Pith. sign in

REVIEW 3 major objections 4 minor 72 references

A state-variable treatment of the trap stiffness eliminates the need for terminal jumps in optimal nano-oscillator protocols.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 12:47 UTC pith:HHZK7ABM

load-bearing objection Worth refereeing: the no-terminal-jump result is a real advance but it is built into the state augmentation in Eq. (2), not a theorem about the earlier stiffness-as-control formulation. the 3 major comments →

arxiv 2510.01823 v1 pith:HHZK7ABM submitted 2025-10-02 cond-mat.stat-mech math-phmath.MP

Optimal Control of Engineered Swift Equilibration of Nanomechanical Oscillators

Julia Sanders , Paolo Muratore-Ginanneschi This is my paper
classification cond-mat.stat-mech math-phmath.MP MSC 49K1582C3193E20
keywords optimal controlstochastic thermodynamicsswift equilibrationunderdamped oscillatorturnpike propertycentre manifoldwork minimizationterminal jumps
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the widely discussed requirement of discontinuous 'terminal jumps' in optimal protocols for swift equilibration is an artifact of treating the mechanical-force parameters as direct controls. Once the trap stiffness and center are promoted to state variables and the actual controls are their time derivatives, the work functional takes the canonical Bolza form and minimum-work transitions impose no explicit boundary condition on the terminal stiffness; that value is selected by a transversality condition. The same reformulation cleanly separates transitions of minimal dissipation between genuine equilibria from transitions of minimal work to non-equilibrium targets. Using centre manifold theory, the paper also shows that optimal protocols generically exhibit a turnpike: in the bulk of the control horizon they stay near a universal manifold set by the running cost, with exponential boundary layers at the endpoints. Numerical solutions of the full and reduced systems support the analysis.

Core claim

For a one-dimensional underdamped oscillator steered by a quadratic potential whose stiffness k_t and offset parameter u_t obey ˙k_t=λ_t, ˙u_t=γ_t, the paper shows by Pontryagin's maximum principle that the first-order conditions for minimum work reduce to a Hamiltonian system with no terminal condition on the stiffness. Stationarity of the terminal cost yields only the transversality relation y^(4)_{t_f} = -x^(1)_{t_f}/2, which implicitly selects the final stiffness. Hence the boundary-value problem is well posed without any jump in the mechanical force, and the overdetermination that produced the jump artifact disappears. A second result is that the optimality conditions admit a universal

What carries the argument

The central construction is the state augmentation of the mechanical potential: stiffness k_t and offset u_t become state variables with controls λ_t=˙k_t and γ_t=˙u_t. This converts the work functional into Bolza form, with a terminal cost that is a pure state function, making the terminal stiffness an optimization variable rather than an assigned boundary condition. The analytic engine is centre manifold reduction in Fenichel coordinates, which yields the universal slow-fast system (49)–(50) whose stable and unstable modes describe boundary layers and whose stationary point is the turnpike.

Load-bearing premise

The paper's conclusion that terminal jumps disappear rests on the modelling choice that the mechanical potential parameters are state variables whose time derivatives are the controls; if one instead treats the stiffness itself as the direct control, the overdetermination and the apparent need for jumps do not disappear.

What would settle it

Measure the trap stiffness during a work-optimal expansion in an underdamped optical trap with finite-bandwidth actuation: if the optimal protocol still exhibits a true discontinuity at the final time as the bandwidth is increased, the central claim fails. Alternatively, construct boundary data for which the transversality condition y^(4)_{t_f} = -x^(1)_{t_f}/2 has no admissible solution, which would limit the claimed disappearance of jumps.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Optimal swift-equilibration protocols do not need to end with a discontinuous change of the mechanical potential; smooth protocols exist with the same thermodynamic cost.
  • In minimum-work transitions to non-equilibrium targets, the final trap stiffness is a prediction of the theory, set by the transversality condition, not an input the experimenter must impose.
  • Minimum-dissipation transitions between equilibria and minimum-work transitions to non-equilibrium states are thermodynamically distinct, and the distinction is fixed by the choice of terminal cost in the Bolza formulation.
  • Optimal controls computed with different penalty mechanisms converge to the same universal turnpike manifold in the bulk, so stiffness-as-control results should be read as turnpike predictions, valid away from the endpoints.
  • The overdamped optimal control conditions are recovered in the ε→0 limit after optimization, confirming that the two limits commute for this detailed-balance dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same state-augmentation argument should apply to other stochastic thermodynamic control problems with actuation delay or memory, such as underdamped bit erasure, where endpoint discontinuities have also been debated.
  • A direct experimental test could compare the measured trap-stiffness protocol against the centre-manifold prediction: for small control penalties, the bulk protocol should be independent of the penalty shape, with deviations confined to short initial and final layers.
  • The transversality condition implies that for work-optimal driving the final stiffness may lie far from the equilibrium stiffness, so the system should undergo a measurable uncontrolled relaxation after t_f; the paper computes the associated heat release, giving a quantitative experimental signature.
  • Treating the potential parameters as states effectively raises the order of the control system; similar jump-elimination may appear more generally when controls act through derivatives rather than directly on physical state variables.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies optimal control of swift equilibration in underdamped Langevin dynamics, focusing on a one-dimensional nanomechanical oscillator. The central reformulation is to treat the parameters of the mechanical potential, k_t and u_t, as state variables obeying first-order control equations (2), with the actual controls being their time derivatives. This places the minimum-work and minimum-dissipation problems in the canonical Bolza form of optimal control, so that terminal costs depend only on state variables. The authors show that in this formulation the terminal stiffness is not prescribed for minimum-work transitions but is selected by the transversality condition y^(4)_tf = -x^(1)_tf/2 (Eq. 28), so no terminal jumps are needed. They derive Pontryagin first-order conditions, analyze the hard-, logarithmic-, and harmonic-penalty cases, and use centre-manifold and multiscale perturbation theory to obtain a universal slow-fast normal form (Eqs. 49-50) describing the turnpike behaviour of optimal protocols. Numerical comparisons between direct and indirect methods support the analytical results.

