{"id":"ff8adbfa-6351-449f-8492-7d992fa29602","arxiv_id":"2608.10267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An asymptotic infinite-horizon optimal stopping time in covariance steering occurs exactly when there is no time or state cost and the system has a positive-definite stationary covariance; otherwise a finite optimal time exists.","lead":"This paper studies how to choose the optimal common stopping time when steering the covariance of a stochastic linear system to a target covariance. It derives a transversality condition for the final time, characterizes when the optimal time is finite or only approached at infinite horizon, and gives an algorithm to compute it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem V.2's dichotomy is conditional on unproved Assumption III.1, which a scalar A=0, D>0 instance violates while still exhibiting H_tf→0 with G∞=∅.","rationale":"The reader identified the same weakest assumption, and the A=0, D>0 example sharpens it: Assumption III.1 is not merely unproven; it is violated by a minimal controllable linear system, and in that system the phenomenon Theorem V.2 purports to characterize (H_tf→0) still occurs. This shows that the theorem's condition is not necessary for asymptotic Hamiltonian-zeros in the natural domain of the paper. The proof of Theorem V.1 also leans on III.1 for compactness and for LaSalle; absent it, periodic or unbounded covariance limits are possible, as the authors themselves note in Remark III.2 via [CW81]. The other flagged issue, the sensitivity formula (20) without derivation, is secondary: even if it needed a derivation, the numerical algorithm is not the central claim. The absence of code and data is a reproducibility weakness, not a correctness attack on the theorem. Credit is due for stating the assumption explicitly and for the clean one-dimensional analysis in Example V.1. Since the theorem is internally consistent under Assumption III.1 and the reader already rendered a CONDITIONAL verdict, this stress-test does not change the verdict.","tokens_in":39096,"tokens_out":32200,"duration_ms":315822,"concrete_test":"Work out the closed form for the scalar problem A=0, B=1, D>0, Q=0, xi=0, R=1, Sigma0=1, Sigmaf=2. Show Pi(t)=1/(C-t) with C>tf; the endpoint condition reads Sigmaf = delta^2 Sigma0/C^2 + D delta - D delta^2/C with delta=C-tf. Taking C->infty gives delta->Sigmaf/D and H_tf = -Sigma0/(2C^2)+D/(2C) -> 0. Also compute the pointwise limit Sigma_*(t)=lim_{tf->infty} Sigma_tf(t)=Sigma0 + D t, verifying that Assumption III.1 fails. If these identities check out, the dichotomy in Theorem V.2 is not a general characterization, and the paper must either prove Assumption III.1 for its setting or restate the main theorem as conditional on a verified regularity condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing point is Assumption III.1, exactly as the reader notes. It is used twice: Proposition B.1(B.1-b) bounds the limiting co-state by bounding Sigma_tf(tau3) uniformly in tf, and Lemma B.2 applies Barbalat to a limiting covariance arc to get Sigma_dot→0. Both uses need pointwise convergence of the horizon-parametrized family and a positive-definite limit. The authors explicitly disclaim a proof of sufficient conditions (Remark III.2). The assumption is not benign. Consider the scalar system A=0, B=1, D=1, Q=0, xi=0, R=1, Sigma0=1, Sigmaf=2. This satisfies Assumption II.1. The fixed-horizon extremal has Pi(t)=1/(C-t) with C>tf, and the Hamiltonian satisfies H_tf = -Sigma0/(2C^2) + D/(2C). The endpoint condition fixes C-tf→Sigmaf/D as tf→∞, hence H_tf→0, so tf=+infty is an asymptotic Hamiltonian-zero. Yet the Lyapunov equation A Sigma_s + Sigma_s A^T + D = 0 reduces to D=0, so no positive-definite stationary covariance exists and G∞=∅. The pointwise limiting covariance is Sigma_*(t)=Sigma0+Dt, unbounded, so Assumption III.1 fails. Thus the necessity direction of Theorem V.2 is false if Assumption III.1 is relaxed; the paper's headline interpretation is not a general characterization of asymptotic Hamiltonian-zeros but a statement about a regularity class that is neither verified nor checked in the numerical examples.