Optimal cost of fast boundary controls for the one-dimensional heat equation

Pierre Lissy

Optimal cost of fast boundary controls for the one-dimensional heat equation

T0 review · 0 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The optimal small-time cost of one-dimensional heat boundary controls is $\exp((\kappa_* L^2+o(1))/T)$ with $\kappa_*\approx 0.6966$.

desk verdict The exact small-time cost constant for the 1D heat equation is settled; the proof is convincing except for one intricate phase-comparison lemma that deserves independent checking. read the letter →

arxiv 2608.08041 v1 pith:VMOOA52V submitted 2026-08-08 math.OC math.AP

Pierre Lissy This is my paper · ORCID

classification math.OCmath.AP MSC 93B0535K0530H1042A3835R11

keywords heatequationnullcontrollabilitycostoffastcontrolsmomentmethodentirefunctionsHardyspacesHilberttransformhalf-Laplacian

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

The paper establishes the exact small-time asymptotic of the minimal $L^2$ norm of a Dirichlet boundary control that drives the one-dimensional heat equation on $(0,L)$ to zero. The main theorem states that the optimal control cost satisfies $C_{\mathrm H}(T,L)=\exp((\kappa_* L^2+o(1))/T)$ as $T\to 0^+$, where $\kappa_* = \Gamma(1/4)^4/(8\pi^3) \approx 0.6966$. This closes a long-standing gap: the best previous lower bound gave the coefficient $1/2$, while the best previous upper bound was a constant $K_0$ given by a convergent series; both are now replaced by the single number $\kappa_*$. The reason this matters is that the constant quantifies exactly how much harder it is to observe sign-changing solutions than the Gaussian heat-kernel estimate suggests. A sympathetic reader should see this as a first sharp answer for a basic control-theoretic quantity in a model geometry.

What carries the argument

The load-bearing mechanism is a phase comparison between inner-outer factorizations in the two half-planes, which converts compact Fourier support into a one-sided inequality on the modified Hilbert transform. The central objects are the canonical product $P_-(\zeta)=\prod_{n\ge 2}(1-i\zeta/\mu_n)$, carrying the shifted heat eigenvalues $\mu_n=\pi^2(n^2-1)/L^2$, and its reflected twin $P_+$. For an admissible multiplier $m$, Lemma 2.2 produces $(H\Omega)'\ge -T/2$ with $\Omega=-\log|m|$. After the rescaling $X=T^2\zeta$, this becomes $|D|u\ge -1/2$ together with $u\ge q_T$, the obstacle inequality; $q_T$ converges to $q_*(X)=L\sqrt{|X|/2}$, a square-root profile. The explicit test function $\rho_b$ is the Green function of the restricted half-Laplacian on $(-b,b)$ with zero exterior condition, and its half-Laplacian has exactly the sign needed to combine the two constraints and optimize over $b$.

What would settle it

One concrete check is to test identity (2.32) on a nontrivial entire Hardy function with real zeros, such as $G(\zeta)=\sin\zeta/\zeta$: both sides must give the same distribution, and the measures $\eta_-$, $\eta_+$, and $\eta_0$ must all be nonnegative as asserted. Alternatively, compute $T\log C_{\mathrm H}(T,L)$ numerically for $L=1$ from a truncated moment problem at several small $T$; if the values do not approach $\kappa_*\approx 0.6966$ to the expected precision, the claimed asymptotic is wrong.

Watch

Extended reading notes

Core claim

The core discovery is that the cost of fast boundary controls for the one-dimensional heat equation is governed by a single explicit constant: $C_{\mathrm H}(T,L)=\exp((\kappa_* L^2+o(1))/T)$, with $\kappa_* = \Gamma(1/4)^4/(8\pi^3) \approx 0.6966$. The paper proves the lower bound by turning the moment problem into an entire interpolation problem, removing the spectral zeros via the canonical product $P_-$, and deriving a necessary one-sided constraint $(H\Omega)'\ge -T/2$ on the logarithmic modulus of the multiplier from the fact that the interpolating function has Fourier support in $[0,T]$. After the natural rescaling $X=T^2\zeta$, this constraint becomes the half-Laplacian inequality $|D|u\ge -1/2$, paired with the pointwise obstacle $u\ge q_T(X)$; the limit obstacle is $q_*(X)=L\sqrt{|X|/2}$. Testing these inequalities against the explicit function $\rho_b(x)=\frac1\pi\log\frac{b+\sqrt{b^2-x^2}}{|x|}$ for $|x|<b$ and optimizing over $b$ yields exactly $\kappa_* L^2$. The matching upper bound comes from evaluating the previously known upper-bound constant, expressed as a convergent series, in closed form and obtaining the same $\kappa_*$.