Significance. If the reformulation is accepted, the paper makes a useful conceptual clarification: the terminal-jump problem discussed in earlier optimal-control treatments of swift equilibration is an artefact of modelling the stiffness itself as the control. Treating the potential parameters as state variables with bounded rates of change removes the overdetermination and yields continuous protocols. The centre-manifold/turnpike analysis provides a compact universal description of the bulk dynamics, and the explicit distinction between minimum-dissipation (equilibrium-to-equilibrium) and minimum-work (equilibrium-to-non-equilibrium) transitions is valuable. The paper also provides reproducible numerical code and compares direct and indirect optimization methods. These are genuine strengths. The main caveat is that the headline conclusion is conditional on the state-augmentation assumption, and some parts of the asymptotic derivation are not fully transparent.

major comments (3)
  1. [Abstract; Sec. 2.1; Sec. 10.2] The statement that the terminal-jump problem 'completely disappears' is a property of the reformulated model with k_t and u_t as state variables obeying (2), not a theorem about the original stiffness-as-control formulation. In that original formulation (case S.II), the paper's own Figs. 5-6 show endpoint jumps. The abstract and conclusions should qualify the claim, e.g., 'in the Bolza formulation with first-order actuator dynamics' or 'for finite control bandwidth'. As written, the claim overreaches and could mislead readers about the scope of the result.
  2. [Sec. 7.4] The derivation of the universal normal form (49)-(50) rests on the solvability condition of the 8x8 system (47), specifically on the claim that ker(A^(1)^T) is spanned by {e_5,...,e_8} and that the omitted block entries 'do not play a role'. Without the explicit entries of A^(1) or a rigorous argument establishing the kernel and the projection of the non-homogeneous term, the reader cannot verify this load-bearing step. Please provide the explicit blocks in an appendix or in supplementary material, or give an independent derivation of the solvability condition.
  3. [Eqs. (10), (11), (13)] The running cost in the work functional is written as ∫(x^(3)_t + x^(6)_t/2)dt - t_f. From the Itô-lemma calculation in Section 3, the running cost should be ∫E[p_t^2]dt - t_f = ∫(x^(3)_t + (x^(6)_t)^2)dt - t_f. The term x^(6)/2 appears dimensionally inconsistent with the cumulant definitions (x^(6) is a first moment, x^(3) a second moment). Although the paper restricts to protocols with x^(6)=0, the displayed formula is a general statement and should be corrected to avoid propagating an error.
minor comments (4)
  1. [Sec. 10.2; Figs. 7-8] For minimum-work (C.II) transitions, the text explains that the terminal stiffness is free and determined by (28), but the figure captions say 'replace the boundary condition for x^(4)_tf with (28)' while referring to the equilibrium boundary conditions (56). Please state explicitly that in C.II the final state is generically not an equilibrium and that the target state is specified only by x^(1)_tf (and x^(3)_tf = 1, zero cross-correlation), not by a terminal stiffness value.
  2. [Sec. 7.2] The scaling h = g^{1/4} is motivated by analogy with [45], but the paper does not explain why this particular scaling is the distinguished limit for the harmonic penalty. A short justification would help readers.
  3. [Sec. 10.1.1] The description of the numerical comparison S.I versus S.II says 'Applying a standard mean filter convolution onto the data uncovers the centre manifold'. This smoothing is a post-processing step; it would be helpful to state explicitly that the raw S.II solution is noisy and the smoothing is only for visual comparison, not part of the optimization.
  4. [Throughout] Minor typographical issues: Fig. 2 legend 'solution of first order conditions direct optimisation' is ambiguous; Sec. 8.2 labels y^(4:1)_0,t2 and x^(4:1)_0,t2 as constants but the text says 'first order corrections to stiffness and corresponding drift are constant', which is consistent; please double-check subscripts in Eqs. (45)-(46) for readability.

