Turnpike property of linear quadratic control problems with unbounded control operators

Emmanuel Tr\'elat; Hoai-Minh Nguyen

REVIEW 2 major objections 4 minor 42 references

Turnpike property of linear quadratic control problems with unbounded control operators

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that the exponential turnpike property holds for linear quadratic control problems with unbounded control operators, provided the control operator is admissible and the observation operator is coercive.

desk verdict Genuine extension with a load-bearing observability gap in the Yosida approximation step. read the letter →

arxiv 2506.01605 v1 pith:EIQP3YPM submitted 2025-06-02 math.OC math.AP

Hoai-Minh Nguyen , Emmanuel Trélat This is my paper

classification math.OCmath.AP MSC 49N1093C2593B0793D20

keywords turnpikepropertylinearquadraticoptimalcontrolunboundedoperatorsadmissibleinfinite-horizonsemigroupYosidaapproximationexponentialdecayHilbertspace

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 proves that the exponential turnpike property holds for linear quadratic optimal control problems in Hilbert spaces even when the control operator is admissible but unbounded, a case previously accessible only in special one-dimensional hyperbolic systems or for analytic semigroups. The turnpike property says that on a long time interval, the optimal trajectory stays exponentially close to the optimal steady state except near the two endpoints. The proof approximates the unbounded operator by bounded Yosida approximations, shows that the optimal solutions of the approximate problems converge to those of the original problem, and then transfers the turnpike estimate across the limit. The argument avoids the Riccati equation theory that is delicate for unbounded operators.

What carries the argument

The machinery has two parts. First, the infinite-horizon optimal semigroup S_{8,opt}(t), which maps an initial state to the state reached at time t by the optimal trajectory of the infinite-horizon LQ problem, is the object whose exponential decay (ensured by (H2)) sets the turnpike rate. Second, the identity of Proposition 1.3: with P the infinite-horizon Riccati operator, the quantity g(t)=y_{T,opt}(T−t)−ȳ−P(x_{T,opt}(T−t)−x̄) evolves as g(t)=S^*_{8,opt}(t)g(0). This identity is proved for bounded B by direct differentiation, and for unbounded B by approximating B with B_k=J_kB, establishing convergence of the optimal controls, states, and adjoints (Lemmas 3.1, 4.3), and passing to the limit. A final energy estimate using the parallelogram identity converts the semigroup decay into the turnpike inequality.

What would settle it

Solve explicitly the linear quadratic problem for a one-dimensional heat equation with boundary control and an observation satisfying (H1) and (H2), e.g., observation over a nonempty open subinterval, and verify whether the quantity sup_{t∈[T0,T−T0]} |x_{T,opt}(t)−x̄| decays like $e^{{−λT}}$ with λ the decay rate of S_{8,opt}; a failure of this decay would falsify the quantitative claim of Theorem 1.1.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.1: under (H1) finite-time observability of (A,C) and (A*,B*) plus kernel conditions, and (H2) the coercivity estimate C*C ≥ δ Id for some δ>0, for every T>T0 and every x0,z∈H the optimal state, adjoint, and control of the finite-horizon problem (1.2) satisfy the exponential turnpike inequality (1.9) with constants c,λ depending only on A,B,C. Here (x̄,ū,ȳ) is the unique stationary triple given by Proposition 1.1. The decay rate λ can be taken as the decay rate of the optimal infinite-horizon semigroup S_{8,opt}, and is optimal in the exponential rate. The proof's core is the convergence of the approximate problems obtained by replacing B with the Yosida approximation B_k=J_kB, and the identity (Proposition 1.3) expressing the finite-horizon error in terms of the adjoint of S_{8,opt}.

Load-bearing premise

The load-bearing premise is assumption (H2), C*C ≥ δ Id, which forces the optimal infinite-horizon semigroup to decay exponentially; without it, the proof's rate and bound do not go through, and this coercivity fails for common point or trace observations.

