Pith. sign in

REVIEW 1 major objections 4 minor 56 references

Intrinsic Lipschitz Regularity of Mean-Field Optimal Controls

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper establishes sufficient conditions under which mean-field optimal controls of continuity equations are intrinsically Lipschitz in space, via a Wasserstein-Hessian coercivity estimate.

desk verdict A plausible and novel approach to intrinsic Lipschitz regularity of mean-field optimal controls, but Proposition 7's global feedback construction rests on an unproved non-collision assertion; major revision needed. read the letter →

arxiv 1908.04183 v6 pith:ISCAJ34L submitted 2019-08-12 math.OC

Benoît Bonnet , Francesco Rossi This is my paper · ORCID
classification math.OC MSC 35B6549J2049J3049Q2258E2593A16
keywords mean-fieldoptimalcontrolcontinuityequationLipschitzregularityWassersteinHessiancoercivityestimatefeedbackempiricalmeasureapproximationPontryaginmaximumprinciple
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

Mean-field optimal control of continuity equations usually requires imposing Lipschitz regularity on the control field by hand, because non-smooth velocity fields may take the equation outside classical well-posedness. This paper proves a sufficient condition for such regularity to be intrinsic: if the control cost is strongly convex with constant larger than a data-dependent threshold, then an optimal control exists that is Lipschitz in space for almost every time. The proof approximates the mean-field problem by finite-agent problems, proves a uniform coercivity estimate expressed in Wasserstein calculus, and passes Lipschitz feedbacks through the variational limit. A one-dimensional example shows the threshold is sharp: Lipschitz regularity holds exactly when the control-cost curvature exceeds the time horizon. The practical interest is that Lipschitz controls make the continuity equation classically well-posed and make infinite-dimensional strategies transferable to finite-agent systems.

What carries the argument

The load-bearing object is the uniform mean-field coercivity estimate (CON): along any optimal mean-field Pontryagin triple for the N-agent problems, the Wasserstein-Hessian second variation of the final cost minus the integrated Wasserstein-Hessian second variations of the Hamiltonian is bounded below by ρ_T times the squared L² norm of the control perturbation, over every linearised trajectory-control pair. This inequality makes the linearised optimality system strongly positive-definite uniformly in N, allowing a known finite-dimensional theorem on locally optimal Lipschitz feedbacks to be applied with a uniform Lipschitz constant. The Wasserstein Hessian is the second-order derivative on the space of probability measures, restricted to empirical measures through the rescaled inner product; the uniformity in N comes from bounding the discrete $C^{{2,1}}$ norms by the ambient second-order Wasserstein norms. In the sharp example, the machinery collapses to the explicit condition λ > T with optimal coercivity constant ρ_T = λ − T.

What would settle it

Find a datum satisfying the paper's hypotheses with λ_ψ > λ(P) where two initially distinct optimal agent trajectories coincide at some time t in [0,T]; then the local feedbacks constructed in Proposition 7 would disagree at the crossing point, so the asserted global feedback map u*_N is not well-defined.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: under the paper's standing hypotheses and the strong convexity condition λ_ψ > λ(P), problem (P) admits an optimal pair (μ*, u*) such that the map x ↦ u*(t,x) is L_U-Lipschitz for $L^{1}$-almost every t. The constant λ(P) is intrinsic, depending only on the support of the initial measure, the horizon T, and the C² norms of the dynamics and costs. The stronger Theorem 2 states that if a uniform mean-field coercivity estimate (CON) holds along optimal Pontryagin triples of the discretised N-agent problems, then the discrete optimal feedbacks are uniformly Lipschitz in space and their weak cluster points are optimal controls for (P). Proposition 8 shows that strong convexity of ψ is a sufficient condition for (CON), while Section 6 gives a variance-maximisation problem where (CON) holds if and only if λ > T and this condition is equivalent to a uniform Lipschitz bound on the optimal controls.

Load-bearing premise

In the key patching step, the authors assume that no two optimal agents' trajectories collide in finite time; that assumption is stated but not proved, and if it fails the local feedbacks may not agree on their overlap.