Circularity Check

2 steps flagged

No-terminal-jump result is built into the state-variable reformulation; the centre-manifold scaling is imported from the authors' own prior work, but the core Pontryagin/numerical analysis is internally consistent.

specific steps
  1. self definitional [Abstract; Section 2.1 Eq. (2); Section 5.1.2 Eq. (28); Section 11]
    "we show that transitions at minimum work do not directly imply explicit boundary conditions on terminal values of parameters of the mechanical force and on control protocols. Thus, the problem often discussed in the literature, that optimal protocols need terminal jumps to satisfy boundary conditions, completely disappears. ... We assume that stiffness and centre obey the first order differential equations \dot k_t=\lambda_t, \dot u_t=\gamma_t ... boundary conditions should be imposed on system state variables rather than controls ... avoiding identifying, by construction, terminal costs with"

    The jump-free conclusion is not a theorem about the original stiffness-as-control problem; it is introduced by the state augmentation (2), under which k_t is an absolutely continuous state variable and cannot jump by definition. The terminal stiffness in case C.II is simply left unprescribed and fixed by the transversality condition (28). The paper's own S.II numerical comparison (Figs. 5-6) shows that endpoint jumps reappear when stiffness is treated as the control. Thus the abstract's 'completely disappears' is a property of the reformulated model, and the paper's conclusion even says 'by construction.'

  2. ansatz smuggled in via citation [Section 7.1-7.2 (Eqs. (41)-(42), h=g^{1/4})]
    "To this end, we draw from the supplementary material of [45]. There, centre (also known as invariant) manifold analysis [52, 53] is used to identify universal properties of mean dissipation minimising protocols in underdamped transition between Gaussian states when the control is the mechanical potential. ... insight from [45, Suppl] suggests to identify h=g^{1/4} as the order parameter of the expansion."

    The key asymptotic scaling (h=g^{1/4}) and the centre-manifold reduction are taken as 'insight' from the same first author's earlier paper [45], which analysed the very 'control = mechanical potential' formulation the present paper claims to supersede. The claimed universal normal form (49)-(50) therefore inherits the prior ansatz through a same-author citation; if [45] is not independently validated, the universality claim rests on that citation chain.

full rationale

The paper does not fit parameters to data, and its Pontryagin first-order conditions, boundary-value calculations, and direct-vs-indirect numerical comparisons are internally self-contained: the centre-manifold equations (49)-(50) are shown to reproduce the stationarity system (40), and the overdamped cell equations are checked against the overdamped optimal-control problem. The main circularity burden is the headline no-jump claim: it follows by construction from choosing k_t,u_t as state variables obeying (2), so it is not a resolution of the stiffness-as-control endpoint-jump problem but a reformulation. The paper itself acknowledges this in Section 11 ('by construction') and in the S.II discussion. A secondary self-citation supplies the asymptotic scaling from [45], but this does not infect the numerical comparisons or the first-order optimality analysis. Overall score 4: partial definitional circularity in the central claim, with substantial independent content elsewhere.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The derivation relies on standard stochastic thermodynamics plus a deliberate state-augmentation (integrator) model for the potential. No new physical entities are introduced. The main free parameters are the penalty weight g and the control bound Λ; formal scaling exponents are chosen to obtain normal forms.

free parameters (3)
  • g, g̃ (control penalty weights) = g = 0.01 in most numerics; 0 for hard penalty; varied 0.001, 0.0001, 0.1
    Introduced in Eq. (14) to make the non-convex work functional well posed; the paper explicitly states that quantitative properties of optimal controls are determined by the penalty form.
  • Λ (admissible control bound / barrier scale) = √2, 9, 1, 10 in figures
    Bounds controls in (15) or scales logarithmic/harmonic penalties (18)-(19); chosen by hand per setup. The turnpike claim is partly about insensitivity to Λ, but the quantitative protocols depend on it.
  • Scaling exponents h = g^{1/4} and g = h̃/ε² = formal, n/a
    The h = g^{1/4} fast-time scaling (Sec. 7.2) and the overdamped scaling (51) are chosen to make the singular perturbation expansion work; they are not derived from data or from a uniqueness theorem.
axioms (7)
  • domain assumption Underdamped Langevin dynamics (1) with additive white noise and linear friction.
    Central model; all cumulant equations follow from it. If the noise or friction model changes, the optimal-control structure changes.
  • standard math Gaussian state / linear dynamics: the cumulant hierarchy closes at second order (Sec. 2.2).
    Exact for linear SDEs, but restricts to harmonic potentials; anharmonic extension is only discussed as future work.
  • ad hoc to paper Mechanical potential parameters are state variables obeying first-order control equations ˙k = λ, ˙u = γ (Eq. 2).
    Load-bearing modeling choice: it converts endpoint potential values into state variables and removes terminal jumps. If the potential is directly controlled, the overdetermination remains.
  • domain assumption Maxwell-Boltzmann boundary conditions (5)-(8) at t = 0 and t_f.
    Defines genuine equilibrium endpoints; if relaxed, the C.I/C.II distinction and the engineered swift equilibration interpretation change.
  • standard math Normal extremals only; abnormal extremals are absent (Appendix A).
    Uses Pontryagin's maximum principle; the appendix gives a compact argument that abnormal extremals lead to empty conditions, though the proof is terse.
  • ad hoc to paper Self-concordant barrier / harmonic penalty models with g > 0 represent experimental control costs (Sec. 4.1).
    The quantitative claim that controls are determined by penalty form assumes these penalty models capture real actuator costs.
  • ad hoc to paper Singular perturbation scalings h = g^{1/4} and g = h̃/ε² (Secs. 7.2, 8).
    Formal asymptotic assumptions needed for the centre-manifold and overdamped normal forms; not proven to hold beyond these models.