Editorial extensions

If this is right

The exponential turnpike property now holds for a general class of infinite-dimensional LQ systems with unbounded admissible control operators, not just for 1D hyperbolic systems or analytic semigroups.
The decay rate λ in the turnpike estimate is the decay rate of the optimal infinite-horizon semigroup, so the estimate is optimal in the exponent whenever that semigroup decays exactly exponentially.
For unbounded B, the control estimate is in L2 on intervals away from the endpoints, not pointwise, reflecting the limited regularity of admissible controls.
The proof avoids differential and algebraic Riccati equations for unbounded B, instead using the infinite-horizon semigroup and convergence of Yosida approximations.
The paper also clarifies that existing extension claims for unbounded B via multiplier techniques rest on an erroneous domain argument, pointing to the Yosida route as a sound alternative.

Reading between the lines

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

If the role of (H2) is only to guarantee exponential decay of S_{8,opt}, as the paper remarks, then any sufficient condition for that decay (for instance, exponential stabilizability and detectability in the appropriate sense) should yield the same turnpike theorem; this could be tested by re-running the proof with such a condition.
The convergence of the Yosida-approximated problems suggests a numerical strategy: compute turnpike bounds for bounded-control approximations B_k and pass to the limit, potentially giving explicit rates for boundary-controlled PDEs.
The h/g identity separates the turnpike mechanism from Riccati theory and indicates that, in other optimal control settings (e.g., with state constraints or nonlinear dynamics), the decay of the infinite-horizon optimal semigroup is the quantity that controls the turnpike rate.
The noted flaw in the multiplier-based extension [12, Lemma 6] suggests that domain intersections like D(A)∩D(BKB) cannot be assumed; the Yosida approximation sidesteps this but may impose a price in terms of the coercivity of the observation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper goes after something real. Exponential turnpike for LQ problems with unbounded admissible control operators has only been treated in special 1D hyperbolic or analytic semigroup settings, so a general theorem would fill an actual gap. The authors also identify a concrete flaw in [12]'s extension to unbounded operators, and their overall route -- approximate B by bounded B_k, prove convergence, then pass through S_{8,opt} -- is credible and avoids Riccati regularity issues. The writing is mostly careful, and the final energy estimates are not the problem.

The problem is in Section 3.2, and it is load-bearing. Corollary 3.1 needs (A, B_k) to inherit finite-time observability from (A, B). The chain in the proof is

||B_k^* e^{tA^*} xi||^2 dt = ||B^* e^{tA^*} J_k^* xi||^2 dt >= c ||e^{T0 A} J_k^* xi||^2,

and then the proof replaces e^{T0 A} J_k^* xi by J_k^* xi and asserts J_k^* is bounded below with constant c_k. Both moves fail for general unbounded A. The RHS of the observability inequality is not ||J_k^* xi||^2. And J_k^* = k(kI - A^*)^{-1} maps into D(A^*), which is dense but not closed; for skew-adjoint A with eigenvalues i n, high-frequency vectors are attenuated by k/(k + i n), so no uniform lower bound exists. With B = I, (A^*, B^*) is perfectly observable, but (A^*, B_k^*) is not: the observability constant for mode n is order k^2/n^2. So the approximate stationary problems, and therefore the h_k, g_k used in Proposition 1.3, are not guaranteed to exist. This is a substantive gap, not a missing detail.

The reader's report is too generous on soundness. I would still send the paper to peer review: the target is important, the literature treatment is honest, and the gap is localized enough that a different approximation (or an extra assumption preserving observability under Yosida) might repair it. But the main theorem is not established as written.

Referee Report

2 major / 4 minor

Summary. The paper proves an exponential turnpike estimate for linear-quadratic optimal control problems whose control operator B is admissible and possibly unbounded. The main result, Theorem 1.1, asserts that, under finite-time observability of (A,C) and (A*,B*), kernel conditions, and the strong observation condition C*C ≥ δ I, the optimal state, adjoint state, and control of the finite-horizon problem (1.2) stay exponentially close to the stationary solution (x̄,ū,ȳ) except near the endpoints. The proof strategy follows Porretta–Zuazua: it replaces B by a Yosida-regularized bounded operator B_k = J_k B, proves convergence of the associated stationary problems and finite-horizon optimal control problems, establishes a relation g(t) = S_{∞,opt}^*(t)g(0) between finite-horizon adjoint shifts and the infinite-horizon optimal semigroup, and then derives the turnpike estimate via observability and parallelogram identities.