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies continuous-time covariance steering with a common free final time, where both the feedback control law and the horizon are optimized. The SDE-constrained problem is reformulated as a deterministic covariance optimal control problem (Section III), from which the authors derive the Hamiltonian transversality condition H=0 for candidate optimal final times (Theorem III.1). Under standing controllability and strict endpoint-covariance assumptions (Assumption II.1) together with an asymptotic pointwise convergence assumption on the family of fixed-horizon optimal covariance trajectories (Assumption III.1), the paper gives sufficient conditions for existence of a finite candidate horizon (Theorem IV.1), introduces Asymptotic Gateways G∞ and Asymptotic Connections, and proves the main dichotomy (Theorem V.2): an asymptotic Hamiltonian-zero exists iff the time penalty and state cost vanish and G∞ is nonempty. The paper then derives final-time sensitivity formulas (Section VI-A) and proposes a trust-region line-search algorithm with an infinite-horizon detector, demonstrated on an illustrative example, a spacecraft maneuver, and Gaussian-mixture steering.","tokens_in":39452,"tokens_out":10099,"duration_ms":100694,"significance":"If the main results are accepted, the paper fills a genuine gap: previous continuous-time covariance steering typically fixes the horizon, whereas here the final time is optimized and the possibility of only asymptotic optimality is characterized. The turnpike-type interpretation via Asymptotic Gateways is conceptually useful, and the numerical examples illustrate the claimed finite, unbounded-gateway, and singular-gateway regimes. The derivation builds on PMP and on the authors' earlier fixed-horizon well-posedness theorem rather than assuming the free-final-time result, so the reliance on [LT24b] is not circular. However, the significance is presently limited because the central dichotomy is conditional on Assumption III.1, which is explicitly not verified in the paper and is not implied by the standing assumptions. Without a verifiable sufficient condition for that assumption, the advertised characterization is a result about a regularity class rather than a complete solution of the general problem.","major_comments":[{"comment":"The 'if and only if' characterization in Theorem V.2 is conditional on Assumption III.1, but Remark III.2 explicitly disclaims any derivation of sufficient conditions for that assumption. The assumption is load-bearing: Proposition B.1(b) uses the bounded positive-definite limit to bound co-states uniformly in the horizon, and Lemma B.2 applies Barbalat's lemma to a limiting covariance arc obtained from that assumption. It is not a consequence of Assumption II.1. For example, the scalar system A=0, B=1, D=1, Q=0, xi=0, R=1, Sigma0=1, Sigmaf=2 satisfies Assumption II.1. The fixed-horizon extremal has Pi(t)=1/(C-t) with C-tf tending to Sigmaf/D=2 as tf goes to infinity, and the Hamiltonian satisfies H_tf tending to 0, so tf=+infinity is an asymptotic Hamiltonian-zero. Yet the Lyapunov equation A Sigma_s + Sigma_s A^T + D=0 reduces to 1=0, so G_infinity is empty, and the limiting covariance is Sigma_star(t)=1+t, unbounded, so Assumption III.1 fails. Thus the unconditional reading of Theorem V.2 is false. The authors should either prove verifiable sufficient conditions for Assumption III.1 or restate Theorem V.2 as a conditional theorem and explicitly discuss the failure mode illustrated by this counterexample.","section":"Section V-C, Theorem V.2 and Remark III.2"},{"comment":"Theorem IV.1 is advertised as giving simple sufficient conditions for the finiteness of the candidate optimal final time, but the data conditions (IV.1-a)--(IV.1-c) are not sufficient on their own: the theorem also assumes Assumption III.1. The scalar counterexample above satisfies condition (IV.1-c), since the Lyapunov equation has no positive-definite solution, yet it has no finite Hamiltonian-zero because H_tf remains strictly positive for every finite tf and only tends to zero asymptotically. The theorem therefore does not deliver the stated data-only sufficiency result. The statement and the accompanying discussion in Section I-C should be revised to separate the data conditions from the unverified asymptotic regularity assumption, and the applicability to the numerical examples should be justified.","section":"Section IV, Theorem IV.1"},{"comment":"The sensitivity formula (20) is stated without derivation and is central to Algorithm 1: the algorithm uses deltaH/delta tf to build the Hessian approximation B^(k), and Remark VI.1 uses (20) to conclude that the first and second derivatives of the value function vanish along Asymptotic Connections. As written, the identity 1/2 trace(deltaPi_{tf;0}/delta tf Sigma_dot(0)) = 1/2 trace(deltaPi_{tf}/delta tf Sigma_dot(tf)) is not evident from the preceding equations, and no proof or citation is supplied. A rigorous derivation, including existence of the endpoint variations and the differentiability of Pi_{tf}(0) with respect to tf under the stated assumptions, is needed. In addition, Eq. (24) refers to 'a solution Y' of a Lyapunov equation but does not discuss