Load-bearing premise

The load-bearing premise is Lemma 2.2's claim that comparing the two Hardy-space factorizations yields the one-sided inequality $(H\Omega)'\ge -T/2$ with the sign and jump terms exactly as stated; a single reversed sign or a missed jump at a real zero would destroy the lower bound.

Editorial extensions

If this is right

The true small-time rate is $\kappa_* L^2/T$, so the previous lower bound $1/2$ was not sharp: oscillatory solutions are exponentially harder to observe than the Gaussian heat-kernel barrier alone predicts.
The upper-bound estimate from [7] is optimal: its constant $K_0$, originally a convergent series, evaluates to exactly $\kappa_*$, so no refined upper construction can improve the exponential rate.
The duality estimate (1.7) now gives a precise small-time stability constant for reconstructing the final state from boundary flux observations, namely $\exp((\kappa_* L^2+o(1))/T)$.
The limiting variational data—the obstacle $q_*(X)=L\sqrt{|X|/2}$ and the constraint $|D|u\ge -1/2$—provide a canonical profile for the small-time control problem in this geometry.
For control problems that reduce to this one-dimensional spectral geometry, the sharp rate $\kappa_* L^2/T$ is the benchmark any quantitative estimate has to match.

Reading between the lines

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

The same Hardy-space phase comparison should apply to other one-dimensional equations with quadratic spectra, such as the Schrödinger equation, by replacing the canonical product with the relevant spectral product; the sharp constant would come from the analogous obstacle profile.
Because the sharp constant is extracted from the Green function of the restricted half-Laplacian, the small-time cost problem is probably equivalent to a genuinely variational fractional obstacle problem; the paper does not pursue that duality.
The closed-form value of the series suggests that an explicit optimal multiplier, not just the constant, may exist; constructing it would give a constructive sharp control for the heat equation.
A numerical study of the truncated moment problem at small $T$ would provide an independent check of the rate and could also reveal the size of the $o(1)$ remainder in the exponential.

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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

Dear colleague,

You should know about this paper: it proves the long-open exact exponential rate for the null-control cost of the 1D heat equation, C_H(T,L) = exp((κ* L^2 + o(1))/T) with κ* = Γ(1/4)^4/(8π^3) ≈ 0.6966. This closes the gap between Lissy's lower bound 1/2 and Dardé–Ervedoza's upper bound K0.

What is genuinely new: the lower bound that matches the upper bound, obtained by a converse of the Beurling–Malliavin phase method. Compact support is converted into the one-sided Hilbert-transform constraint (HΩ)' ≥ −T/2 instead of using phase to construct a multiplier. The paper also gives a closed form for the Dardé–Ervedoza series, S = 2Γ(1/4)/Γ(3/4) − 2πΓ(3/4)/Γ(1/4), which is elegant and correct. I checked the hypergeometric summation and the optimization of J*(b); both are internally consistent. The writing is clear and the organization is sensible.

The soft spot is Lemma 2.2, specifically the distributional identity (2.32) comparing boundary phases of G and Ψ. This requires careful bookkeeping of real-zero jumps (η0), inner-factor phase derivatives, and the signs in the modified Hilbert transform. I rechecked (2.34) and the algebra leading to (2.35); it is consistent as far as I can see. But the treatment at real zeros is delicate, and this is the one step I could not fully verify at a glance. An independent expert should examine it. If there is an error, it would likely change the constant; but I found no concrete flaw, and the rest of the lower-bound argument—the rescaling, the half-Laplacian obstacle inequality, and the test function—holds up.

This paper is for control theorists and complex analysts. It deserves a serious referee; the central claim is likely correct, and the method is reusable. Send it to review, with a referee specifically asked to check Lemma 2.2.

