Pith. sign in

REVIEW 2 major objections 5 minor 62 references

Optimal control of mean-field limit of multiagent systems with and without common noise

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper proves Γ-convergence of finite-particle optimal control problems for a herd-herder system to a McKean-Vlasov mean-field limit, and convergence of the minima as the herd size tends to infinity.

desk verdict Solid mean-field limit with a repairable but real gap in the control compactness argument. read the letter →

arxiv 2505.16721 v1 pith:2NBHAXVG submitted 2025-05-22 math.OC

Giuseppe La Scala This is my paper · ORCID
classification math.OC MSC 49J2049J5560H10
keywords mean-fieldlimitpropagationofchaoscommonnoiseoptimalcontrolGamma-convergenceSDE-ODEsystemMcKean-VlasovequationFokker-PlanckSPDE
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 studies optimal control of a two-species particle system: a large "herd" whose members follow stochastic differential equations, and a fixed small number of "herders" whose motions are controlled ordinary differential equations. The main claim is that as the herd size $N$ tends to infinity, the finite-particle optimal control problems converge, in the sense of $\Gamma$-convergence, to a mean-field optimal control problem described by a McKean-Vlasov equation coupled to the herders' ODEs. The limit is established both without common noise, where herd particles become independent in the limit, and with common noise, where they become conditionally independent given the common noise. The payoff is that the infinite-dimensional limit problem is a tractable surrogate for designing controls for very large herds.

What carries the argument

The central object is the McKean-Vlasov limit: for a fixed number $M$ of herders, the empirical measure $\mu_N(t)$ of the herd is replaced by the conditional law $\mu^{(i)}(t)$ of a single representative herd particle given the common noise, turning the $N$-particle SDE-ODE system into one McKean-Vlasov SDE coupled with $M$ ODEs. The convergence proof combines Gronwall estimates on pathwise differences, the Burkholder-Davis-Gundy inequality to control moments of the multiplicative idiosyncratic and common noises, and a quantitative law of large numbers for Wasserstein distances to control the random empirical measure. The $\Gamma$-convergence step is carried by convexity of the running cost in the control variable, which gives lower semicontinuity with respect to weak $L^1$, and by uniform continuity of the transient and endpoint costs, which is funneled through Jensen's inequality.

What would settle it

Test the claimed rates in Theorem 4.1 on a system with a singular interaction kernel, for instance $K_1(x)=x/|x|^3$ in dimension three: if the pathwise or Wasserstein error bounds (4.1)-(4.2) fail or degenerate, the quantitative mean-field limit is refuted. A second check is to replace the convex running cost by a nonconvex $\Psi_\rho$ and verify whether the liminf inequality in Theorem 6.10 still holds.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the discrete optimal control problems (6.7) $\Gamma$-converge to the mean-field problem (6.18) in the product topology of weak $L^1$ for the time-dependent control factor and strong $C((\mathbb{R}^d)^M \times W_1(\mathbb{R}^d);\mathbb{R}^d)$ for the state-dependent control factor, and that the minima converge, $\lim_N \min F_N = \min F$. This rests on the mean-field limit Theorem 4.1, which gives quantitative convergence of the $N$-particle trajectories and empirical measures to the McKean-Vlasov system, and on conditional propagation of chaos. The paper also proves existence of minimizers for both problems, derives the stochastic Fokker-Planck equation satisfied by the conditional law of the limit herd, and establishes uniqueness of measure-valued solutions in the zero-common-noise case via a Feynman-Kac duality argument.

Load-bearing premise

The load-bearing premise is that all interaction kernels, noise coefficients, and controls are globally Lipschitz with moderate interaction intensity, meaning no singular or long-range strong forces; if any kernel becomes singular, the quantitative Gronwall estimates that drive the mean-field limit no longer apply.