unique solvability; this should be clarified.","section":"Section VI-A, Eq. (20)"}],"minor_comments":[{"comment":"There is a typo in the sentence introducing the vector field: 'Let hte vector field' should read 'Let the vector field'.","section":"Definition IV.1"},{"comment":"The switch from the general sensitivity identity (20) to the special case D=kappa B B^T is abrupt; please state explicitly that (21)--(24) are taken from [CGP18a] with the relevant sign conventions, since a reader comparing with the cited formulas will otherwise have difficulty verifying the signs in (21) and (23).","section":"Section VI-A, Eqs. (21)--(24)"},{"comment":"The output flag in Line 12 is returned as '†=1' but the flag values are never defined; please document what the flag indicates. In addition, Line 23 uses the symbol 'epsilon' for the step-size threshold while the convergence tolerance is denoted 'varepsilon', which is confusing.","section":"Algorithm 1"},{"comment":"The condition 'any(Sigma(·) in G_infinity)' in Algorithm 2 is implemented through sampled trajectory points and tolerances, but there is no discussion of how the sampling grid and threshold eta_G affect the reliability of detecting an asymptotic Hamiltonian-zero; a brief comment on this approximation would be useful.","section":"Algorithm 2 and Section VI-B"},{"comment":"The claim that (20) implies lim_{tf to infinity} deltaH/delta tf = 0 requires uniform boundedness of the co-state sensitivity deltaPi_{tf}/delta tf as tf tends to infinity; this boundedness is not established in the paper and should be stated as an additional condition or proved.","section":"Remark VI.1"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript is self-referential to the authors' earlier fixed-horizon theorem [LT24b], but I do not see a circularity problem, since the free-final-time characterization is not contained in that theorem. The main concern is that the paper's headline dichotomy is conditional on an assumption that is neither proved nor verified, and the paper's own Remark III.2 concedes this. I believe this is fixable by substantially revising the claims and adding a rigorous discussion of Assumption III.1, rather than by rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a genuinely useful paper. It gives the first systematic treatment of common free-final time in continuous-time covariance steering, and it does so with the right tools: a deterministic reformulation, a PMP-based transversality condition expressed as a Hamiltonian-zero, and a clean dichotomy between finite and asymptotic final times. The asymptotic gateway and connection concepts are new, and the trust-region line-search algorithm with detector is a practical bridge from the theory to computation. The proofs are substantial and mostly standard, and the reliance on [LT24b] for fixed-horizon well-posedness is legitimate, not circular.\n\nNow the soft spots, in proportion. The load-bearing issue is Assumption III.1, the assumption that the family of fixed-horizon optimal covariance trajectories has a bounded positive-definite pointwise limit. The authors themselves flag in Remark III.2 that they don't prove sufficient conditions, but the assumption is used twice in essential ways: to get uniform boundedness of the limiting arcs (Proposition B.1) and to apply Barbalat to reach \\dot\\Sigma\\to 0 (Lemma B.2). The stress-test counterexample is telling: for scalar A=0, B=1, D=1, Q=0, xi=0, R=1, Sigma0=1, Sigmaf=2, the Hamiltonian does tend to zero as tf→∞, so there is an asymptotic Hamiltonian-zero, yet G∞ is empty because the Lyapunov equation reduces to D=0. The limiting covariance is unbounded, so III.1 fails. That means the necessity direction of Theorem V.2 is genuinely condition-dependent, and the paper's headline interpretation should be read as a statement about a regularity class, not a general characterization. The numerical examples likely satisfy III.1, but the authors don't verify it, and they don't provide code or data.\n\nA secondary issue: the sensitivity formula (20) is stated without derivation, though the algorithm can fall back to secant updates, so this is addressable rather than fatal.\n\nWho gets value from this? Researchers working on covariance steering, optimal stopping with distributional constraints, and mission design where maneuver duration is optimized. It deserves a serious referee. I'd recommend conditional acceptance: either prove sufficient conditions for III.1 or restate Theorem V.2 explicitly as conditional and check the assumption in the examples, and either derive (20) or label it as a numerical approximation.