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Joint Identifiability and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics

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Pith's one-line read Gain variation lets finite-horizon data reveal dynamics and LQR cost

desk verdict 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. read the letter →

arxiv 2608.11932 v1 pith:XL3ZSDCA submitted 2026-08-12 math.OC

Meiling Yu , Yuan-Hua Ni , Lei Jiang This is my paper · ORCID
classification math.OC MSC 49N1093B3093E12
keywords inverseoptimalcontrollinearquadraticregulatorunknowndynamicsidentifiabilityconditioningindicesfinitehorizonconsistentestimationsemidefiniteprogramming
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 5.2, Lemma 5.9, Eq. (68)] 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.
  2. [Abstract and Section 3, Theorem 3.8] 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.
minor comments (5)
  1. [Section 7, conclusion] The phrase 'We proposed CR-IOC' should be 'we propose CR-IOC' to match the present-tense summary of the paper's contribution.
  2. [Theorem 3.3, necessity part] 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.
  3. [Definition 2.7(iii)] 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.
  4. [Section 5.1, after Theorem 5.4] 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.
  5. [Section 6.5, Table 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identifiability conditions are stated as properties of the true system and gain, not as fitted inputs, and the reconstruction and perturbation analysis are self-contained.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. Lemma 3.2 recovers the gain K(t) from the algebraic relation U(t) = -K(t)X(t) under full-rank state richness; Theorem 3.3 separates (A,B) from the functional identity A_c(t) = A - B K(t) using the gain-variation Gramian; Theorem 3.7 and Theorem 3.8 concatenate these steps through the stationarity-residual operator M_alpha. The conditions c_AB > 0 and c_QRH > 0 are conditions on the true tuple (A*,B*,Q*,R*,H*) and the induced optimal gain, not parameters fitted to data, and the empirical diagnostics (c_hat_X, c_hat_AB, c_hat_QRH) are used only as computed indicators. The definition of c_QRH as the restricted minimum gain of M_alpha does make Theorem 3.7 a direct reformulation of injectivity, but the paper states this equivalence explicitly and does not present it as an empirical prediction; the cost-recovery SDP, the closed-form dynamics recovery, and the non-asymptotic perturbation bounds contain independent content. Assumption 2.2 (H = alpha Q) is an admitted structural restriction, and Proposition 3.5 explicitly shows non-identifiability outside it, so it is a scope limitation rather than circular reasoning. The unquantified err_ODE term in Lemma 5.9 is a gap in the non-asymptotic guarantee, but it is an external numerical error assumed to vanish and is not a circular reuse of the target result. No load-bearing self-citation chain or ansatz-smuggling is present. The identifiability theory is therefore not circular. 0/10.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's theoretical claims rest on standard LQR and Lyapunov theory, sub-Gaussian concentration, and the explicit assumptions on stabilizability, rank, terminal-cost structure, noise, and smoothness. No fitted constants or invented entities are introduced; the conditioning indices are defined in terms of the true model and are not free parameters licensed to the reader.

assumptions (7)
  • standard math Standard finite-horizon LQR optimality: the optimal control is unique and has the form u(t) = -K(t)x(t) with K(t) = R^{-1}B^T P(t) and P solving the Riccati differential equation.
    Invoked in equations (4) and (5); the starting point of the entire formulation.
  • standard math Representation of terminal-value Lyapunov equations via variation of constants (Lemma 2.4).
    Proved in the paper; foundational for the operator formulation of the cost stage.
  • standard math Sub-Gaussian concentration inequalities for sums of independent sub-Gaussian vectors (Vershynin, Section 2.5).
    Used in Lemma 5.2 to obtain the tail bound on the denoising error.
  • domain assumption Assumptions A1-A3: (A,B) stabilizable, rank(B)=m, and rank(X(0))=n.
    Imposed in Assumption 2.1 to ensure well-posedness and state richness for the gain recovery.
  • domain assumption Assumption 2.2: H = alpha*Q for a known scalar alpha >= 0.
    The structural restriction that makes the cost identifiable in the underactuated case; discussed in Remark 2.3 and used throughout Sections 3 and 4.
  • domain assumption Assumption 4.1: observation noises are zero-mean, independent, and sub-Gaussian.
    Standard noise model for the perturbation analysis and consistency proof.
  • domain assumption Assumption 5.1: boundedness and Lipschitz regularity of trajectories, gain, and closed-loop matrix.
    Required for the discretization and denoising bounds in Section 5.

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Cite this review

Pith. "Pith review of Joint Identifiability and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics." pith.science (2026). https://pith.science/paper/XL3ZSDCA

@misc{pith2026260811932,
  author       = {Pith},
  title        = {Pith review of: Joint Identifiability and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XL3ZSDCA}},
  note         = {Machine review of arXiv:2608.11932}
}
abstract

Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper studies the finite-horizon continuous-time inverse LQR problem from closed-loop state--input trajectories, where both the system matrices and the quadratic cost are unknown. The finite horizon induces a time-varying optimal gain, and this endogenous excitation serves as the structural mechanism that makes joint recovery possible. We quantify this mechanism through three computable conditioning indices, which measure state richness, gain-variation richness, and injectivity of a structured cost operator. Using these indices, we establish joint identifiability conditions for the inverse problem considered here. Crucially, these conditions guarantee recovery of the ground-truth system matrices $(A,B)$ and the true cost weighting matrices, rather than merely a behaviorally equivalent surrogate. We also develop a conditioning-aware sampled-data reconstruction method that reconstructs the gain $K(\cdot)$ and the closed-loop dynamics matrix $A_c(\cdot)$ from noisy measurements, recovers $(A,B)$ in closed form, and identifies the quadratic weights through a convex semidefinite program. We further establish the non-asymptotic perturbation bounds and the consistency of the full reconstruction method under sub-Gaussian observation noise, with explicit dependence on the same conditioning indices. Numerical experiments support the theory and illustrate the diagnostic value of the conditioning indices.

Figures

Figures reproduced from arXiv: 2608.11932 by the authors.

Figure 1
Figure 1. Reference two-mass spring–damper system used to anchor the experiments. Two actuated masses [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Recoverability sweep on the mass–spring–damper family. Fig. [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Matched noisy comparison on the six-mass chain ( [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Closed-loop reconstruction on the fixed dense reference system ( [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]
Figure 5
Figure 5. Figure 5: Diagnostic conditioning study on the fixed 4 [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]

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