Editorial extensions

If this is right

  • If Theorem 1 is correct, every mean-field optimal control problem satisfying the hypotheses and λ_ψ > λ(P) has an optimal closed-loop policy whose spatial Lipschitz bound is known a priori, so the continuity equation is classically well-posed along that pair.
  • Theorem 2 shows that optimal feedbacks for finite-agent approximants are uniformly Lipschitz and converge, up to subsequences, to a mean-field optimal control, giving a quantitative bridge between the discrete and infinite-dimensional problems.
  • When the intrinsic constant λ(P) vanishes, for instance under displacement-convex costs with zero final cost and linear dynamics, any strictly convex control cost already yields Lipschitz regularity, with no small-horizon condition.
  • The variance-maximisation example demonstrates that the coercivity threshold is not an artifact: below the threshold the discrete optimal controls do not admit a uniform Lipschitz bound, so λ(P) is a genuine structural constant.
  • A Lipschitz optimal control prevents Lavrentiev-type instabilities and makes numerical methods such as semi-Lagrangian schemes well-posed on the optimal trajectory, which is a stated motivation of the paper.

Reading between the lines

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

  • A load-bearing step in Proposition 7 is the assertion that no finite-time collisions occur between optimal agents, which allows local feedbacks to be patched into one global map; since this assertion is not proved, a collision case could break the construction even if the final Lipschitz statement remains true.
  • The same Wasserstein-Hessian coercivity mechanism suggests a quantitative route to regularity in mean-field games: value functions and optimal velocity fields should be Lipschitz when cost curvature dominates the product of the horizon and the data seminorms, connecting to existing master-equation regularity results.
  • A direct numerical test is available in the paper's example: for symmetric empirical initial measures and λ slightly above T, the predicted uniform bound |u_i(t) − u_j(t)| ≤ |x_i(t) − x_j(t)|/(λ − T) should hold uniformly in N, while for λ ≤ T it should fail.
  • The proof only needs strong positive-definiteness of the second variation, so the strong convexity of ψ could likely be replaced by any uniform coercivity condition on the Hamiltonian's Hessian, which would widen the class of admissible control costs.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper studies the mean-field optimal control problem (P) for a non-local continuity equation, with running cost L(t,μ)+∫ψ(u)dμ and final cost φ(μ), without imposing a priori spatial regularity on the control vector field. It claims two main results. Theorem 2 asserts that, under hypotheses (H) and a uniform mean-field coercivity estimate (CON) along optimal discrete Pontryagin triples, the N-agent approximations admit optimal feedbacks u*_N(t,·) with a common Lipschitz constant L_U, and that cluster points of these feedbacks are optimal controls for (P). Theorem 1 derives (CON) from strong convexity of the control cost ψ with λψ > λ(P), where λ(P) is an intrinsic constant, and thereby yields existence of an intrinsically Lipschitz-in-space optimal control for (P). The proof combines the Γ-convergence result of [38], the locally optimal Lipschitz feedback theorem of [33], and a Wasserstein-Hessian reformulation of coercivity. Section 6 develops a one-dimensional variance example to show that (CON) is necessary and sufficient for uniform Lipschitz regularity in that special case.

Significance. The question addressed is significant: intrinsic spatial regularity of optimal controls for continuity equations is known to fail in general, and this paper proposes a general sufficient condition expressed through an intrinsic constant λ(P). The two-step strategy—through empirical approximations and locally optimal feedbacks with uniform Lipschitz bounds—is original and combines recent tools such as Wasserstein Hessians and metric regularity in a productive way. The paper also provides a sharp one-dimensional example relating coercivity to Lipschitz bounds, which is a useful contribution in itself. However, the main theorems are contingent on the patching step in Proposition 7, and that step is not established in the present text. Because the gap is load-bearing, the current version does not yet prove the advertised results; with a repaired argument the conclusions would be a valuable contribution to mean-field control and Wasserstein calculus.