\n\nBest,\n[You]","headline":"A solid first systematic treatment of free-final-time covariance steering with a clean transversality condition, but the main dichotomy in Theorem V.2 is conditional on an unverified regularity assumption that stress-testing shows is not benign.","tokens_in":40001,"tokens_out":2323,"would_cite":true,"duration_ms":23813,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E20","93E03","49K20","49L99","58E25","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"When a free final time for covariance steering exists only at infinity, this paper pins down the exact condition.","keywords":["covariance steering","free-final time","stochastic optimal control","transversality condition","Hamiltonian-zero","asymptotic gateway","turnpike","trust-region line search"],"falsifier":"Construct an LTI system satisfying the standing controllability assumptions with Q = 0, omega = 0, and a nonempty gateway set, but pick endpoint covariances so that the family of fixed-horizon optimal covariance trajectories oscillates periodically in the large-horizon limit (for instance, by including a center-mode structure in A that excites oscillation without violating boundedness). If the family then admits no pointwise limit Sigma_infinity > 0, the equivalence in Theorem V.2 would fail even though the gateway is nonempty, exposing Assumption III.1 as indispensable.","tokens_in":38883,"feed_emoji":"🎲","tokens_out":1625,"duration_ms":17953,"temperature":0.7,"pith_summary":"This paper studies the problem of steering the covariance of a stochastic linear system from one prescribed Gaussian to another, where the final time itself is a decision variable shared by all realizations. It establishes the correct transversality condition for the optimal common final time, shows when that time is finite, and characterizes when the optimality condition is achieved only as the horizon goes to infinity. The central dichotomy is governed by an 'Asymptotic Gateway' set G∞: if the uncontrolled covariance dynamics admit a positive-definite stationary covariance on which the state cost vanishes, then the Hamiltonian-zero is approached only asymptotically; otherwise a finite optimal time exists. The paper also provides a trust-region line-search algorithm that detects both regimes.","feed_headline":"Free final time in covariance steering: when the optimum lives only at infinity","feed_subtitle":"A new condition pins down when the optimal stopping time is finite and when it is approached only as the horizon grows unbounded.","key_machinery":"The Asymptotic Gateway is the set G∞ of positive-definite stationary covariances of the uncontrolled dynamics on which the state cost trace(Q Sigma) vanishes: G∞ = { Sigma > 0 : A Sigma + Sigma A^T + D = 0 and trace(Q Sigma) = 0 }. It carries the argument by identifying the only states near which a long-horizon trajectory can linger without accumulating running cost, and its nonemptiness is equivalent to the existence of an asymptotic Hamiltonian-zero.","core_discovery":"Under controllability and a regularity assumption on the limiting covariance trajectory, the paper proves that an optimal common final time is approached only in the infinite-horizon limit exactly when the time penalty is zero (omega = 0) and there exists a positive-definite stationary covariance Sigma_s of the uncontrolled dynamics, A Sigma_s + Sigma_s A^T + D = 0, on which the state cost vanishes, trace(Q Sigma_s) = 0. In that regime the Hamiltonian of the fixed-horizon problem tends to zero as the horizon tends to infinity, defining an asymptotic Hamiltonian-zero, while the finite-horizon optimal trajectories approach the gateway set G∞ and remain near it for most of the horizon before departing to meet the terminal constraint. If the gateway is empty or the time penalty is positive, the large-horizon Hamiltonian remains strictly positive, and since the small-horizon Hamiltonian diverges to -infinity, a finite Hamiltonian-zero exists by continuity.","pith_inferences":["The same dichotomy likely governs discrete-time covariance steering with free final time, where the Lyapunov equation is replaced by its discrete analogue and the gateway would be the set of stationary covariances under the uncontrolled one-step map; the paper does not treat this case.","The asymptotic Hamiltonian-zero suggests a candidate definition of overtaking optimality for covariance steering in which the terminal constraint is approached rather than met, and the gateway point plays the role of a Skorokhod-like boundary; this interpretation is not made by the authors beyond a remark.","A testable consequence is that when (A, B) is controllable, D is nearly singular, and Q = 0, the approach to a singular gateway drives the feedback gain to blow up near the terminal constraint; this