pith-pipeline@v1.3.0-alltime-deepseek · 30104 in / 15343 out tokens · 130430 ms · 2026-08-04T12:47:53.216816+00:00 · methodology

0 comments
read the original abstract

We propose a reformulation of the problem of optimally controlled transitions in stochastic thermodynamics. We impose that any terminal cost specified by a thermodynamic functional should depend only on state variables and not on control protocols, according to the canonical Bolza form. In this way, we can unambiguously discriminate between transitions at minimum dissipation between genuine equilibrium states, and transitions at minimum work driving a system from a genuine equilibrium to a non-equilibrium state. For underdamped dynamics subject to a mechanical force, genuine equilibrium means a Maxwell-Boltzmann probability distribution defining a vanishing current velocity. Transitions at minimum dissipation between equilibria are a model of optimal swift engineered equilibration. Remarkably, we show that transitions at minimum work do not directly imply explicit boundary conditions on terminal values of parameters of the mechanical force and on control protocols. Thus, the problem often discussed in the literature, that optimal protocols need terminal jumps to satisfy boundary conditions, completely disappears. The quantitative properties of optimal controls are entirely determined by the form of the penalty modelling an experimental setup. More generally, we use centre manifold theory to analytically account for the tendency of optimal controls to exhibit a turnpike property: optimal protocols in the bulk of the control horizon tend to converge to a universal centre manifold determined only by the running cost. Exponential deviations from the centre manifold occur at the ends of the control horizon in order to satisfy the boundary conditions. Our findings are supported numerically.

Figures

Figures reproduced from arXiv: 2510.01823 by Julia Sanders, Paolo Muratore-Ginanneschi.

Figure 1
Figure 1. Figure 1: Engineered swift equilibration minimising the entropy production (20), subject to the harmonic penalty (19). The solution is computed using a direct optimisation on the cost functional (20) (blue) with InfiniteOpt.jl [61] and solved using Interior Point Optimisation IPOpt [33] and using g = 0.01. The central manifold solution is shown for decreasing values of g: g = 0.01 orange, g = 0.001 green, g = 0.0001… view at source ↗
Figure 2
Figure 2. Figure 2: Engineered swift equilibration minimising the entropy production (20). We find the solution by a direct optimisation of the cost functional with a hard penalty (15) (blue) and the solution of system of differential equations specifying the first order optimality conditions (29) with a logarithmic penalty (18) (orange-dashed). We fix Λ = 9, ε = 1, tf = 3 and use g = 0.001 and g = 0 for the logarithmic and h… view at source ↗
Figure 3
Figure 3. Figure 3: Engineered swift equilibration minimising the entropy production (20) for a contraction. We use a harmonic penalty (19) and find the solution with a direct optimisation of the cost functional (blue lines) and as a solution of the first order conditions (orange, dashed). We fix Λ = √ 2, ε = 1, tf = 3 and g = 0.01. Boundary conditions are given by (28), with σ 2 0 = 1 and σ 2 tf = 1/2. Numerical methods are … view at source ↗
Figure 4
Figure 4. Figure 4: Entropy production as a function of the time horizon tf where the stiffness is a state S.I (blue, triangle) and where the stiffness is a control S.II (orange, circle). When the stiffness is a state, we use a hard penalty −Λ ≤ λt ≤ Λ. Panels (a) and (b) constrain the stiffness kt in the interval 0.2 ≤ kt ≤ 1.2. Panels (c) and (d) have the constraint −0.1 ≤ kt ≤ 1.2, allowing for negative values of kt . To m… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of engineered swift equilibration at minimum entropy production when the stiffness is the control S.II (orange) and when the stiffness is a state S.I (blue, green). The stiffness kt is constrained in the interval 0.2 ≤ kt ≤ 1.2 in both cases. We use a hard penalty (15) to model the case when the stiffness is a state, with Λ = 1 (blue) and Λ = 10 (green). Results are computed by a direct optimisa… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of engineered swift equilibration at minimum entropy production when the stiffness is the control S.II (orange) and when the stiffness is a state S.I (blue, green). The stiffness kt is constrained in the interval −0.1 ≤ kt ≤ 1.2 in both cases, allowing for negative values. We use a hard penalty (15) to model the case when the stiffness is a state, with Λ = 1 (blue) and Λ = 10 (green). Results ar… view at source ↗
Figure 7
Figure 7. Figure 7: Expansion at minimum work. We use a harmonic penalty and find the solution using a direct optimisation (blue) and as the solution of the first order conditions (orange, dashed). We use tf = 3, ε = 1, g = 0.01 and Λ = √ 2. We use boundary conditions (56) and replace the boundary condition for x (4) tf with (28), where σ 2 0 = 1 and σ 2 tf = 2. Numerical methods are used as those in [PITH_FULL_IMAGE:figures… view at source ↗
Figure 8
Figure 8. Figure 8: Contraction at minimum work. We use a harmonic penalty and find the solution using a direct optimisation (blue) and as the solution of the first order conditions (orange, dashed). We use tf = 7, ε = 1, g = 0.1 and Λ = √ 2. We use boundary conditions (56) and replace the boundary condition for x (4) tf with (28), where σ 2 0 = 4 and σ 2 tf = 1. Numerical methods are used as those in [PITH_FULL_IMAGE:figure… view at source ↗
Figure 9
Figure 9. Figure 9: Thermodynamic costs of an expansion for engineered swift equilibration (a) and at minimum work (b). We show the mean work (Wtf (10), blue triangle); mean heat release (Qtf , orange cross); mean entropy production (Etf (9), green dot) as functions of the time horizon. Inset in panel (a) shows the difference in entropy production between the engineered swift equilibration and minimal work transition, showing… view at source ↗
Figure 10
Figure 10. Figure 10: Thermodynamic costs of an expansion for engineered swift equilibration (a) and at minimum work (b). We use parameters and boundary conditions as [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