major comments (1)
  1. [Section 5.2, Proposition 8] This is a completeness issue in the derivation of the main sufficient condition; the preceding comment about Proposition 7 is the primary obstacle.
minor comments (4)
  1. [Section 4, definition of U] The set U is defined as L∞([0,T], L1(R^d,U; μ(t))), but μ(t) depends on the unknown control, so this is not a fixed vector space. The rigorous measure-control framework is introduced later via (Pmeas); please clarify that the L∞-type definition is only formal.
  2. [Section 5.1, Proposition 7] Even assuming that optimal trajectories do not collide, the text should justify that the projected neighbourhoods N_i can be shrunk to be pairwise disjoint while still containing (t,x*_i(t)) for all t. This follows from compactness and the positive separation of finitely many disjoint compact graphs, but it is not automatic from the definition of N_i.
  3. [Section 6, Proposition 9] The final step from the discrete pairwise estimate (b) to the asserted necessity and sufficiency for the Lipschitz regularity of the mean-field optimal control is compressed into one sentence. Since Section 6 is presented as a sharpness result, please expand the limiting argument connecting the discrete inequalities to the existence or non-existence of a Lipschitz mean-field optimal control.
  4. [Throughout] There are several typographical issues: 'Li pschitz' in the abstract, 'mean-feld' in the opening of Section 4, 'Charaterisation' in the title of Lemma 7, and the citation '[56, Theorem 12. 50)' in the introduction should be '[56, Theorem 12.50]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central derivation is independent of the authors' prior conclusions, and the flagged collision step is a proof gap rather than a circular reduction.

full rationale

I find no step in which the paper's own equations force the main result to be equivalent to its inputs. Theorem 1 is derived as a corollary of Theorem 2 via Proposition 8; Proposition 8 derives the coercivity estimate (CON) from strong convexity of psi and a uniform lower bound on the mean-field Hessians (Lemma 6), using only the stated C^{2,1}_{loc}-Wasserstein regularity hypotheses and Gronwall estimates. Theorem 2 in turn relies on Theorem 4 of [33] (external, not self-authored) for locally optimal Lipschitz feedbacks, on [38] for Gamma-convergence, and on [25] for Wasserstein Hessian computations; none of these citations is replaced by an unverified self-referential premise. The authors' own PMP results ([7,8,10,11]) are used only to write the optimality system (40), not to produce the Lipschitz regularity, so their self-citations are not load-bearing. The only substantive concern in the proof is the sentence in Proposition 7 (Section 5.1, Step 2): if x*_j(tau) lies in N_i, then local optimality of u~_i necessarily implies u*_j(t) = u~_i(t,x*_j(t)), and therefore 'no finite-time collisions can occur between agents' so that the sets N_i can be chosen disjoint and u*_N is well-defined. This assertion is not proven and is needed to make u*_N well-defined; however it is a correctness gap, not a circular reduction. It does not identify the target Lipschitz feedback with an input parameter or with a self-cited theorem by construction; it claims a factual property of the optimal trajectories. Under the stated rules, unproven premises of this kind are correctness risks and do not raise the circularity score. I therefore score 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on external Gamma-convergence and Lipschitz feedback theorems plus a second-order Wasserstein calculus. No fitted constants or new entities are introduced; the only unquantified quantity is the intrinsic constant λ(P) arising in the sufficient condition of Theorem 1.

free parameters (1)
  • lambda(P)
    Intrinsic constant in Theorem 1 defined via C^2 Wasserstein norms and T, shown to exist but not explicitly computed. The theorem's hypothesis λ_ψ > λ(P) is therefore not directly checkable from the paper.
assumptions (4)
  • domain assumption Theorem 6 from [38]: Gamma-convergence of finite-agent optimal control problems to the mean-field problem (P), yielding existence of optimal mean-field controls as limits of empirical measures.
    External result from Fornasier-Lisini-Orrieri-Savaré, used as the starting point for the approximation sequence in Theorem 2.
  • domain assumption Theorem 4 from [33]: existence of locally optimal Lipschitz feedbacks for finite-dimensional control problems under a uniform coercivity estimate.
    External result from Dontchev-Krastanov-Veliov, applied to the discrete problems (P_N) in Proposition 7.
  • standard math Wasserstein Hessian calculus from [25]: second-order expansion formula (Proposition 3) and the mean-field Hessian representation (Proposition 4).
    Borrowed from Chow-Gangbo, providing the analytic expression of Wasserstein Hessians used in the coercivity estimate (CON).
  • domain assumption Theorem 5 from [52]: well-posedness of non-local continuity equations with Lipschitz velocity fields in W1 metric.
    External result from Piccoli-Rossi, cited to justify the well-posedness framework for (P).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Intrinsic Lipschitz Regularity of Mean-Field Optimal Controls." pith.science (2026). https://pith.science/paper/ISCAJ34L

@misc{pith2026190804183,
  author       = {Pith},
  title        = {Pith review of: Intrinsic Lipschitz Regularity of Mean-Field Optimal Controls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISCAJ34L}},
  note         = {Machine review of arXiv:1908.04183}
}
read the original abstract

