varepsilon_c=0$) and small fill distance $h_X$, the surrogate's cost-controllability constants $B^\\varepsilon_N$ converge to the nominal constants $B_N$. With a horizon $N$ chosen so that the contraction factor $\\alpha_N\\in(0,1)$ from the nominal cost-controllability analysis remains valid, the EDMD-based MPC controller renders the origin $\\varepsilon$-practically asymptotically stable on a sublevel set of its value function, meaning every trajectory eventually enters and stays in a ball whose radius tends to zero as the fill distance $h_X$ tends to zero.","pith_inferences":["Extension: the same proof pattern would work for any surrogate whose error is proportional to the state and uniformly continuous in the state, so the result is a template for stability certificates beyond kernel EDMD.","Extension: treating sampling noise would require a quantitative smallness condition on $\\varepsilon_c$; a probabilistic version of the bound could yield high-probability practical stability with noisy data.","Extension: the exponential decay seen in the simulations suggests practical exponential stability may hold under a strengthened uniformity condition on the cost-controllability constants, which the paper does not claim."],"forward_implications":["Every control-affine system satisfying cost controllability admits a kernel EDMD surrogate for which MPC without terminal ingredients is practically asymptotically stable; the same horizon that works for the nominal system continues to work once the fill distance is small enough.","The ultimate bound on the closed-loop trajectory can be made arbitrarily small by refining the cluster-point grid, since the error bound and hence $\\varepsilon$ shrink with $h_X$.","State constraints in the surrogate optimization must be tightened by the error bound, so the feasible region shrinks as data density decreases.","Because the error bound is pointwise and vanishes at the cluster points, the stability neighborhood is centered at the origin and does not require invariance of the dictionary."],"supporting_citations":[{"why":"Supplies the kernel EDMD construction for control systems with the pointwise proportional error bound that the stability argument is built on.","marker":"[4]"},{"why":"Provides the $L_\\infty$ error bound for autonomous kernel EDMD and the fill-distance dependence used in the error analysis.","marker":"[12]"},{"why":"Gives the earlier EDMD-MPC stability proof under invariance assumptions, whose structure is reused in Theorem 3.","marker":"[3]"},{"why":"Supplies the practical asymptotic stability framework and the relaxed Lyapunov inequality conditions that the proof verifies.","marker":"[9]"},{"why":"Establishes that cost controllability without terminal ingredients suffices for stability of MPC with a sufficiently long horizon.","marker":"[2]"},{"why":"Provides the Wendland kernel properties, RKHS interpolation estimates, and derivative bounds used in Lemma 1 and Theorem 2.","marker":"[20]"}],"fun_headline_variants":["Kernel EDMD gives Koopman MPC stability without terminal costs","Stable Koopman MPC from kernel EDMD, no terminal costs needed","Data-driven Koopman MPC stability via kernel EDMD, no terminal conditions","Kernel EDMD enables stable Koopman MPC without terminal restrictions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantee rests on the surrogate inheriting cost controllability with constants close enough to the nominal ones that the horizon-$N$ contraction factor $\\alpha_N^\\varepsilon$ in Theorem 3 stays in $(0,1)$; the paper does not quantify how small $h_X$ must be to ensure this, and the proof requires exact data at the cluster points ($\\varepsilon_c=0$).","fun_headline_variants_meta":{"raw":{"variants":["Kernel EDMD gives Koopman MPC stability without terminal costs","Stable Koopman MPC from kernel EDMD, no terminal costs needed","Data-driven Koopman MPC stability via kernel EDMD, no terminal conditions","Kernel EDMD enables stable Koopman MPC without terminal restrictions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":2863,"prompt_tokens":784,"completion_tokens":2079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":400,"completion_tokens_details":{"reasoning_tokens":2000}},"tokens_in":400,"tokens_out":2079,"duration_ms":13305,"temperature":1.0,"reasoning_tokens":2000,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:20:00.146437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the van der Pol example with $\\varepsilon_c=0$, refine the Chebyshev grid (increase $d$) and record the asymptotic offset of the closed-loop trajectory: the theorem predicts the offset goes to zero as $h_X\\to 0$. If the offset does not shrink toward zero with increasing $d$, or if, for a fixed small $h_X$, the value-function decrease $\\alpha$ computed from the surrogate violates $\\alpha\\in(0,1)$, the claimed guarantee fails.","supporting_citations":[{"cited_title":"Data-driven MPC with stability guarantees using extended dynamic mode decomposition","cited_arxiv_id":null,"evidence_quote":"Gives the earlier EDMD-MPC stability proof under invariance assumptions, whose structure is reused in Theorem 3."},{"cited_title":"Nonlinear model predictive control","cited_arxiv_id":null,"evidence_quote":"Supplies the practical asymptotic stability framework and the relaxed Lyapunov inequality conditions that the proof verifies."},{"cited_title":"Stability and feasibility of state constrained MPC without stabilizing terminal constraints","cited_arxiv_id":null,"evidence_quote":"Establishes that cost controllability without terminal ingredients suffices for stability of MPC with a sufficiently long horizon."},{"cited_title":"Scattered data approximation, volume 17","cited_arxiv_id":null,"evidence_quote":"Provides the Wendland kernel properties, RKHS interpolation estimates, and derivative bounds used in Lemma 1 and Theorem 2."}],"review_version":1}