72 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    Mart ´ınez I A, Rold´an E, Dinis L, Petrov D, Parrondo J M R and Rica R A 2016Nature Physics1267–70 URL https://doi.org/10.1038/nphys3518

    work page doi:10.1038/nphys3518

  2. [2]

    Mart ´ınez I A, Petrosyan A, Gu ´ery-Odelin D, Trizac E and Ciliberto S 2016Nature Physics12843–846 ISSN 1745-2481 URLhttps://doi.org/10.1038/nphys3758

    doi:10.1038/nphys3758

  3. [3]

    org/10.1063/1.5143602

    Albay J A C, Lai P Y and Jun Y 2020Applied Physics Letters116ISSN 1077-3118 URLhttp://dx.doi. org/10.1063/1.5143602

    work page doi:10.1063/1.5143602

  4. [4]

    Gonzalez-Ballestero C, Aspelmeyer M, Novotny L, Quidant R and Romero-Isart O 2021Science374ISSN 1095-9203 URLhttp://dx.doi.org/10.1126/science.abg3027

    doi:10.1126/science.abg3027

  5. [5]

    Dago S, Ciliberto S and Bellon L 2023Proceedings of the National Academy of Sciences120ISSN 1091-6490 URLhttp://dx.doi.org/10.1073/pnas.2301742120

    work page doi:10.1073/pnas.2301742120

  6. [6]

    Cunuder A L, Martinez I, Petrosyan A, Gu ´ery-Odelin D, Trizac E and Ciliberto S 2016Applied Physics Letters 109113502 URLhttps://doi.org/10.1063/1.4962825

    work page doi:10.1063/1.4962825

  7. [7]

    Barros N, Ciliberto S and Bellon L 2024Physical Review Letters133057101 ISSN 1079-7114 URLhttps: //doi.org/10.1103/PhysRevLett.133.057101

    work page doi:10.1103/physrevlett.133.057101

  8. [8]

    Raynal D, de Guillebon T, Gu ´ery-Odelin D, Trizac E, Lauret J S and Rondin L 2023Physical Review Letters 131087101 URLhttps://doi.org/10.1103/PhysRevLett.131.087101

    doi:10.1103/physrevlett.131.087101

  9. [9]

    A Concise Introduction(Princeton, New Jersey: Princeton University Press) ISBN 978-0-691-15187-8 URLhttp://press.princeton.edu/ titles/9760.html

    Liberzon D 2012Calculus of Variations and Optimal Control Theory. A Concise Introduction(Princeton, New Jersey: Princeton University Press) ISBN 978-0-691-15187-8 URLhttp://press.princeton.edu/ titles/9760.html

  10. [10]

    Bechhoefer J 2021Control Theory for Physicists(Cambridge: Cambridge University Press) ISBN 9781009028493 URLhttps://www.cambridge.org/gb/academic/subjects/physics/ mathematical-methods/control-theory-physicists?format=AR

  11. [11]

    Aurell E, Gawe ¸dzki K, Mej ´ıa-Monasterio C, Mohayaee R and Muratore-Ginanneschi P 2012Journal of Statistical Physics147487–505 URLhttps://doi.org/10.1007/s10955-012-0478-x

    doi:10.1007/s10955-012-0478-x

  12. [12]

    Villani C 2009Optimal transport: old and new(Grundlehren der mathematischen Wissenschaftenvol 338) (Springer) URLhttps://link.springer.com/book/10.1007/978-3-540-71050-9

    doi:10.1007/978-3-540-71050-9

  13. [13]

    Chupeau M, Ciliberto S, Gu ´ery-Odelin D and Trizac E 2018New Journal of Physics20075003 ISSN 1367-2630 URLhttp://dx.doi.org/10.1088/1367-2630/aac875

    work page doi:10.1088/1367-2630/aac875

  14. [14]

    1103/PhysRevE.96.012144

    Li G, Quan H T and Tu Z C 2017Physical Review E96ISSN 2470-0053 URLhttp://dx.doi.org/10. 1103/PhysRevE.96.012144

