{"id":"05922796-b3eb-4c57-ace5-6b012e26f37e","arxiv_id":"2608.11932","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a known proportional terminal cost, finite-horizon gain variation makes the open-loop dynamics and quadratic cost jointly identifiable in continuous time, and the CR-IOC algorithm recovers them consistently from noisy samples.","lead":"This paper shows when the unknown dynamics and cost weights of a finite-horizon linear-quadratic control problem can be recovered together from observed trajectories, and provides a practical algorithm with statistical guarantees. It matters because inverse optimal control, which infers an agent's objective from its behavior, previously lacked a systematic joint identifiability and error analysis for this continuous-time setting.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.9's bound (68) adds an unquantified err_ODE, so Theorem 5.10 is not a fully non-asymptotic consistency guarantee; the end-to-end claim depends on an unspecified Lyapunov-solver error.","rationale":"I read the paper in good faith. The core identifiability argument in Section 3 is coherent: given full-row-rank state data and time-varying gain, K(t) is recovered, then (A,B) via the gain-variation Gramian, then (Q,R) via the structured stationarity operator. The restriction H=alphaQ is explicit, and the reader's concern about it is legitimate, but the paper does not claim identifiability outside that class; the central theorem is internally consistent within its stated assumptions. The more load-bearing issue for the strongest claim is the gap in the perturbation analysis. Lemma 5.9 is presented as the key end-to-end operator perturbation bound, and Eq. (68) is the only place where the numerical solver error enters. Leaving this term unquantified means the 'non-asymptotic' bound is not actually a bound, and Theorem 5.10's consistency is conditional on a black-box err_ODE -> 0 rather than on algorithm parameters. This is fixable with a concrete solver analysis, so it does not require rejecting the paper; it supports the CONDITIONAL verdict already given. The concrete test above would settle whether the gap is merely cosmetic or causes the consistency claim to fail.","tokens_in":35700,"tokens_out":14295,"duration_ms":153679,"concrete_test":"Fix a concrete Lyapunov solver, e.g., backward Euler on the same grid with step h = Delta t / p for p >= 1, and derive an explicit bound for err_ODE in Lemma 5.9 in terms of h, T, the Lipschitz constants M_Ac, L_K, and the condition numbers involved. Then verify that the final bound in Theorem 5.10 tends to zero under the stated limits (eta_x, eta_u -> 0, Delta -> 0, eta_x/Delta -> 0, h -> 0). If no such bound can be derived, the consistency theorem should be restated as conditional on an oracle ODE solver and the 'non-asymptotic' terminology should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper advertises non-asymptotic perturbation bounds and end-to-end consistency, and Theorem 5.10 is one of the two central results. In Lemma 5.9, the Frobenius estimate for ||hat M_L - M_L|| is derived from deterministic terms, and then Eq. (68) simply adds 'err_ODE' for the numerical solution of the terminal-value Lyapunov equations that define the columns of M_L and hat M_L. No bound for err_ODE in terms of the solver step, problem constants, Delta, or sample size is provided, and no specific solver is required. Theorem 5.10 then assumes err_ODE -> 0 as part of the consistency hypotheses. This makes the advertised guarantee conditional on an external numerical error that is not controlled by the paper's assumptions. If the Lyapunov solves are not refined in a way that is uniform over the vanishing Delta, eta_x, eta_u, the final estimate (70) may diverge or be vacuous. The identifiability theory (Theorem 3.8) is not affected, but the statistical reconstruction claim is incomplete at the point where it is supposed to be fully non-asymptotic.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies finite-horizon continuous-time inverse LQR when both the system matrices and the quadratic cost are unknown. It introduces three conditioning indices (c_X, c_AB, c_QRH) and proves that, within the normalized structured class H=αQ with known α and tr(R)=m, positivity of c_AB and c_QRH is sufficient for global identifiability, with c_QRH also necessary when Q*≻0. The paper then proposes an estimator (CR-IOC) that denoises sampled trajectories, reconstructs K and A_c, separates (A,B) in closed form via a gain-variation Gramian, and recovers (Q,R) through a convex semidefinite program. Staged perturbation bounds are derived, and an end-to-end consistency theorem is stated under sub-Gaussian observation noise. Numerical experiments on mass-spring and dense 4x2 benchmarks