Significance. If the main theorem is correct, the paper is a substantial contribution: it extends the exponential turnpike property to unbounded admissible control operators without relying on the differential Riccati equation, and it provides explicit dependence of the constants on the initial data and target. The proof is structurally transparent, with the core Proposition 1.3 treated in detail, and the paper is careful to distinguish pointwise-in-time estimates for the control from the L2 estimate that is natural in the unbounded case. The explicit critique of the multiplier technique in [12] is also useful. The main caveat is the restrictive assumption C*C ≥ δ I, which the authors themselves note can be replaced by any condition guaranteeing exponential decay of S_{∞,opt}; as stated, the theorem excludes many natural observation operators such as point or trace observations.

major comments (2)

[Section 3.2, Corollary 3.1] The proof that (A, B_k) is finite-time observable whenever (A, B) is, is not valid. The displayed chain uses the observability inequality (1.6) for (A*,B*) with ξ replaced by J_k^*ξ, but (1.6) has ||e^{T0 A}ξ|| on the right-hand side, not ||ξ||; even ignoring that mismatch, the final step ||J_k^*ξ||^2 ≥ c_k||ξ||^2 requires J_k^* = k(kI - A*)^{-1} to be bounded below. This is false for unbounded A*: for example, if A = -diag(n) on ℓ², then ||J_k^* e_n|| → 0 as n → ∞. Consequently the existence of the approximate Lagrange multipliers ȳ_k, which underpins Lemma 3.1 and the construction in Section 4.2, is not established. Since Proposition 1.3 and hence Theorem 1.1 rely on this approximation, the gap is load-bearing and needs to be repaired, either by proving observability of (A, B_k) under additional assumptions or by choosing a different approximation scheme.
[Section 4.2, Lemma 4.2] Lemma 4.2 is used to obtain the differential equation for h_k that is essential for the passage from the approximate problems to the limit g(t) = S_{∞,opt}^*(t)g(0). Its proof, however, is summarized only by 'The details are omitted.' The omitted approximation argument is precisely the kind of delicate step needed when B is unbounded, and the lemma has no external reference. The authors should supply the full argument or a precise reference to a proof in the literature.

minor comments (4)

[Section 1.1, Eq. (1.6)] The observability inequality for (A*,B*) is written with the right-hand side ||e^{T0 A}ξ||, but the proofs in Sections 3 and 5 appear to use the stronger inequality with ||ξ|| on the right-hand side. If the displayed form is not a typo, the intended definition should be clarified; if it is a typo, it should be corrected.
[Remark 1.4] The pointwise control estimate for bounded B is written with e^{-pT-t}; this appears to be a typo for e^{-λ(T-t)} or a similarly defined exponential factor.
[Abstract and Remark 1.3] The abstract describes the assumptions as 'quite general and natural', but Assumption (H2) C*C ≥ δI excludes many standard observations, including point and trace observations. Remark 1.3 appropriately notes that H2 can be replaced, but the wording of the abstract should be tempered or the theorem stated with a weaker sufficient condition.
[Throughout] There are several typographical errors and OCR artifacts, e.g., 'TURNPIKE PROPER TY' and 'QUADRA TIC' in the running title and abstract. These should be corrected in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem follows from external Riccati/observability inputs and an approximation argument; no fitted quantity is relabeled as a prediction.

full rationale

The central claim (Theorem 1.1) is an estimate bounding optimal trajectories by the stationary triple, with decay rate λ taken as the decay rate of the infinite-horizon optimal semigroup S8,opt from Proposition 1.2. That decay rate is not tuned to the turnpike inequality; it is obtained from the algebraic Riccati operator P imported from [10] and the sufficient condition C*C ≥ δ id. The paper does not fit parameters to the turnpike data and then 'predict' the same quantity. Approximate stationary triples (xbar_k, ubar_k, ybar_k) are defined through Corollary 3.1 and shown in Lemma 3.1 to converge to the original triple using observability and optimality; this is an approximation step, not a definitional identity. The only self-citation is Lemma 2.1 from [24] (same first author), an integration-by-parts/duality lemma used repeatedly; it is a general tool published independently, and its content is not the present turnpike conclusion, so by Rule 4 it does not raise the circularity score. The manuscript also states its own limitation (Remark 1.3) that (H2) is only a sufficient condition for exponential decay of S8,opt; that is an honest scoping statement, not a circularity. The skeptic concern about Corollary 3.1 (whether Yosida approximation preserves observability) is a mathematical correctness risk, not a circular-reasoning defect: even if the approximation argument failed, the claimed reduction would not be equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented entities. The central argument imports standard LQ theory (Riccati operator P), the duality lemma from [24], and Yosida approximation; these are external results, not constructed to force the conclusion.