  15. [15]

    Frim A G, Zhong A, Chen S F, Mandal D and DeWeese M R 2021Physical Review E103ISSN 2470-0053 URL http://dx.doi.org/10.1103/PhysRevE.103.L030102

    work page doi:10.1103/physreve.103.l030102

  16. [16]

    Gu ´ery-Odelin D, Jarzynski C, Plata C A, Prados A and Trizac E 2023Reports on Progress in Physics86035902 ISSN 1361-6633 URLhttp://dx.doi.org/10.1088/1361-6633/acacad

    doi:10.1088/1361-6633/acacad

  17. [17]

    Barros N, Whitelam S, Ciliberto S and Bellon L 2025Physical Review E111044114 ISSN 2470-0053 (Preprint 2409.15050) URLhttps://doi.org/10.1103/PhysRevE.111.044114

    Pith/arXiv arXiv

  18. [18]

    Collin D, Ritort F, Jarzynski C, Smith S B, Jr I T and Bustamante C 2005Nature437231–234 URL https://doi.org/10.1038/nature04061

    doi:10.1038/nature04061

  19. [19]

    Chetrite R, Muratore-Ginanneschi P and Schwieger K 2021The European Physical Journal H461–29 URL https://doi.org/10.1140%2Fepjh%2Fs13129-021-00032-7

  20. [20]

    Plata C A, Prados A, Trizac E and Gu ´ery-Odelin D 2021Physical Review Letters127ISSN 1079-7114 URL http://dx.doi.org/10.1103/PhysRevLett.127.190605

    doi:10.1103/physrevlett.127.190605

  21. [21]

    com/book/10.1007/978-1-4939-1323-7

    Pavliotis G A 2014Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations(New York: Springer New York) ISBN 978-1-4939-1322-0 URLhttps://link.springer. com/book/10.1007/978-1-4939-1323-7

    doi:10.1007/978-1-4939-1323-7

  22. [22]

    Schmidt R, Carusela M F, Pekola J P, Suomela S and Ankerhold J 2015Physical Review B91224303 URL https://doi.org/10.1103/PhysRevB.91.224303 Optimal control of engineered swift equilibration of nanomechanical oscillators40

    work page doi:10.1103/physrevb.91.224303

  23. [23]

    Muratore-Ginanneschi P, Mej´ıa-Monasterio C and Peliti L 2013Journal of Statistical Physics150181–203 URL https://link.springer.com/article/10.1007/s10955-012-0676-6

    work page doi:10.1007/s10955-012-0676-6

  24. [24]

    Lent C S, Orlov A O, Porod W and Snider G L (eds) 2019Energy Limits in Computation(Cham: Springer International Publishing) URLhttps://doi.org/10.1007%2F978-3-319-93458-7

  25. [25]

    Schmiedl T and Seifert U 2008EPL (Europhysics Letters)8120003 ISSN 1286-4854 URLhttps:// iopscience.iop.org/article/10.1209/0295-5075/81/20003

    work page doi:10.1209/0295-5075/81/20003

  26. [26]

    Muratore-Ginanneschi P and Schwieger K 2015EPL (Europhysics Letters)11220002 URLhttps:// iopscience.iop.org/article/10.1209/0295-5075/112/20002

    work page doi:10.1209/0295-5075/112/20002

  27. [27]

    Gu ´ery-Odelin D, Ruschhaupt A, Kiely A, Torrontegui E, Mart ´ınez-Garaot S and Muga J G 2019Review of Modern Physics91045001 URLhttps://link.aps.org/doi/10.1103/RevModPhys.91. 045001

    doi:10.1103/revmodphys.91

  28. [28]

    Hirose M and Cappellaro P 2018Quantum Information Processing171–17 (Preprint1510.06801) URL https://doi.org/10.1007/s11128-018-1845-6

    Pith/arXiv arXiv

  29. [29]

    Chupeau M, Besga B, Gu ´ery-Odelin D, Trizac E, Petrosyan A and Ciliberto S 2018Physical Review E98ISSN 2470-0053 URLhttp://dx.doi.org/10.1103/PhysRevE.98.010104

    work page doi:10.1103/physreve.98.010104

  30. [30]

    1016/0022-0396(79)90152-9

    Fenichel N 1979Journal of Differential Equations3153–98 ISSN 0022-0396 URLhttps://doi.org/10. 1016/0022-0396(79)90152-9

  31. [31]

    Guckenheimer J and Kuehn C 2009SIAM Journal on Applied Dynamical Systems8854–879 ISSN 1536-0040 URLhttps://doi.org/10.1137/080741999

    work page doi:10.1137/080741999

  32. [32]

    Cardin P T and Teixeira M A 2017SIAM Journal on Applied Dynamical Systems161425–1452 ISSN 1536-0040 URLhttp://dx.doi.org/10.1137/16M1067202

    work page doi:10.1137/16m1067202

  33. [33]

    W ¨achter A 2009 Short Tutorial: Getting Started With Ipopt in 90 MinutesCombinatorial Scientific Computing (Dagstuhl Seminar Proceedings (DagSemProc)vol 9061) ed Naumann U, Schenk O, Simon H D and Toledo S (Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum f ¨ur Informatik) pp 1–17 ISSN 1862-4405 URL https://drops.dagstuhl.de/entities/document/10.423...