could be verified by observing the norm of the computed gain along the trajectory in the numerical experiments.","The finiteness criterion Theorem IV.1 could be sharpened to an explicit upper bound on the optimal final time in terms of the spectrum of A and the data (Sigma_0, Sigma_f, Q, R, D), since the intermediate value argument only shows existence; the paper does not provide such a bound."],"forward_implications":["If the gateway condition holds, the optimal final time is not achieved at any finite horizon, and any numerical solver that targets a finite Hamiltonian-zero must be modified to detect the asymptotic regime.","If the gateway is empty or the time penalty is positive, the transversality condition H_tf = 0 has a finite root, and the trust-region line-search algorithm provably locates it.","The characterization extends the turnpike picture to stochastic covariance steering: for long horizons the optimal covariance path is composed of a forward arc from the initial covariance to the gateway and a backward arc from the terminal covariance to the gateway, stitched together near G∞.","For matched control and noise channels, the co-state and Hamiltonian sensitivities can be computed analytically, enabling a Hessian-like update in the algorithm and second-order information about the value function.","In Gaussian-mixture steering, the per-component free-final-time solution is no worse than the common-final-time solution, and the gap (the 'price of synchronization') vanishes as the mixture components merge."],"supporting_citations":[{"why":"Provides the fixed-horizon well-posedness result (Fact III.1) and the homeomorphism property of the initial co-state map, on which the continuity and compactness arguments rest.","marker":"[LT24b]"},{"why":"Supplies the analytical expressions for the initial co-state and covariance trajectories in the matched diffusion case, which the sensitivity derivations in Section VI-A differentiate.","marker":"[CGP18a]"},{"why":"Establishes the completion-of-squares argument used in the proof of Proposition III.1 to show the optimal controller is linear state feedback.","marker":"[CGP16]"},{"why":"Provides the Lyapunov equation and modal controllability results used to analyze boundedness and convergence of the uncontrolled covariance dynamics in the LaSalle step.","marker":"[Hes18]"},{"why":"Supplies LaSalle's invariance principle and Barbalat's lemma used in Theorem V.1 and Lemma B.2 to show convergence of the limiting arcs to the gateway set and of the Hamiltonian to zero.","marker":"[Kha02]"},{"why":"The classical reference for turnpike behavior in large-time optimal control, cited to frame the Asymptotic Connection as a stochastic analogue of the turnpike.","marker":"[AK87]"},{"why":"The trust-region line-search framework on which Algorithm 1 is based.","marker":"[NW06]"}],"fun_headline_variants":["Covariance steering: the infinity-only optimum","When covariance steering's optimum lives only at infinity","Zero time penalty can push covariance steering optimum to infinity","New condition splits finite vs infinite horizon in covariance steering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dichotomy hinges on Assumption III.1: that the limiting covariance trajectory, as the horizon goes to infinity, actually converges to a bounded positive-definite limit rather than oscillating or becoming singular.","fun_headline_variants_meta":{"raw":{"variants":["Covariance steering: the infinity-only optimum","When covariance steering's optimum lives only at infinity","Zero time penalty can push covariance steering optimum to infinity","New condition splits finite vs infinite horizon in covariance steering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001946,"raw_usage":{"total_tokens":7599,"prompt_tokens":920,"completion_tokens":6679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":6618}},"tokens_in":536,"tokens_out":6679,"duration_ms":49796,"temperature":1.0,"reasoning_tokens":6618,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:11:24.955706+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an LTI system satisfying the standing controllability assumptions with Q = 0, omega = 0, and a nonempty gateway set, but pick endpoint covariances so that the family of fixed-horizon optimal covariance trajectories oscillates periodically in the large-horizon limit (for instance, by including a center-mode structure in A that excites oscillation without violating boundedness). If the family then admits no pointwise limit Sigma_infinity > 0, the equivalence in Theorem V.2 would fail even though the gateway is nonempty, exposing Assumption III.1 as indispensable.","supporting_citations":[],"review_version":1}