assumptions (4)

domain assumption Finite cost condition and exponential decay of the infinite-horizon optimal semigroup S_8,opt under C*C >= delta*Id (Proposition 1.2, from [10]).
Used to give decay of g(t) in Proposition 1.3 and hence the final turnpike estimate; in Theorem 1.1 the finite cost condition follows from (H1) via null-controllability.
standard math Lemma 2.1 (duality identity for weak solutions, cited from [24, Lemma 3.1], the first author's earlier work).
Invoked repeatedly (Sections 2, 4, 5) to derive (2.5), the boundedness of B* z, and the energy estimates (5.1)-(5.3).
standard math Algebraic Riccati equation theory: existence of P in Lemma 1.1 and the identity (2.13) for bounded B, cited from [10].
The operator P defined by (1.12) is the algebraic Riccati solution; Corollary 2.1 and Lemma 4.1 rely on it.
standard math Yosida approximation properties: J_k defined by (1.16) is a sequence of bounded operators converging strongly to the identity with J_k* commuting with A*; observability of (A,B_k) inherited from (A,B).
The approximation scheme B_k = J_k B is defined in Section 1.3 and used in Section 3.2 and Section 4.

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Pith. "Pith review of Turnpike property of linear quadratic control problems with unbounded control operators." pith.science (2026). https://pith.science/paper/EIQP3YPM

@misc{pith2026250601605,
  author       = {Pith},
  title        = {Pith review of: Turnpike property of linear quadratic control problems with unbounded control operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIQP3YPM}},
  note         = {Machine review of arXiv:2506.01605}
}

read the original abstract

We establish the turnpike property for linear quadratic control problems for which the control operator is admissible and may be unbounded, under quite general and natural assumptions. The turnpike property has been well studied for bounded control operators, based on the theory of differential and algebraic Riccati equations. For unbounded control operators, there are only few results, limited to some special cases of hyperbolic systems in dimension one or to analytic semigroups. Our analysis is inspired by the pioneering work of Porretta and Zuazua \cite{PZ13}. We start by approximating the admissible control operator with a sequence of bounded ones. We then prove the convergence of the approximate problems to the initial one in a suitable sense. Establishing this convergence is the core of the paper. It requires to revisit in some sense the linear quadratic optimal control theory with admissible control operators, in which the roles of energy and adjoint states, and the connection between infinite-horizon and finite-horizon optimal control problems with an appropriate final cost are investigated.

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [12]

Differential Equations 268 (2020), no

Lars Gr¨ une, Manuel Schaller, and Anton Schiela, Exponential sensitivity and turnpike analysis for linear qua- dratic optimal control of general evolution equations , J. Differential Equations 268 (2020), no. 12, 7311–7341. MR 4079009

work page 2020

[2] [1]

Veljko Aˇ skovi´ c, Emmanuel Tr´ elat, and Hasnaa Zidani,Linear quadratic optimal control turnpike in finite and infinite dimension: two-term expansion of the value function , Systems & Control Letters 188 (2024), 105803

work page 2024

[3] [2]

Delfour, and Sanjoy K

Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, Representation and control of infinite dimensional systems , second ed., Systems & Control: Foundations & Applications, Birkh¨ auser Boston, Inc., Boston, MA, 2007. MR 2273323

work page 2007

[4] [3]

MR 2759829

Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations , Universitext, Springer, New York, 2011. MR 2759829

work page 2011

[5] [4]

Marco Cirant and Alessio Porretta, Long time behavior and turnpike solutions in mildly non-monotone mean field games , ESAIM Control Optim. Calc. Var. 27 (2021), Paper No. 86, 40. MR 4294187

work page 2021

[6] [5]

136, American Mathematical Society, Providence, RI, 2007

Jean-Michel Coron, Control and nonlinearity , Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. MR 2302744

work page 2007

[7] [6]

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Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. MR 1351248

work page 1995

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