In this article, we provide sufficient conditions under which the controlled vector fields solution of optimal control problems formulated on continuity equations are Lipschitz regular in space. Our approach involves a novel combination of mean-field approximations for infinite-dimensional multi-agent optimal control problems, along with a careful extension of an existence result of locally optimal Lipschitz feedbacks. The latter is based on the reformulation of a coercivity estimate in the language of Wasserstein calculus, which is used to obtain uniform Lipschitz bounds along sequences of approximations by empirical measures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [38]

    Fornasier, S

    M. Fornasier, S. Lisini, C. Orrieri, and G. Savaré. Mean -Field Optimal Control as Gamma-Limit of Finite Agent Contr ols. European Journal of Applied Mathematics , 30(6):1153–1186, 2019

    work page 2019

  2. [33]

    Dontchev, M.I

    A.L. Dontchev, M.I. Krastanov, and V.M. Veliov. On the E xistence of Lipschitz Continuous Optimal Feedback Control s. Vietnam Journal of Mathematics , 47:579–597, 2019

    work page 2019

  3. [1]

    Achdou and M

    Y. Achdou and M. Laurière. On the System of Partial Differe ntial Equations Arising in Mean Field type Control. Discrete and Continuous Dynamical Systems , 35(9):3879–3900, 2015

    work page 2015

  4. [2]

    Ambrosio

    L. Ambrosio. Transport Equation and Cauchy Problem for B V Vector Fields. Inventiones Mathematicae , 158(2):227–260, 2004

    work page 2004

  5. [3]

    Ambrosio and G

    L. Ambrosio and G. Crippa. Continuity Equations and ODE F lows with Non-Smooth Velocities. Proceedings of the Royal Society of Edinburgh , 144(6):1191–1244, 2014

    work page 2014

  6. [4]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variations and Free Discontinuity Pro blems. Oxford Mathe- matical Monographs, 2000

    work page 2000

  7. [5]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probabili ty Measures . Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, 2008

    work page 2008

  8. [6]

    Bellomo, P

    N. Bellomo, P. Degond, E. Tadmor, et al. Active Particles, Volume 1: Advances in Theory, Models, and Applications. Springer, 2017

    work page 2017

Show all 56 references
  1. [7]

    Bongini, M

    M. Bongini, M. Fornasier, F. Rossi, and F. Solombrino. Me an Field Pontryagin Maximum Principle. Journal of Optimization Theory and Applications , 175:1–38, 2017

    2017

  2. [8]

    B. Bonnet. A Pontryagin Maximum Principle in W asserstei n Spaces for Constrained Optimal Control Problems. ESAIM COCV, 25(52), 2019

    2019

  3. [9]

    Bonnet and H

    B. Bonnet and H. Frankowska. Differential Inclusions in W asserstein Spaces: The Cauchy-Lipschitz Framework. Journal of Differential Equations , 271:594–637, 2021

    2021

  4. [10]

    Bonnet and H

    B. Bonnet and H. Frankowska. Necessary Optimality Cond itions for Optimal Control Problems in W asserstein Spaces. In revision, 2021

    2021

  5. [11]

    Bonnet and F

    B. Bonnet and F. Rossi. The Pontryagin Maximum Principl e in the W asserstein Space. Calculus of Variations and Partial Differential Equations , 58:11, 2019

    2019

  6. [12]

    H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differenti al Equations . Universitext. Springer, 2010

    2010

  7. [13]

    Bullo, J

    F. Bullo, J. Cortés, and S. Martines. Distributed Control of Robotic Networks . Applied Mathematics. Princeton University Press, 2009

    2009

  8. [14]

    Burger, R

    M. Burger, R. Pinnau, O. Totzeck, O. Tse, and A. Roth. Ins tantaneous Control of Interacting Particle Systems in the Mean-Field Limit. Journal of Computational Physics , 405:109–181, 2020

    2020

  9. [15]

    Camazine, J.-L

    S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G . Theraulaz, and E. Bonabeau. Self-Organization in Biological Systems. Princeton University Press, 2001

    2001

  10. [16]

    Caponigro, M

    M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. S parse Stabilization and Control of Alignment Models. Mathematical Models and Methods in Applied Sciences , 25 (3):521–564, 2015

    2015

  11. [17]