illustrate the predicted trends and the diagnostic value of the indices.","tokens_in":35915,"tokens_out":13424,"duration_ms":141785,"significance":"The identifiability results are the main strength of the paper. The authors cleanly identify finite-horizon gain variation as the structural mechanism that separates (A,B) from the closed-loop dynamics, and they make the conditions explicit and computable. The proofs of Theorems 3.3, 3.7, and 3.8 are detailed and appear correct, and Proposition 3.5 correctly delineates why the H=αQ assumption is needed for cost identifiability. The perturbation analysis is also largely explicit, with each reconstruction stage controlled by its conditioning index, and the empirical indices provide a falsifiable diagnostic for experimental design. However, the advertised fully non-asymptotic end-to-end guarantee is not complete: Lemma 5.9 leaves the numerical Lyapunov-solver error unquantified, so Theorem 5.10 is conditional on an external error term that the paper does not control. If this gap is closed, the paper would be a substantial contribution to inverse optimal control; even as it stands, the structural identifiability theory is a significant step.","major_comments":[{"comment":"The end-to-end non-asymptotic guarantee is incomplete because the bound for ||M_hat_L - M_L|| is written as sqrt(T(d_Q beta_Q^2 + d_R beta_R^2)) + err_ODE, and err_ODE is never quantified. No bound is given in terms of the numerical integrator, step size, horizon constants, Delta, sample size, or any solver parameter, and no specific solver is required. Theorem 5.10 then simply assumes err_ODE -> 0 as part of the consistency hypotheses. Consequently, the final estimate (70) is conditional on a numerical error that is external to the paper's assumptions; if the Lyapunov solves are not refined uniformly as eta_x, eta_u, and Delta vanish, the condition mu_M < c_QRH,L in (69) may fail and the advertised guarantee becomes vacuous. Please provide a quantitative bound for err_ODE (for example, by specifying a Lyapunov ODE solver and its step-size/tolerance error) or explicitly reformulate Theorem 5.10 as a conditional statement with err_ODE as an additional hypothesis that the user must control.","section":"Section 5.2, Lemma 5.9, Eq. (68)"},{"comment":"The paper's central claim that the true cost weighting matrices are recovered is only valid within the structured family H=alpha Q with a known scalar alpha, as stated in Assumption 2.2. Proposition 3.5 shows that without this assumption the cost is not identifiable from noiseless closed-loop data for underactuated systems (rank(B)=m<n), so this is not a technical convenience but a genuine scope limitation. The abstract and introduction should state this caveat prominently when they say the method recovers 'the true cost weighting matrices'; as written, the abstract can be read as claiming unconditional recovery. The mathematical results are not affected, but the presentation should not overstate the scope.","section":"Abstract and Section 3, Theorem 3.8"}],"minor_comments":[{"comment":"The phrase 'We proposed CR-IOC' should be 'we propose CR-IOC' to match the present-tense summary of the paper's contribution.","section":"Section 7, conclusion"},{"comment":"In the necessity construction, E = b p^T may in principle reduce the rank of B = B* + E, which would violate Assumption (A2). For a generic nonzero b this does not happen, and stabilizability is preserved for small perturbations, but the proof should state this explicitly or choose b so that rank(B)=m is maintained.","section":"Theorem 3.3, necessity part"},{"comment":"The sentence beginning 'Indeed, if two structured normalized pairs...' is grammatically awkward and interrupts the definition of V_alpha; it would be clearer as a separate remark after the definition of c_QRH.","section":"Definition 2.7(iii)"},{"comment":"The practical rule Delta ≍ eta_x^{1/2} is stated as a dominant-balance heuristic, but the displayed bound (59) also contains terms such as M_f eta_x and eta_x^2/Delta that are not included in that balance; a short clarification of which terms are being balanced would improve precision.","section":"Section 5.1, after Theorem 5.4"},{"comment":"The conditioning study reports mean errors over 30 trials but does not provide error bars or quantiles for the three settings; adding them would make the stage-selective degradation in R2 and R3 easier to assess.","section":"Section 6.5, Table 3"}],"recommendation":"major_revision","confidential_remarks":"The identifiability theory is solid and the three conditioning indices are a genuinely useful contribution. The main obstacle is the unquantified err_ODE term in Lemma 5.9 and Theorem 5.10, which prevents the paper from delivering the advertised fully non-asymptotic end-to-end guarantee. I would be willing to re-review after the authors provide a quantitative treatment of the numerical Lyapunov-solver error or explicitly restrict the consistency claim. The scope caveat regarding H=alpha Q should also be reflected in the abstract. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look. The paper's real contribution is showing, cleanly and constructively, that finite-horizon gain variation separates the unknown (A,B) from the time-varying closed-loop dynamics in continuous time. The three conditioning indices (c_X, c_AB, c_QRH) are a genuinely useful decomposition: they appear in the identifiability theorems, the algorithm, and the error bounds, so the paper has a unified structure rather than a grab bag of results. Theorem 3.8 is the centerpiece and it looks correct — under H=αQ and tr(R)=m, c_AB>0 and c_QRH>0 give global identifiability of the whole tuple, and c_QRH>0 is necessary when Q⋆≻0. The closed-form recovery of (A,B) via the gain-variation Gramian is elegant, and the staged perturbation analysis is careful about where each index enters. The experiments support the qualitative story, especially the diagnostic value of the indices in the R2/R3 failure modes.\n\nThe soft spots are in the statistical claims, not the identifiability theory. Lemma 5.9 adds an unquantified err_ODE to the bound on ‖M̂_L−M_L‖; Theorem 5.10 then assumes err_ODE→0 as part of its hypotheses. That is honest but undercuts the word “non-asymptotic” in the abstract and the claim of end-to-end error propagation — the final bound really depends on a numerical error the paper does not control. A referee should ask the authors to either quantify err_ODE for a specific Lyapunov solver (step size, tolerance, and dependence on c_QRH and the horizon) or drop the fully non-asymptotic phrasing. The H=αQ assumption is strong, but the paper’s Proposition 3.5 shows that without it the cost is not identifiable in underactuated systems, so at least the limitation is explicit rather than hidden. Experiments are not accompanied by code or data, which makes the numbers hard to check; for a numerical paper, that should be fixed.\n\nMy bottom line: this is a genuine contribution to continuous-time inverse optimal control, well worth sending to serious referees. It needs revision — quantify or properly scope err_ODE, and provide code/data — but the core identifiability results look solid and should stand.","headline":"Solid continuous-time joint identifiability theory for finite-horizon inverse LQR, held back by an unquantified solver-error term in the advertised non-asymptotic bounds and by the strong H=αQ assumption.","tokens_in":36461,"tokens_out":1681,"would_cite":true,"duration_ms":19365,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49N10","93B30","93E12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Gain variation lets finite-horizon data reveal dynamics and LQR cost","keywords":["inverse optimal control","linear quadratic regulator","unknown dynamics","identifiability","conditioning indices","finite horizon","consistent estimation","semidefinite programming"],"falsifier":"For the central theorem, generate noiseless finite-horizon trajectories from a system satisfying $H=\\alpha Q$ with $c_{AB}>0$ and $c_{QRH}>0$, and search for two distinct admissible normalized tuples that produce exactly the same state and input curves; if such a pair exists, Theorem 3.8 is false. For the boundary of the claim, take a system with $\\mathrm{rank}(B)=m<n$ and any nonzero symmetric $S$ with $SB=0$; Proposition 3.5 predicts that for small $\\tau<0$ the triple $(Q+\\tau(A^\\top S+SA), R, H-\\tau S)$ produces exactly the same closed-loop trajectories as $(Q,R,H)$ while being a different admissible cost, and direct simulation can verify or refute that prediction.","tokens_in":35471,"feed_emoji":"🎛️","tokens_out":8881,"duration_ms":85219,"temperature":0.7,"pith_summary":"This paper asks whether observations of an agent's optimal closed-loop trajectories can recover both the unknown linear dynamics and the unknown quadratic cost that generated them. In finite-horizon continuous-time LQR, the optimal gain varies with time, and the paper shows that this variation is the key structural resource: it allows the constant open-loop matrices $(A,B)$ to be separated from the time-varying closed-loop matrix $A_c(t)=A-BK(t)$. Within the admissible class where the terminal cost is proportional to the running state cost ($H=\\alpha Q$) and the input weight is normalized by $\\mathrm{tr}(R)=m$, the paper proves that two computable conditions, gain-variation richness and injectivity of a structured residual operator, guarantee global identifiability of the true tuple, and it gives necessary conditions when $Q^\\star\\succ 0$. It also supplies a sampled-data reconstruction algorithm with non-asymptotic perturbation bounds and statistical consistency under sub-Gaussian noise. A sympathetic reader cares because the recovered cost is the true cost weighting, not merely a behaviorally equivalent surrogate.","feed_headline":"Gain variation lets finite-horizon data reveal dynamics and LQR cost","feed_subtitle":"Three computable indices predict when closed-loop trajectories pin down the true (A,B,Q,R,H), with a consistent estimator.","key_machinery":"The paper's organizing devices are three conditioning indices. $c_X=\\inf_t \\lambda_{\\min}(X(t)X(t)^\\top)$ measures state richness and controls recovery of the gain $K(t)$ and the closed-loop matrix $A_c(t)$; $c_{AB}=\\lambda_{\\min}\\big(\\int_0^T \\Delta K(t)\\Delta K(t)^\\top dt\\big)$ measures the directional richness of the gain variation and controls separation of the constant $(A,B)$ from $A_c(t)=A-BK(t)$; $c_{QRH}$ is the restricted minimum gain of the structured stationarity residual operator $\\mathcal{M}_\\alpha$ on the trace-normalized difference space $V_\\alpha$, and it controls injectivity of the map from normalized $(Q,R)$ with $H=\\alpha Q$ to the residual $B^\\top P(t)-RK(t)$. The same three scalars appear in the identifiability theorem, in the weighting of the reconstruction stages, and in the non-asymptotic perturbation bounds.","core_discovery":"The central claim is Theorem 3.8: from noiseless finite-horizon closed-loop trajectories, the true tuple $(A^\\star,B^\\star,Q^\\star,R^\\star,H^\\star)$ is globally identifiable within the admissible class $H=\\alpha Q$, $\\mathrm{tr}(R)=m$ whenever the gain-variation Gramian condition $c_{AB}>0$ and the structured cost injectivity condition $c_{QRH}>0$ hold; if $Q^\\star\\succ 0$, $c_{QRH}>0$ is also necessary. The finite-horizon Riccati terminal condition makes the optimal gain $K(t)$ time-varying, and the paper proves from $A_c(t)=A-BK(t)$ that the directional richness of $\\Delta K(t)=K(t)-\\bar K$ separates $A$ from $B$, while the stationarity residual operator $\\mathcal{M}_\\alpha(Q,R)=B^\\top P(t)-RK(t)$, restricted to trace-normalized perturbations, has trivial kernel exactly when $c_{QRH}>0$. The paper further proves (Theorem 5.10) that the CR-IOC estimator built on these stages is consistent as observation noise and sampling step vanish together, with the same indices controlling error propagation.","pith_inferences":["Because all identifiability results require the known proportionality $H=\\alpha Q$, a natural testable extension is to treat $\\alpha$ as an unknown structural parameter and select it by comparing recovery errors across candidate values; the paper's residual operator gives a ready-made objective for that selection.","The same operator decomposition should transfer to other structured cost families, for example block-diagonal $R$ or terminal costs $H=\\gamma I$, with the difference space $V_\\alpha$ replaced accordingly; $c_{QRH}$ then becomes a ready-made injectivity measure for each family.","The necessity of $c_{QRH}>0$ is proven only under $Q^\\star\\succ 0$, so the boundary case where $Q^\\star$ is only positive semidefinite is not covered; characterizing the kernel of $\\mathcal{M}_\\alpha$ on that boundary is a direct open problem suggested by Theorem 3.7."],"forward_implications":["Closed-loop trajectory data alone can identify the true open-loop matrices and true normalized cost weights, without assuming known dynamics or settling for a behaviorally equivalent surrogate.","Finite-horizon effects should be treated as an information source: short horizons or terminal penalties that make $K(t)$ nearly constant shrink $c_{AB}$, so experiments should be designed to excite gain variation.","The empirical indices provide actionable diagnostics: a small $\\hat c_{AB}$ flags insufficient gain variation rather than solver failure, while a small $\\hat c_{QRH}$ points to a cost-stage degeneracy and suggests retaining late-horizon information or changing the cost parametrization.","The non-asymptotic bounds give quantitative error control with explicit dependence on the indices, so a user can predict how accuracy degrades as conditions weaken."],"supporting_citations":[{"why":"Establishes that infinite-horizon closed-loop data identify only the closed-loop pair and that finite horizons can restore identifiability, supplying the problem this paper answers in continuous time.","marker":"[21]"},{"why":"Proves identifiability and consistency for the discrete-time finite-horizon linear-quadratic tracking problem, the closest prior result generalized here.","marker":"[22]"},{"why":"Gives a discrete-time finite-horizon inverse-LQR system identification framework whose continuous-time separation