Editorial extensions

If this is right

  • For large herds, a control that is optimal for the mean-field problem is asymptotically optimal for the finite-particle system, with the minimal-cost gap tending to zero.
  • The quantitative rates in Theorem 4.1 give explicit error bounds for approximating $N$-particle simulations by the limiting McKean-Vlasov dynamics and for comparing empirical measures with the conditional law.
  • In the common-noise case the limit object is a conditional law rather than a deterministic measure, so the effective control problem inherits randomness from the environment.
  • Observables of distinct herd particles become independent in the limit, unconditionally without common noise and conditionally on the common noise otherwise.
  • The existence of minimizers for both the finite and the limit control problems is part of the claim, so the $\Gamma$-convergence result is not vacuous.

Reading between the lines

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

  • A natural extension, not pursued here, would let the number of herders $M$ grow slowly with $N$; the fixed-$M$ assumption is what keeps the herders in the limiting system as ODEs rather than as a second mean-field species.
  • The convexity of the running cost appears to be the essential condition for the liminf inequality; replacing it by a weaker lower-semicontinuity assumption would test whether the $\Gamma$-convergence mechanism is sharper than the proof's use of convexity.
  • The uniqueness argument for the zero-common-noise Fokker-Planck equation suggests a conditional version of the same Feynman-Kac duality might prove uniqueness for the common-noise SPDE under stronger coefficient regularity, a question the paper leaves open.
  • The convergence of minima suggests practical algorithms could optimize the mean-field problem and then project the obtained controls back to finite $N$, but the paper does not discuss rates for the control functions themselves, only for the minima.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the SDE-ODE system (1.1)-(1.3) describing N 'herd' particles and M 'herders', with pairwise interaction kernels H_i, K_i, multiplicative idiosyncratic noise sigma^(i)_* dW^(i)_n, common noise sigma^(c)_* dW^(c), and controls of the separated form u_m = h_m(t) g_m(Y, mu_N). It proves strong well-posedness for the finite-particle system and for the McKean-Vlasov limit (Lemmas 3.5 and 3.6), a quantitative mean-field limit with rates taken from Fournier-Guillin and p-th moment convergence (Theorem 4.1), (conditional) propagation of chaos (Corollary 4.4), the stochastic/deterministic Fokker-Planck equation for the limit law (Theorem 5.1 and Corollary 5.3), existence of optimal controls for the discrete and mean-field costs (Lemmas 6.7 and 6.9), and Gamma-convergence of the discrete optimal control problems to the mean-field problem together with convergence of minima (Theorem 6.10).

Significance. If the gaps discussed below are repaired, this is a solid and useful extension of Ascione-Castorina-Solombrino [9] in two directions: the noise class is enlarged to multiplicative idiosyncratic noise plus common noise, and the control class is enlarged to the separated form h_m(t) g_m(y, nu). The proofs are largely self-contained and contain genuine quantitative content: the Burkholder-Davis-Gundy based moment estimates in Theorem 4.1, the explicit Wasserstein rates in (4.1)-(4.2), the conditional propagation of chaos in Corollary 4.4, and the Feynman-Kac uniqueness argument in Section 5, which avoids the finite-entropy condition used in [9]. The Gamma-convergence framework for finite-particle optimal control problems is a valuable contribution. The main caveat is that the compactness arguments in Section 6 require an additional convexity assumption on the control set U and a corrected justification of the extraction of the limit g.