    Caponigro, B

    M. Caponigro, B. Piccoli, F. Rossi, and E. Trélat. Mean- Field Sparse Jurdjevic-Quinn Control. Mathematical Moddels and Methods in Applied Sciences , 27(7):1223–1253, 2017. 24

    2017

  12. [18]

    Poretta, and D

    P Cardaliaguet, A. Poretta, and D. Tonon. Sobolev Regul arity for the First Order Hamilton–Jacobi Equation. Calculus of Variations and Partial Differential Equations , 54:3037–3065, 2015

    2015

  13. [19]

    Cardaliaguet and L

    P. Cardaliaguet and L. Silvester. Hölder Continuity to Hamilton-Jacobi Equations with Super-Quadratic Growth in the Gradient and Unbounded Right-Hand Side. Communications in Partial Differential Equations , 37(9):1668–1688, 2012

    2012

  14. [20]

    Carlini and F.S

    E. Carlini and F.S. Silva. A Fully Discrete Semi-Lagran gian Scheme for a First Order Mean Field Game Problem. SIAM Journal on Numerical Analysis , 52(1):45–67, 2014

    2014

  15. [21]

    Carrillo, M

    J.A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani. Asymptotic Flocking for the Kinetic Cucker-Smale Model. SIAM Journal on Mathematical Analysis , 42(1):218–236, 2010

    2010

  16. [22]

    Cavagnari, A

    G. Cavagnari, A. Marigonda, K.T. Nguyen, and F.S. Priul i. Generalized Control Systems in the Space of Probability M easures. Set-Valued and Var. Analysis , 26(3):663–691, 2018

    2018

  17. [23]

    Cavagnari, A

    G. Cavagnari, A. Marigonda, and B. Piccoli. Generalize d Dynamic Programming Principle and Sparse Mean-Field Cont rol Problems. Journal of Mathematical Analysis and Applications , 481(1):123437, 2020

    2020

  18. [24]

    Chang and G

    J.S. Chang and G. Cooper. A Practical Difference Scheme f or Fokker-Planck Equations. Journal of Computational Physics , 6(1):1–16, 1970

    1970

  19. [25]

    Chow and W

    Y.T. Chow and W. Gangbo. A Partial Laplacian as an Infinit esimal Generator on the W asserstein Space. Journal of Differential Equations, 267(10):6065–6117, 2019

    2019

  20. [26]

    Cibulka, A.L

    R. Cibulka, A.L. Dontchev, M.I. Krastanov, and V.M. Vel iov. Metrically Regular Differential Generalized Equation s. SIAM Journal on Control and Optimization , 56(1):316–342, 2018

    2018

  21. [27]

    Functional Analysis, Calculus of Variations and Optimal Co ntrol

    F Clarke. Functional Analysis, Calculus of Variations and Optimal Co ntrol. Springer, 2013

    2013

  22. [28]

    Cristiani, B

    E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Pedestrian Dynamics , volume 12. Springer, 2014

    2014

  23. [29]

    Cucker and S

    F. Cucker and S. Smale. Emergent Behavior in Flocks. IEEE Transactions on Automatic Control , 52(5):852–862, 2007

    2007

  24. [30]

    De Phillipis and A

    G. De Phillipis and A. Figalli. Regularity for Solution s of the Monge-Ampère Equation. Inventiones Mathematicae, 192(1):55– 69, 2013

    2013

  25. [31]

    Di Perna and Lions P.-L

    R.L. Di Perna and Lions P.-L. Ordinary Differential Equa tions, Transport Theory and Sobolev Spaces. Inventiones Mathe- maticae, 98(3):511–548, 1989

    1989

  26. [32]

    Dontchev and W.W

    A.L. Dontchev and W.W. Hager. Lipschitzian Stability i n Nonlinear Control and Optimization. SIAM Journal on Control and Optimization , 31(3):569–603, 1993

    1993

  27. [34]

    Duprez, M

    M. Duprez, M. Morancey, and F. Rossi. Approximate and Ex act Controllability of the Continuity Equation with a Local ized Vector Field. SIAM Journal on Control and Optimization , 57(2):1284–1311, 2019

    2019

  28. [35]

    Duprez, M

    M. Duprez, M. Morancey, and F. Rossi. Minimal Time Probl em for Crowd Models with a Localized Vector Field. Journal of Differential Equations , 269(1):82–124, 2020

    2020

  29. [36]