Referee Report

0 major / 4 minor

Summary. The paper studies the optimal L^2 null-control cost C_H(T,L) for the one-dimensional heat equation on (0,L), with a Dirichlet boundary control at one endpoint, homogeneous Dirichlet condition at the other, and initial data in H^{-1}(0,L). The main result, Theorem 1.1, asserts that T log C_H(T,L) tends to κ_* L^2 as T tends to 0^+, with κ_* = Γ(1/4)^4/(8π^3) ≈ 0.6966, and that this matches the previously known Dardé–Ervedoza upper-bound constant. The proof combines the moment method with a Paley–Wiener characterization of compact support through two Hardy-space conditions, then compares inner–outer factorizations to derive the one-sided Hilbert-transform inequality (HΩ)' ≥ -T/2 of Lemma 2.2. After a small-time rescaling this becomes a fractional obstacle-type inequality, which is tested against an explicit function ρ_b to obtain the sharp lower bound. The upper bound is taken from the independent result of Dardé and Ervedoza, and Section 3.2 evaluates the series defining their constant in closed form via hypergeometric transformations.

Significance. If Theorem 1.1 stands, it closes a long-standing gap between the earlier lower bound 1/2 and the known upper bound K_0, giving the exact exponential rate of the control cost for a fundamental parabolic model. The lower bound is self-contained and uses no fitted parameters: the constant emerges from an explicit optimization involving the half-Laplacian of the square-root profile q_*. The paper also gives a closed-form evaluation of the Dardé–Ervedoza series, so the constant is identified in a transparent way. I checked the main auxiliary computations, including the sign bookkeeping in Lemma 2.2: for a real zero of multiplicity m, the difference between the upper and lower boundary arguments is -2πm δ, consistent with the factor 2π in η_0 and with the final identity (HΩ)' + T/2 = 1/2(η_- + eη_+ + η_0). The hypergeometric summation in Theorem 3.1 also checks out. This is an important and genuinely quantitative contribution to the control theory of parabolic equations.

minor comments (4)

[Abstract] The phrase "obtained in by Lissy" should read "obtained by Lissy".
[§2.2] In the displayed definition of A_T, "mentire" should be "entire".
[§2.3, Eq. (2.32)] The jump relation d arg_+ Ψ_G = T dx + d arg_- G - η_0 is correct, but it would help the reader if the proof stated explicitly that the lower-half-plane boundary argument uses the branch tending to -π on the negative real axis and the upper-half-plane branch uses the branch tending to +π, so that a real zero of multiplicity m contributes exactly 2πm δ to the difference.
[§2.1, Definition 2.2] Since only derivatives of the Hilbert transform are used, the modified transform H and the usual transform H differ by a constant and are interchangeable in (HΩ)'; a one-line reminder would prevent sign confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bound is derived self-containedly from the moment method and Paley–Wiener conditions; the upper bound rests on the independent Dardé–Ervedoza theorem, matched to κ* by an explicit series evaluation.

full rationale

The derivation is not circular. The lower-bound chain starts from the minimax definition (1.2) and uses the moment method to obtain Proposition 2.3, an extremal Fourier-multiplier problem. The support condition is converted into the two Hardy conditions of Proposition 2.1, which are standard Paley–Wiener facts. Lemma 2.2 derives the one-sided constraint (HΩ)' ≥ −T/2 by comparing the inner–outer factorizations of G and e^{iTζ}G; the proof is written out in the paper and invokes no result of the author as a black box. Proposition 2.4 rescales the constraint into u_T ≥ q_T and |D|u_T ≥ −1/2, and Lemma 2.3 computes the limiting obstacle q* = L√(|X|/2) by an elementary integral over Fa. Proposition 3.1 then tests these inequalities against the explicit function ρ_b, obtaining max_b J*(b) = κ*L^2 with κ* = Γ(1/4)^4/(8π^3); no fitted parameter enters. The upper bound is taken from Dardé and Ervedoza [7], an independent external reference, and Theorem 3.1 evaluates its series using standard Whipple and Dixon identities. Self-citations ([22], [23], [24], [25]) are contextual or methodological comparisons; the phase-construction remark explicitly states that the direction of the argument is reversed relative to [15,25], and the needed phase comparison is proved in Lemma 2.2 rather than imported. No load-bearing step reduces to its own inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No physical constants are fitted; the only auxiliary parameter b is an optimization variable in the lower-bound test, and the final inequality is maximized over b. The proof imports standard complex analysis and two external control-theory results, all explicitly cited. The mathematical objects P_-, P_+, and rho_b are tools, not new physical entities.