major comments (2)
  1. [§6, Lemma 6.7, Eq. (6.15)] Assumptions 6.1(i) only assume U and G compact; U is not assumed convex. In the proof of Lemma 6.7 a minimizing sequence h^(j) is extracted with h^(j) converging weakly in L^1, and it is asserted 'with no loss of generality' that the weak limit h_* belongs to (L^1([0,T];U))^M because U is compact. This inference is false for nonconvex U: the weak closure of L^1([0,T];U) is contained in L^1([0,T];conv U) and can contain functions not taking values in U. For example, with U={0,1}, the sequence h^(j)(t)=1_{[0,1/2]}(j t mod 1) on [0,1] converges weakly in L^1 to the constant 1/2. Since the running cost is only assumed convex in its first argument and no convexity of U is stated, the existence conclusions of Lemma 6.7 and Lemma 6.9 and the extraction step in Theorem 6.10(ii) are not established as written. Please either add an explicit convexity assumption on U or formulate the limiting problem over relaxed controls valued in conv U; with either repair the subsequent argument goes through.
  2. [§6, Lemma 6.7, Eq. (6.16)] The proof states that W_p(R^d) is sigma-compact and uses Ascoli-Arzela to extract a pointwise convergent subsequence of g^(j). W_p(R^d) is not sigma-compact: it is a complete metric space that is not locally compact, so it cannot be a countable union of compact sets. The stated justification is therefore incorrect. The extraction can be repaired, because the family G is equi-Lipschitz and W_p(R^d) is separable: a diagonal subsequence over a countable dense set converges pointwise, and equicontinuity upgrades this to uniform convergence on compact sets. Please replace the argument or cite a correct compactness criterion.
minor comments (5)
  1. [Assumptions 2.9(iii), Eq. (2.27)] Condition (2.27) repeats the continuity assumption for sigma^(i)_*; it should state continuity of sigma^(c)_*.
  2. [Remark 6.3] The text refers to 'Lemmas 6.7, 6.7 and Theorem 6.10'; the second mention should be Lemma 6.9.
  3. [Proof of Theorem 6.10(i), last paragraph] The sentence 'one can apply Fatou Lemma to get that (II.1) <= 0' refers to the term labelled (I.1); the label should be corrected.
  4. [Corollary 5.3, Eq. (5.8)] The displayed equation has a missing closing parenthesis in the term involving the trace; the bracket should be closed after the sigma_* argument.
  5. [Abstract and Section 1] The phrase 'the control is applied not only on the herd dynamics, but also on the herd one' should read 'on the herder dynamics'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper proves its mean-field limit and Gamma-convergence results from stated assumptions and standard external theorems, with no fitted inputs or self-citation chain.

full rationale

No circular step could be exhibited. The central derivation chain is: Assumptions 2.9 (Lipschitz kernels, Lipschitz noise coefficients, admissible controls) are used to prove well-posedness of the discrete and mean-field systems (Lemmas 3.5 and 3.6); Theorem 4.1 then proves the mean-field limit directly via Gronwall estimates, BDG inequalities, and the quantitative law of large numbers of Fournier-Guillin [37], without assuming the desired convergence. Corollary 4.4 derives (conditional) propagation of chaos from Theorem 4.1. The Fokker-Planck equations in Section 5 are obtained by applying Ito's formula to the already-constructed mean-field process and conditioning, not by postulating the measure evolution. The optimal-control existence results (Lemmas 6.7 and 6.9) are proved by minimizing sequences, weak/pointwise compactness, and the continuity estimates of Lemmas 6.5 and 6.8; the Gamma-convergence Theorem 6.10 is then established by explicit liminf and recovery-sequence arguments using the mean-field limit and stability estimates. None of these steps defines a target object in terms of itself or fits a parameter to a data subset and then relabels it as a prediction. The control ansatz u_m(t,y,nu)=h_m(t)g_m(y,nu) is an explicitly stated modeling restriction, not a disguised form of the Gamma-convergence conclusion. The paper openly flags limitations, e.g. that singular interactions are outside the Sznitman-style quantitative approach (Open problems, p. 5), and that Pontryagin/Dynamic Programming principles remain open; these are scope or completeness issues, not circularity. A separate correctness concern is that Lemma 6.7 uses weak L1 compactness to claim the weak limit h_* still takes values in U without explicitly assuming U convex; even if this gap is real, it is an unproved compactness assertion, not a reduction of the theorem to its own assumptions. No load-bearing self-citation occurs: reference [9] provides the baseline framework, but the present proofs re-derive the needed estimates in the new multiplicative/common-noise setting, and the other load-bearing external results (BDG, Gronwall, quantitative LLN, Feynman-Kac) are standard and independent.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All results rest on the standing Lipschitz and compactness assumptions listed above; the paper introduces no free parameters or invented entities.