    Evans and R.F

    L.C. Evans and R.F. Gariepy. Measure Theory and Fine Properties of Functions . CRC Press, 1992

    1992

  30. [37]

    Figalli, Y.H

    A. Figalli, Y.H. Kim, and R.J. McCann. Hölder Continuit y and Injectivity of Optimal Maps. Archives of Rational Mechanics and Analysis , 209(3):747–795, 2013

    2013

  31. [39]

    Fornasier, B

    M. Fornasier, B. Piccoli, and F. Rossi. Mean-Field Spar se Optimal Control. Philosophical Transactions of the Royal Society A., 372(2028), 2014

    2014

  32. [40]

    Fornasier and F

    M. Fornasier and F. Solombrino. Mean Field Optimal Cont rol. ESAIM COCV , 20(4):1123–1152, 2014

    2014

  33. [41]

    Gangbo and A.R

    W. Gangbo and A.R. Mészáros. Global W ell-Posedness of M aster Equations for Deterministic Displacement Convex Pot ential Mean Field Games. arXiv preprint arXiv:2004.01660 , 2020

    2004 arXiv

  34. [42]

    Gangbo and A

    W. Gangbo and A. Swiech. Existence of a Solution to an Equ ation Arising in the Theory of Mean Field Games. Journal of Differential Equations , 259(11):6573–6643, 2015

    2015

  35. [43]

    Gangbo and A

    W. Gangbo and A. Tudorascu. On Differentiability in the W asserstein Space and W ell-Posedness for Hamilton-Jacobi E qua- tions. Journal de Mathématiques Pures et Appliquées , 00:1–47, 2018

    2018

  36. [44]

    Ha and J.G

    S.-Y. Ha and J.G. Liu. A Simple Proof of the Cucker-Smale Flocking Dynamics and Mean-Field Limit. Comm. Math. Sci. , 7(2):297–325, 2009

    2009

  37. [45]

    Huang, R

    M.Y. Huang, R. Malhamé, and P.E. Caines. Large Populati on Stochastic Dynamic Games : Closed-Loop McKean-Vlasov Systems and the Nash Certainty Equivalence Principle. Communications in Information and Systems , 6(3):221–252, 2006

    2006

  38. [46]

    Jimenez, A

    C. Jimenez, A. Marigonda, and M. Quincampoix. Optimal C ontrol of Multiagent Systems in the W asserstein Space. Calculus of Variations and Partial Differential Equations , 59:58, 2020

    2020

  39. [47]

    Lasry and P.-L

    J-M. Lasry and P.-L. Lions. Mean Field Games. Japanese Journal of Mathematics , 2(1):229–260, 2007

    2007

  40. [48]

    Lavrentiev

    M. Lavrentiev. Sur Quelques Problèmes du Calcul des Var iations. Annali di Matematica Pura e Applicata , 4(1):7–28, 1927

    1927

  41. [49]

    S. Mayorga. Short Time Solution to the Master Equation o f a First Order Mean Field Game. Journal of Differential Equations , 268(10):6251–6318, 2020

    2020

  42. [50]

    Mesbahi and M

    M. Mesbahi and M. Egerstedt. Graph Theoretic Multi-Agent Systems . 2010

    2010

  43. [51]

    Muntean, J

    A. Muntean, J. Rademacher, and A. Zagaris. Macroscopic and Large Scale Phenomena: Coarse Graining, Me an Field Limits and Ergodicity. Springer, 2016

    2016

  44. [52]

    Piccoli and F

    B. Piccoli and F. Rossi. Transport Equation with Nonloc al Velocity in W asserstein Spaces : Convergence of Numerica l Schemes. Acta Applicandae Mathematicae , 124(1):73–105, 2013

    2013

  45. [53]

    Piccoli, F

    B. Piccoli, F. Rossi, and E. Trélat. Control of the kinet ic Cucker-Smale model. SIAM Journal on Mathematical Analysis , 47(6):4685–4719, 2015

    2015

  46. [54]

    Pogodaev

    N. Pogodaev. Optimal Control of Continuity Equations. Nonlinear Differential Equations and Applications , 23:21, 2016

    2016

  47. [55]

    Santambrogio

    F. Santambrogio. Optimal Transport for Applied Mathematicians , volume 87. Birkhauser Basel, 2015

    2015

  48. [56]

    C. Villani. Optimal Transport : Old and New . Springer-Verlag, Berlin, 2009. 25

    2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.