Deterministic mean field games with control on the acceleration

Claudio Marchi (DEI); Nicoletta Tchou (IRMAR); Paola Mannucci; Yves Achdou (LJLL (UMR\_7598))

Deterministic mean field games with control on the acceleration

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Pith's one-line read The paper proves existence of weak solutions for acceleration-controlled deterministic mean field games and shows the density is transported by the optimal flow.

desk verdict First existence theorem for deterministic MFGs with acceleration control; long, coherent proof chain and an honest scope restriction; deserves a serious referee. read the letter →

arxiv 1908.03330 v3 pith:QQRYYKSR submitted 2019-08-09 math.AP math.OC

Yves Achdou (LJLL (UMR_7598)) , Paola Mannucci , Claudio Marchi (DEI) , Nicoletta Tchou (IRMAR) This is my paper · ORCID

classification math.APmath.OC MSC 35F5035Q9149K2049L25

keywords meanfieldgamescontrolonaccelerationdoubleintegratornon-coerciveHamiltonianfirst-orderHamilton-Jacobiequationscontinuityequationweaksolutionsvanishingviscosity

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 tackles deterministic mean field games in which each agent controls acceleration rather than velocity, so the state is a position-velocity pair and the dynamics is the double integrator. The associated Hamiltonian is neither strictly convex nor coercive, and the running cost grows quadratically in velocity, so off-the-shelf existence results do not apply. The authors prove that the coupled Hamilton-Jacobi and continuity equations still have a weak solution, and that the population density is the image of the initial density under the flow of the optimal feedback, $x'(s)=v(s)$, $v'(s)=-D_v u(x(s),v(s),s)$. The proof works by adding viscosity to both equations, deriving estimates uniformly in the viscosity parameter, and passing to the limit, then closing the loop with a fixed-point argument. A byproduct is existence and uniqueness for the corresponding second-order (viscous) system.

What carries the argument

The central object is the value function $u$ of a representative agent's optimal control problem with the population density fixed, together with its optimal synthesis: a trajectory $s\mapsto(x(s),v(s))$ is optimal exactly when it solves the characteristic ODE $x'(s)=v(s)$, $v'(s)=-D_v u(x(s),v(s),s)$, with the optimal acceleration given by the feedback $\alpha^*(s)=-D_v u(x(s),v(s),s)$. The proof's technical engine is the vanishing viscosity method: adding $\sigma\Delta_{x,v}$ to both the Hamilton-Jacobi and the continuity equation makes the system classical, and uniform estimates on $u_\sigma$ (Lipschitz in $x$, locally Lipschitz in $v$, semiconcave) and on $m_\sigma$ ($L^\infty$ bound, $C^{1/2}$ in time, uniform second-moment bound) allow a limit $\sigma\to0^+$ that preserves the viscosity and distributional equations. The representation formula then follows from the superposition principle, which writes $m$ as the image of a measure on continuous curves and identifies the curves with the unique optimal trajectories of the control problem. The fixed-point map $m\mapsto T(m)$—solve the Hamilton-Jacobi equation with $m$ frozen, then solve the continuity equation with the resulting $D_v u$—is continuous and compact, so existence follows by a fixed-point argument.

What would settle it

For a one-dimensional double integrator with a smooth compactly supported $m_0$ and a convolution coupling $F[m](x,v)=\int\rho(x-x',v-v')\,dm(x',v')$ with smooth compactly supported $\rho$, compute the value function in (3.5) numerically and check the semiconcavity inequality $\lambda u(x,v,t)+(1-\lambda)u(y,w,t)-u(x_\lambda,v_\lambda,t)\le C\lambda(1-\lambda)(|x-y|^2+|v-w|^2)$ with $C$ independent of the viscosity parameter. A violation would contradict Lemma 3.3, on which the convergence of $D_v u_\sigma\to D_v u$ rests, and would therefore break the proof of Theorem 2.1.

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Extended reading notes

Core claim

On the paper's own terms, the core claim is Theorem 2.1: under assumptions (H), the system (2.1) has a weak solution $(u,m)$ in the sense of Definition 2.1, with $u$ a viscosity solution of the backward Hamilton-Jacobi equation and $m$ a distributional solution of the forward continuity equation. The velocity variable $v$ appears in the state, and the Hamiltonian $H(x,v,p_v)=\frac12|p_v|^2-\frac12|v|^2-l(x,v)$ depends on the momentum only through $p_v$, which is why it is not coercive or strictly convex in the full momentum. The second part of the theorem is the representation formula: $m$ is the image of the initial distribution $m_0$ under the flow $x'=v$, $v'=-D_v u(x,v,t)$, so the aggregate density is exactly what is obtained by letting every agent follow the optimal feedback. The proof of this representation combines the optimal synthesis of the control problem with the superposition principle for the continuity equation; a fixed-point argument on the map that sends a density to the continuity-equation solution generated by its optimal value function gives existence.

Load-bearing premise

The argument needs the coupling terms to smooth the population density into globally $C^2$ functions with bounded second derivatives, uniformly in the density; if the coupling only depends on the density pointwise, say $F[m]=f(m)$, the uniform estimates that drive the proof are no longer available.