assumptions (5)
  • domain assumption Interaction kernels H_i, K_i are globally Lipschitz continuous (Assumption 2.9(i), (2.22)).
    Used in all Gronwall estimates and contraction arguments to bound differences of convolutions and pairwise interactions.
  • domain assumption Noise coefficients sigma^(i)_*, sigma^(c)_* are continuous and Lipschitz in (y,x,nu) (Assumptions 2.9(ii)-(iii), (2.23)-(2.28)).
    Needed for well-posedness and for the BDG-based moment estimates in the convergence proof.
  • ad hoc to paper Control functions satisfy the separated form u_m = h_m(t)g_m(y,nu) with h_m in L1([0,T];U) and g_m in a compact Lipschitz set G (Assumptions 6.1, (1.7)).
    This product structure is imposed to obtain compactness of admissible controls and existence of minimizers; it restricts the class of controls compared to more general settings.
  • domain assumption Initial herd positions are i.i.d. and independent of Brownian motions; herder initial positions are deterministic (Theorem 4.1).
    Propagation of chaos and the quantitative law of large numbers rely on independence of the initial data.
  • standard math Standard probabilistic facts: Ito formula, Burkholder-Davis-Gundy inequality (Prop 2.3), Gronwall inequality (Prop 3.4), quantified LLN of Fournier-Guillin (2.19), Feynman-Kac formula (Lemma 5.2).
    Background results invoked throughout, cited to [10], [37], [55].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal control of mean-field limit of multiagent systems with and without common noise." pith.science (2026). https://pith.science/paper/2NBHAXVG

@misc{pith2026250516721,
  author       = {Pith},
  title        = {Pith review of: Optimal control of mean-field limit of multiagent systems with and without common noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2NBHAXVG}},
  note         = {Machine review of arXiv:2505.16721}
}
abstract

We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $\Gamma-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 59 canonical work pages

  1. [9]

    Ascione, D

    G. Ascione, D. Castorina, F. Solombrino , Mean-Field Sparse Optimal Control of Sys- tems with Additive White Noise , SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)

    work page 2023

  2. [1]

    G. Albi, S. Almi, M. Morandotti, F. Solombrino , Mean-field selective optimal control via transient leadership , Appl. Math. Optim., 85 (2022)

    work page 2022

  3. [2]

    G. Albi, M. Bongini, E. Cristiani, D. Kalise , Invisible control of self-organizing agents leaving unknown environments , SIAM J. Appl. Math., 76 (2016)

    work page 2016

  4. [3]

    Albi, Y.-P

    G. Albi, Y.-P. Choi, M. Fornasier, D. Kalise, Mean field control hierarchy, Appl. Math. Optim., 76 (2017)

    work page 2017

  5. [4]

    S. Almi, M. Morandotti, F. Solombrino, Optimal control problems in transport dynamics with additive noise , J. Differential Equations 373 (2023)

    work page 2023

  6. [5]

    Ambrosio, E

    L. Ambrosio, E. Bru`e, D. Semola, Lectures on Optimal Transport, Springer, Unitext 130 (2021)

    work page 2021

  7. [6]

    Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

    F. Anceschi, G. Ascione, D. Castorina, F. Solombrino , Optimal control problems driven by nonlinear degenerate Fokker-Planck equations , preprint, arXiv:2410.24000 (2024)

    work page Pith review arXiv 2024

  8. [7]

    Anceschi, G

    F. Anceschi, G. Ascione, D. Castorina, F. Solombrino , Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift, Journal of Mathematical Analysis and Applications, Volume 543, Issue 2, Part 1 (2025)

    work page 2025

Show all 62 references
  1. [8]

    Auletta, D

    F. Auletta, D. Fiore, M. J. Richardson, M. di Bernardo , Herding stochastic auton- omous agents via local control rules and online target selection strategies , Auton. Robot. (2022)

    2022

  2. [10]

    Baldi, Stochastic Calculus

    P. Baldi, Stochastic Calculus. An Introduction through Theory and Exercises , Universitext, Springer (2017)

    2017

  3. [11]