free parameters (1)

test-function width b = b* = L^2 B(1/4,1/2)^2/(2 pi^2)
Auxiliary parameter in the lower-bound test function rho_b. The final lower bound is maximized over b, so no particular value is assumed a priori; it is an optimization variable, not a fitted constant.

assumptions (6)

standard math Hardy-space factorization theory: boundary values, inner-outer factorization, Blaschke products, Paley-Wiener for half-planes.
Used throughout Section 2.1; cited to Duren [8], Koosis [17], Mashreghi [27].
standard math Properties of the Kober modified Hilbert transform and the identity |D| = (H .)' as the 1D half-Laplacian.
Definition 2.2 and Lemma 3.1; cited to Kober [16] and Kwasnicki [19].
domain assumption Moment method reduction for 1D heat null controllability with boundary control.
Section 2.2, Proposition 2.3; cited to Fattorini-Russell [9,10].
domain assumption Duality between null controllability and observability, with observability constant equal to C_H(T,L).
Eq. (1.7); cited to Tucsnak-Weiss [38, Theorem 11.2.1].
domain assumption Darde-Ervedoza one-sided observability upper bound [7, Theorem 4.5] with constant K0.
Section 3.2; external theorem supplies the limsup half of Theorem 1.1.
standard math Whipple quadratic transformation and Dixon summation formula for 3F2 series.
Theorem 3.1; cited to NIST DLMF [31].

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Pith. "Pith review of Optimal cost of fast boundary controls for the one-dimensional heat equation." pith.science (2026). https://pith.science/paper/VMOOA52V

@misc{pith2026260808041,
  author       = {Pith},
  title        = {Pith review of: Optimal cost of fast boundary controls for the one-dimensional heat equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VMOOA52V}},
  note         = {Machine review of arXiv:2608.08041}
}

abstract

We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If $C_{\mathrm H}(T,L)$ denotes the optimal $L^2$ null-control cost for initial data in $H^{-1}(0,L)$, we prove that \[ C_{\mathrm H}(T,L) = \exp\left(\frac{\kappa_*L^2+o(1)}{T}\right), \qquad \kappa_* = \frac{\Gamma(\frac14)^4}{8\pi^3} \simeq 0.696601964842838, \qquad T\to0^+. \] The constant $\kappa_*$ coincides with the upper-bound constant obtained by Dard\'e and Ervedoza (2019, ANPDE), which was expressed there through a convergent series. This closes the gap between the lower bound obtained in by Lissy (2015, JDE) and the upper bound established by Dard\'e and Ervedoza.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces

V. Boulard, A. Hayat and E. Tr\'elat, Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces, preprint, arXiv:2607.13279, 2026

work page Pith review arXiv 2026

[2] [2]

Boyer, F

F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, J. Math. Pures Appl. (9) 93 (2010), no. 3, 240--276

work page 2010

[3] [3]

Boyer, F

F. Boyer, F. Hubert and J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations, Numer. Math. 118 (2011), no. 4, 601--661

work page 2011

[4] [4]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425--461

work page 2008

[5] [5]

Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007

work page 2007

[6] [6]

Coron and S

J.-M. Coron and S. Guerrero, Singular optimal control: a linear 1-D parabolic--hyperbolic example, Asymptot. Anal. 44 (2005), no. 3--4, 237--257

work page 2005

[7] [7]

Dard\'e and S

J. Dard\'e and S. Ervedoza, On the cost of observability in small times for the one-dimensional heat equation, Anal. PDE 12 (2019), no. 6, 1455--1488

work page 2019

[8] [8]

P. L. Duren, Theory of H^p Spaces, Pure and Applied Mathematics, vol. 38, Academic Press, New York--London, 1970

work page 1970