must be rederived.","marker":"[24]"},{"why":"Derives continuous-time inverse quadratic optimal control identities that the structured residual operator builds on.","marker":"[25]"},{"why":"Reconstructs the full continuous-time cost from the control cost matrix; its Lyapunov-stationarity machinery is reused in the cost stage.","marker":"[26]"},{"why":"Provides the Richardson extrapolation used to cancel discretization bias in the local reconstruction of $A_c$.","marker":"[23]"},{"why":"Supplies the finite-horizon LQR and Riccati theory that produces the time-varying gain mechanism at the center of the paper.","marker":"[27]"},{"why":"Provides the sub-Gaussian tail bounds used in the denoising and perturbation analysis.","marker":"[32]"}],"fun_headline_variants":["Gain variation reveals true dynamics and cost in finite-horizon inverse LQR","Time-varying gain makes hidden dynamics and LQR cost identifiable","Finite-horizon inverse LQR: conditions to recover true (A,B,Q,R,H)","Conditioning indices guarantee true LQR recovery from noisy trajectories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire cost-identifiability result assumes the terminal cost is a known scalar multiple of the running state cost, $H=\\alpha Q$; without that assumed proportionality, underactuated systems admit infinitely many distinct costs that generate identical optimal behavior, so the true cost cannot be recovered.","fun_headline_variants_meta":{"raw":{"variants":["Gain variation reveals true dynamics and cost in finite-horizon inverse LQR","Time-varying gain makes hidden dynamics and LQR cost identifiable","Finite-horizon inverse LQR: conditions to recover true (A,B,Q,R,H)","Conditioning indices guarantee true LQR recovery from noisy trajectories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":3043,"prompt_tokens":1062,"completion_tokens":1981,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":1901}},"tokens_in":678,"tokens_out":1981,"duration_ms":13749,"temperature":1.0,"reasoning_tokens":1901,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:21:47.060779+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the central theorem, generate noiseless finite-horizon trajectories from a system satisfying $H=\\alpha Q$ with $c_{AB}>0$ and $c_{QRH}>0$, and search for two distinct admissible normalized tuples that produce exactly the same state and input curves; if such a pair exists, Theorem 3.8 is false. For the boundary of the claim, take a system with $\\mathrm{rank}(B)=m<n$ and any nonzero symmetric $S$ with $SB=0$; Proposition 3.5 predicts that for small $\\tau<0$ the triple $(Q+\\tau(A^\\top S+SA), R, H-\\tau S)$ produces exactly the same closed-loop trajectories as $(Q,R,H)$ while being a different admissible cost, and direct simulation can verify or refute that prediction.","supporting_citations":[{"cited_title":"Inferring system and opti- mal control parameters of closed-loop systems from partial observations,","cited_arxiv_id":null,"evidence_quote":"Establishes that infinite-horizon closed-loop data identify only the closed-loop pair and that finite horizons can restore identifiability, supplying the problem this paper answers in continuous time."},{"cited_title":"Data-driven inverse optimal control for linear quadratic tracking with unknown target states,","cited_arxiv_id":null,"evidence_quote":"Proves identifiability and consistency for the discrete-time finite-horizon linear-quadratic tracking problem, the closest prior result generalized here."},{"cited_title":"System identification approach for inverse optimal control of finite-horizon discrete-time LQR,","cited_arxiv_id":null,"evidence_quote":"Gives a discrete-time finite-horizon inverse-LQR system identification framework whose continuous-time separation must be rederived."},{"cited_title":"Continuous-time inverse quadratic optimal control problem,","cited_arxiv_id":null,"evidence_quote":"Derives continuous-time inverse quadratic optimal control identities that the structured residual operator builds on."},{"cited_title":"Inverse continuous-time linear quadratic regulator: From control cost matrix to entire cost reconstruction,","cited_arxiv_id":null,"evidence_quote":"Reconstructs the full continuous-time cost from the control cost matrix; its Lyapunov-stationarity machinery is reused in the cost stage."},{"cited_title":"The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,","cited_arxiv_id":null,"evidence_quote":"Provides the Richardson extrapolation used to cancel discretization bias in the local reconstruction of $A_c$."},{"cited_title":"Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science","cited_arxiv_id":null,"evidence_quote":"Provides the sub-Gaussian tail bounds used in the denoising and perturbation analysis."}],"review_version":1}