    V. Betz, F. Hiroshima, J. L ¨orinczi, Feynman-Kac-Type theorems and Gibbs measures on path space with applications to rigorous Quantum Field Theory , De Gruyter Studies in Mathematics 34 (2011)

    2011

  4. [12]

    Bolley, J

    F. Bolley, J. A. Canizo, J. A. Carrillo , Stochastic mean-field limit: Non-Lipschitz forces and swarming , Math. Models Methods Appl. Sci., 21 (2011)

    2011

  5. [13]

    Boltzmann, Weitere studien ¨ uber das w¨ armegleichgewicht unter gasmolek¨ ulen, pp

    L. Boltzmann, Weitere studien ¨ uber das w¨ armegleichgewicht unter gasmolek¨ ulen, pp. 115– 225, Vieweg+Teubner Verlag, Wiesbaden (1870)

  6. [14]

    Bongini, M

    M. Bongini, M. Fornasier, F. Rossi, F. Solombrino , Mean-field Pontryagin maximum principle, J. Optim. Theory Appl., 175 (2017). 34

    2017

  7. [15]

    Bonnet, A Pontryagin maximum principle in Wasserstein spaces for constrained optimal control problems, ESAIM Control Optim

    B. Bonnet, A Pontryagin maximum principle in Wasserstein spaces for constrained optimal control problems, ESAIM Control Optim. Calc. Var., 25 (2019)

    2019

  8. [16]

    Bonnet, H

    B. Bonnet, H. Frankowska, Necessary optimality conditions for optimal control problems in Wasserstein spaces , Appl. Math. Optim., 84 (2021)

    2021

  9. [17]

    Bonnet, F

    B. Bonnet, F. Rossi , The Pontryagin maximum principle in the Wasserstein space , Calc. Var. Partial Differential Equations, 58 (2019)

    2019

  10. [18]

    B¨otcher, L

    B. B¨otcher, L. Partzsch, R.L. Schilling, Brownian Motion. An Introduction to Stochas- tic Processes, Walter de Gruyter GmbH and Co. KG, Berlin/Boston (2012)

    2012

  11. [19]

    Burger, R

    M. Burger, R. Pinnau, C. Totzeck, O. Tse, A. Roth , Instantaneous control of inter- acting particle systems in the mean-field limit , J. Comput. Phys., 405 (2020)

    2020

  12. [20]

    Camazine, J.-L

    S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraula, E. Bonabeau, Self-Organization in Biological Systems , Princeton University Press, Princeton, NJ (2020)

    2020

  13. [21]

    Carmona, F

    R. Carmona, F. Delarue , Probabilistic Theory of Mean Field Games with Applications , Vols. 1 and 2, Springer-Verlag, Berlin (2018)

    2018

  14. [22]

    Carmona, F

    R. Carmona, F. Delarue, D. Lacker , Mean field games with common noise , The Annals of Probability (2016)

    2016

  15. [23]

    J. A. Carrillo, E. A. Pimentel, V. K. Voskanyan , On a mean field optimal control problem, Nonlinear Anal., 199 (2020)

    2020

  16. [24]

    J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani , Asymptotic flocking dynamics for the kinetic Cucker-Smale model , SIAM J. Math. Anal. 42 (2010)

    2010

  17. [25]

    Cherny, On the uniqueness in law and the pathwise uniqueness for stochastic differential equations, Translated from Russian Journal, Theory Probab

    A.S. Cherny, On the uniqueness in law and the pathwise uniqueness for stochastic differential equations, Translated from Russian Journal, Theory Probab. Appl., Vol. 46, No. 3 (2001)

    2001

  18. [26]

    Chuang, Y

    Y.-L. Chuang, Y. R. Huang, M. R. D’Orsogna, A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, Proceedings of the IEEE International Conference on Robotics and Automation, IEEE, New York (2007)

    2007

  19. [27]

    Coghi, J.-D

    M. Coghi, J.-D. Deuschel, P. K. Friz, M. Maurelli , Pathwise McKean-Vlasov theory with additive noise , Ann. Appl. Probab., 30 (2020)

    2020

  20. [28]