    work page doi:10.4230/dagsemproc.09061.16 2009

  34. [34]

    Exploring Benefits of Linear Solver Parallelism on Modern Nonlinear Optimization Applications

    Tasseff B, Coffrin C, W ¨achter A and Laird C 2019eprint arXiv:1909.08104URLhttps://doi.org/10. 48550/arXiv.1909.08104

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1909.08104 1909

  35. [35]

    Boscain U, Sigalotti M and Sugny D 2021PRX Quantum2URLhttps://doi.org/10.1103% 2Fprxquantum.2.030203

  36. [36]

    Nelson E 2001Dynamical Theories of Brownian Motion2nd ed (Princeton University Press) ISBN-13: 978-0- 691-07950-9 URLhttps://web.math.princeton.edu/ ˜nelson/books.html

  37. [37]

    Peliti L and Pigolotti S 2020Stochastic Thermodynamics(Princeton and Oxford: Princeton University Press) ISBN 9780691215525 URLhttps://press.princeton.edu/books/ebook/9780691215525/ stochastic-thermodynamics

    arXiv

  38. [38]

    Ch ´etrite R and Gawe ¸dzki K 2008Communications in Mathematical Physics282469–518 (Preprint0707

  39. [39]

    Baldovin M, Ben Yedder I, Plata C A, Raynal D, Rondin L, Trizac E and Prados A 2024Physical Review Letters 135097102 (Preprint2408.00043) URLhttps://doi.org/10.1103/pss4-b69w

    Pith/arXiv arXiv

  40. [40]

    anilvrao.com/Publications/ConferencePublications/trajectorySurveyAAS.pdf

    Rao A V 2009 A Survey of Numerical Methods for Optimal Control AAS 09-334)Astrodynamics 2009: proceedings of the AAS/AIAA Astrodynamics Specialist Conference held August 9-13 2009, Pittsburgh, Pennsylvania(Adavnces in Astronautical Sciencesno 135) (American Astronautical Society / American Institute of Aeronautics and Astronautics) pp 497–528 ISBN 9780877...

    2009

  41. [41]

    Teo K L, Li B, Yu C and Rehbock V 2021Applied and Computational Optimal Control(Springer Optimization and Its Applicationsvol 171) (Springer) ISBN 978-3-030-69913-0 URLhttps://doi.org/10.1007/ 978-3-030-69913-0

  42. [42]

    Karmarkar N 1984Combinatorica4373–395 ISSN 1439-6912 URLhttp://dx.doi.org/10.1007/ BF02579150

  43. [43]

    9781611970791

    Nesterov Y and Nemirovskii A 1994Interior-Point Polynomial Algorithms in Convex Programming(Society for Optimal control of engineered swift equilibration of nanomechanical oscillators41 Industrial and Applied Mathematics) ISBN 9781611970791 URLhttp://dx.doi.org/10.1137/1. 9781611970791

    doi:10.1137/1

  44. [44]

    org/10.1017/S0962492906370018

    Nemirovski A S and Todd M J 2008Acta Numerica17191–234 ISSN 1474-0508 URLhttp://dx.doi. org/10.1017/S0962492906370018

    work page doi:10.1017/s0962492906370018

  45. [45]

    Muratore-Ginanneschi P and Schwieger K 2014Physical Review E90060102(R) URLhttps://doi.org/ 10.1103/PhysRevE.90.060102

    work page doi:10.1103/physreve.90.060102

  46. [46]

    Sanders J, Baldovin M and Muratore-Ginanneschi P 2024Journal of Statistical Physics191117 URLhttps: //doi.org/10.1007/s10955-024-03320-w

    work page doi:10.1007/s10955-024-03320-w

  47. [47]

    Boscain U and Piccoli B 2005 An Introduction to Optimal ControlControle non lineaire et applicationsLe cours du CIMPA ed Sari T (Hermann) pp 19–66 URLhttp://www.cmapx.polytechnique.fr/ ˜boscain/

    2005

  48. [48]

    Gomez-Marin A, Schmiedl T and Seifert U 2008The Journal of Chemical Physics129024114 URLhttps: //doi.org/10.1063/1.2948948

    work page doi:10.1063/1.2948948

  49. [49]

    org/prl/abstract/10.1103/PhysRevLett.111.260501

    Hegerfeldt G C 2013Physical Review Letters111260501 ISSN 1079-7114 URLhttps://journals.aps. org/prl/abstract/10.1103/PhysRevLett.111.260501

    work page doi:10.1103/physrevlett.111.260501

  50. [50]

    Muratore-Ginanneschi P and Schwieger K 2017Entropy19379 URLhttps://doi.org/10.3390/ e19070379

  51. [51]

    Lucente D, Manacorda A, Plati A, Sarracino A and Baldovin M 2025Entropy27268 ISSN 1099-4300 (Preprint 2501.07735) URLhttp://dx.doi.org/10.3390/e27030268

    Pith/arXiv arXiv

  52. [52]