    Coghi, F

    M. Coghi, F. Flandoli , Propagation of chaos for interacting particles subject to environ- mental noise , Ann. Appl. Probab. 26 (2016)

    2016

  21. [29]

    Cucker, S

    F. Cucker, S. Smale , Emergent behavior in flocks , IEEE Trans. Automat. Control, 52 (2007)

    2007

  22. [30]

    Delarue, R

    F. Delarue, R. Maillet, E. Tanr´e, Ergodicity of some stochastic Fokker-Planck equations with additive common noise , preprint, arXiv:2405.09950 (2024)

    2024 arXiv

  23. [31]

    Delarue, J

    F. Delarue, J. Inglis, S. Rubenthaler, E. Tanr ´e, Particle systems with a singular mean-field self-excitation: Application to neuronal networks , Stochastic Process. Appl., 125 (2015)

    2015

  24. [32]

    D’Onofrio, A.M

    G. D’Onofrio, A.M. Hernandez , A large multi-agent system with noise both in position and control, preprint, arXiv:2503.10543 (2025)

    2025 arXiv

  25. [33]

    D ¨uring, P

    B. D ¨uring, P. Markowich, J.-F. Pietschmann, M.-T. Wolfram , Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders , Proc. R. Soc. A Math. Phys. Eng. Sci., 465 (2009)

    2009

  26. [34]

    Flandoli, E

    F. Flandoli, E. Priola, G. Zanco , A mean-field model with discontinuous coefficients for neurons with spatial interaction , Dyn. Syst., 39 (2019)

    2019

  27. [35]

    Fornasier, B

    M. Fornasier, B. Piccoli, F. Rossi , Mean-field sparse optimal control , Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 372 (2014)

    2014

  28. [36]

    Fornasier, F

    M. Fornasier, F. Solombrino , Mean-field optimal control , ESAIM Control Optim. Calc. Var., 20 (2014)

    2014

  29. [37]

    Fournier, A

    N. Fournier, A. Guillin , On the rate of convergence in Wasserstein distance of the em- pirical measure, Probab. Theory Related Fields, 162, pp.707-738 (2015)

    2015

  30. [38]

    Galeati, K

    L. Galeati, K. Lˆe, A. Mayorcas, Quantitative propagation of chaos for singular interact- ing particle systems driven by fractional Brownian motion , preprint, arXiv:2403.05454 (2024)

    2024

  31. [39]

    Ikeda, S

    N. Ikeda, S. W atanabe, Stochastic Differential Equations and Diffusion Processes , North- Holland Mathematical Library, Second Edition, Volume 24 (1989). 35

    1989

  32. [40]

    Jabin, Z

    P.-E. Jabin, Z. W ang, Mean field limit for stochastic particle systems , in Active Particles, Vol. 1, Springer-Verlag, Berlin (2017)

    2017

  33. [41]

    Jabin, Z

    P.-E. Jabin, Z. W ang, Quantitative estimates of propagation of chaos for stochastic systems with W −1,∞ kernels, Inventiones Mathematicae, Volume 214, pages 523–591 (2018)

    2018

  34. [42]

    Kac , Foundations of kinetic theory , Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol

    M. Kac , Foundations of kinetic theory , Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3: Contributions to Astronomy and Physics, University of California, Berkeley (1956)

    1956

  35. [43]

    E. F. Keller, L. A. Segel , Initiation of slime mold aggregation viewed as an instability , J. Theoret. Biol., 26 (1970)

    1970

  36. [44]

    Kolokoltsov, M

    V.N. Kolokoltsov, M. Troeva , On the mean field games with common noise and the McKean-Vlasov SPDEs, Stochastic Analysis and Applications Volume 37 (2019)

    2019

  37. [45]

    N. E. Leonard, E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, Proceedings of the 40th IEEE Conference on Decision and Control, IEEE, New York (2001)

    2001

  38. [46]

    J.-M. Lien, O. B. Bayazit, R. T. Sowell, S. Rodriguez, N. M. Amato , Shepherding behaviors, Proceedings of the IEEE International Conference on Robotics and Automation, Vol. 4, IEEE, New York (2004)

    2004

  39. [47]

    L.Li, J.-G. Liu, P. Yu , On the mean field limit for Brownian particles with Coulomb inter- action in 3D , J. Math. Phys. 60, 111501 (2019)

    2019

  40. [48]

    J.-M. Lien, S. Rodriguez, J.-P. Malric, N. M. Amato , Shepherding behaviors with multiple shepherds , Proceedings of the IEEE International Conference on Robotics and Au- tomation, IEEE, New York (2005)

    2005

  41. [49]

    T. S. Lim, Y. Lu, J. H. Nolen, Quantitative propagation of chaos in a bimolecular chemical reaction-diffusion model, SIAM J. Math. Anal., 52 (2020)

    2020

  42. [50]

    H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations, Lecture Series in Differential Equations, Session 7, Catholic University of America, Washington, DC (1967)

    1967

  43. [51]

    S. M´el´eard, Asymptotic behaviour of some interacting particle systems: McKean-Vlasov and Boltzmann models, Probabilistic Models for Nonlinear Partial Differential Equations, Springer- Verlag, Berlin (1996)

    1996

  44. [52]

    Mozgunov, M

    P. Mozgunov, M. Beccuti, A. Horvath, T. Jaki, R. Sirovich, E. Bibbona , A review of the deterministic and diffusion approximations for stochastic chemical reaction networks , Reaction Kinetics Mech. Catalysis, 123 (2018)

    2018

  45. [53]

    Oelschl ¨ager, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes , Probab

    K. Oelschl ¨ager, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes , Probab. Theory Related Fields, 82 (1989)

    1989

  46. [54]

    Øksendal , An introduction with applications, in Stochastic Differential Equations: An Introduction with Applications, 6th ed., Universitext, Springer-Verlag, Berlin (2003)

    B. Øksendal , An introduction with applications, in Stochastic Differential Equations: An Introduction with Applications, 6th ed., Universitext, Springer-Verlag, Berlin (2003)

    2003

  47. [55]

    Pavliotis, Stochastic Processes and Applications, Springer (2014)

    G.A. Pavliotis, Stochastic Processes and Applications, Springer (2014)

    2014

  48. [56]

    Perea, G

    L. Perea, G. G `omez, P. Elosegui , Extension of the Cucker-Smale control law to space flight formations , J. Guidance Control Dyn., 32 (2009)

    2009

  49. [57]

    Pierson, M

    A. Pierson, M. Schwager , Controlling noncooperative herds with robotic herders , IEEE Trans. Robot., 34 (2017)

    2017

  50. [58]

    Revuz, M

    D. Revuz, M. Yor , Continuous martingales and Brownian motion , Grundlehren der math- ematischen Wissenschaften, Vol. 293, 3rd ed., Springer-Verlag, Berlin (1999)

    1999

  51. [59]

    Sebasti `an, E

    E. Sebasti `an, E. Montijano , Multi-robot implicit control of herds , IEEE International Conference on Robotics and Automation, IEEE, New York (2021)

    2021

  52. [60]

    Sznitman, Topics in propagation of chaos , Ecole d’´ et´ e de probabilit´ es de Saint-Flour XIX-1989, Springer-Verlag, Berlin (1991)

    A.-S. Sznitman, Topics in propagation of chaos , Ecole d’´ et´ e de probabilit´ es de Saint-Flour XIX-1989, Springer-Verlag, Berlin (1991)

    1991

  53. [61]

    Toscani, Kinetic models of opinion formation , Commun

    G. Toscani, Kinetic models of opinion formation , Commun. Math. Sci., 4 (2006)

    2006

  54. [62]

    Villani , Optimal Transport

    C. Villani , Optimal Transport. Old and New , Grundlehren der mathematischen Wis- senschaften, Springer (2009). 36 Giuseppe La Scala Mathematical and Physical Sciences for Advanced Materials and Technologies Scuola Superiore Meridionale Via Mezzocannone, 4, 80138 Naples, Italy...

    2009

Pith tools

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