    Tin S K, Kopell N and Jones C K R T 1994SIAM Journal on Numerical Analysis311558–1576 ISSN 1095-7170 URLhttps://doi.org/10.1137/0731081

    work page doi:10.1137/0731081

  53. [53]

    Hayes M, Kaper T J, Kopell N and Ono K 1998International Journal of Bifurcation and Chaos08189–209 ISSN 1793-6551 URLhttps://doi.org/10.1142/S0218127498000140

    work page doi:10.1142/s0218127498000140

  54. [54]

    Verhulst F 2005Methods and applications of singular perturbations: boundary layers and multiple timescale dynamicsTexts in applied mathematics (New York: Springer) ISBN 978-0387-22966-9 URLhttps: //doi.org/10.1007/0-387-28313-7

    work page doi:10.1007/0-387-28313-7

  55. [55]

    1007/978-0-387-73829-1

    Pavliotis G A and Stuart A M 2008Multiscale methods: averaging and homogenization(Texts in applied mathematicsvol 53) (New York: Springer) ISBN 978-0-387-73828-4 URLhttps://doi.org/10. 1007/978-0-387-73829-1

  56. [56]

    Hartmann C, Latorre J C, Zhang W and Pavliotis G A 2014Journal of Computational Dynamics1279–306 ISSN 2158-2491 URLhttps://www.aimsciences.org/article/doi/10.3934/jcd.2014. 1.279

    doi:10.3934/jcd.2014 2014

  57. [57]

    Bo S and Celani A 2017Physics Reports6701–59 ISSN 0370-1573 URLhttp://dx.doi.org/10.1016/ j.physrep.2016.12.003

    2016

  58. [58]

    Risken H 1996The Fokker-Planck equation: methods of solution and applications3rd ed (Series in Synergeticsvol 18) (Springer) ISBN 978-3-540-61530-9 URLhttps://doi.org/10.1007/ 978-3-642-61544-3

  59. [59]

    functionaldifferentialequations.com/index.php/fde/article/view/570

    Halperin I and Agranovich G 2014Functional Differential Equations21119–136 URLhttps://www. functionaldifferentialequations.com/index.php/fde/article/view/570

  60. [60]

    Ascher U 2015Collocation Methods(Berlin, Heidelberg: Springer Berlin Heidelberg) p 224–227 ISBN 978-3- 540-70529-1 URLhttps://doi.org/10.1007/978-3-540-70529-1_102

    work page doi:10.1007/978-3-540-70529-1_102

  61. [61]

    Pulsipher J L, Zhang W, Hongisto T J and Zavala V M 2022Computers & Chemical Engineering156107567 ISSN 0098-1354 URLhttps://doi.org/10.1016/j.compchemeng.2021.107567

    arXiv 2021

  62. [62]

    Rackauckas C and Nie Q 2017Journal of Open Research Software515 URLhttps:// openresearchsoftware.metajnl.com/articles/10.5334/jors.151

    doi:10.5334/jors.151

  63. [63]

    Jay L O 2015Lobatto Methods(Springer Berlin Heidelberg) p 817–826 ISBN 9783540705291 URLhttp: //dx.doi.org/10.1007/978-3-540-70529-1_123

    work page doi:10.1007/978-3-540-70529-1_123

  64. [64]

    Sanders J 2025GitHub repositoryURLhttps://github.com/julia-sand/swift_equilibrium

  65. [65]

    Sanders J, Baldovin M and Muratore-Ginanneschi P 2025Physical Review E111ISSN 2470-0053 URL Optimal control of engineered swift equilibration of nanomechanical oscillators42 http://dx.doi.org/10.1103/PhysRevE.111.034127

    work page doi:10.1103/physreve.111.034127

  66. [66]

    Schmiedl T and Seifert U 2007Physical Review Letters98108301 (Preprintcond-mat/0701554) URL https://doi.org/10.1103/PhysRevLett.98.108301

    arXiv

  67. [67]

    Aurell E, Mej ´ıa-Monasterio C and Muratore-Ginanneschi P 2011Physical Review Letters106250601 URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.250601

    doi:10.1103/physrevlett.106.250601

  68. [68]

    Aurell E, Mej ´ıa-Monasterio C and Muratore-Ginanneschi P 2012Physical Review E85(2) 020103(R) (Preprint 1111.2876)

    Pith/arXiv arXiv

  69. [69]

    Samuelson P A 1965The American Economic Review55486–496 URLhttps://www.jstor.org/ stable/1814560

    arXiv

  70. [70]

    org/stable/2296111

    V on Neumann J 1945The Review of Economic Studies131 ISSN 0034-6527 URLhttp://www.jstor. org/stable/2296111

    arXiv

  71. [71]

    Tr ´elat E and Zuazua E 2015Journal of Differential Equations25881–114 ISSN 0022-0396 URLhttp: //dx.doi.org/10.1016/j.jde.2014.09.005

    work page doi:10.1016/j.jde.2014.09.005 2014

  72. [2725]

    URLhttps://doi.org/10.1007/s00220-008-0502-9

    doi:10.1007/s00220-008-0502-9