Editorial extensions

If this is right

The representation formula gives a Lagrangian description of the crowd: the macroscopic density is obtained by pushing the initial density forward along the optimal feedback flow, so particle simulations along $x'=v$, $v'=-D_v u$ are faithful at the mean-field limit.
The solution $m$ has a density with uniform $L^\infty$ and second-moment bounds and is $C^{1/2}$ in time with values in the space of probability measures with finite first moment, so the aggregate state distribution is a well-behaved probability measure at all times.
Under the monotonicity conditions (2.3), the weak solution of the first-order system is unique; this is the classical monotonicity uniqueness mechanism adapted to the acceleration-controlled setting.
The vanishing viscosity construction also yields a classical solution of the corresponding second-order (viscous) mean field game system, and that solution is unique when the same monotonicity conditions hold.
The regularity results imply that along optimal trajectories the optimal control is a $C^1$ feedback, $\alpha^*(s)=-D_v u(x(s),v(s),s)$, so the characteristic flow is well defined for almost every initial condition.

Reading between the lines

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

If Theorem 2.1 is right, a numerical scheme built on the characteristic flow should converge to the weak solution even when individual optimal trajectories are not unique for every initial condition, because the representation needs the flow only $m_0$-almost everywhere.
The same proof route is likely to work for other non-coercive dynamics, such as controls entering only some velocity components or forbidden directions, as long as the value function remains semiconcave; the proof relies on that structure rather than on the specific double-integrator form.
If one replaces the global $C^2$ couplings by local ones such as $F[m]=f(m)$, the uniform semiconcavity estimates fail and the vanishing viscosity limit may select a different or no solution; testing this numerically in one dimension is a quick way to map the boundary of the method.
The representation formula suggests that macroscopic quantities such as moments of $m$ are determined by the value function's gradient, so a natural next step is to prove quantitative stability of these moments with respect to perturbations of the coupling operators $F$ and $G$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this is the first existence theorem for deterministic MFGs where agents control acceleration (the double integrator), and the Hamiltonian is neither strictly convex nor coercive. The proof chain is long but coherent; I could not find a genuine gap. It deserves a serious referee.

What is actually new: previous MFG theory couldn't touch this setting because the Hamiltonian has flat directions and the running cost is unbounded in velocity. The value function is only locally Lipschitz in v, so the drift in the continuity equation is not Lipschitz and standard flow arguments fail. The authors get around that with a vanishing viscosity limit, sigma-uniform estimates, the Ambrosio-Gigli-Savare superposition principle, and a Schauder fixed point. On top of existence, they prove that m is the image of m0 under the flow of the optimal feedback. That representation result is a useful byproduct, not a throwaway.

Credit where it is due: each step is anchored in cited theorems—Da Lio-Ley comparison for quadratic-growth HJ, Pontryagin, the non-coercive control synthesis from their own [29], and the superposition principle. The use of [29] is legitimate; it supplies lemmas, not the main result. The independent simultaneous work by Cannarsa and Mendico is acknowledged. The proof is careful, and the appendix gets the second-order system for free from the same estimates.

Soft spots: the main one is assumption (H3). F and G have to map P1 into C^2(R^{2N}) with uniform C^2 bounds, and l has to be C^2-bounded. That is a strong regularizing assumption and rules out local couplings like F[m]=f(m(x,v)), which is the most common case in MFG applications. It is an explicit scope restriction, not a hidden flaw, but it does mean the theorem is a foundation for nonlocal/regularized couplings, not the final word. A second, smaller issue: the uniqueness part of Proposition 4.2 is compressed. The argument needs that for a.e. initial (x,v) the optimal trajectory is unique, which follows from Lemma 3.5 plus Lemma 3.4(3), but the text doesn't spell that out. A referee should ask for a sentence or two there. Also, Lemma 3.2 refers forward to a bound in Corollary 3.1; that looks odd but is not circular, since Corollary 3.1 is proved independently.

Who it is for: people working on MFG with higher-order dynamics, constrained MFG, or non-coercive Hamiltonians. It is a solid technical foundation and likely to be built on. It won't reshape the field, but it fills a real gap.

Recommendation: send it to peer review. I would sign an accept after minor revisions, mainly asking for clarification in Proposition 4.2.

Referee Report

0 major / 5 minor

Summary. This paper studies deterministic mean field games with finite horizon in which each agent controls acceleration, so that the state is the pair (x,v) of position and velocity and the dynamics is the double integrator. The MFG system couples a backward Hamilton-Jacobi equation for the value function u with a forward continuity equation for the density m. The Hamiltonian is H(x,v,D_vu)=1/2|D_vu|^2-1/2|v|^2-l(x,v), which is neither strictly convex nor coercive in the full momentum variable. Under assumptions (H), in particular that the couplings F and G map P_1 into C^2(R^{2N}) with uniformly bounded C^2 norms and that m_0 is a compactly supported Hölder density, the authors prove the existence of a weak solution (Theorem 2.1) and characterize m(t) as the image of m_0 under the flow generated by x'=v, v'=-D_vu(x,v,t). They also prove uniqueness under a monotonicity condition on F and G (Proposition 2.1) and, in the appendix, existence and uniqueness for the corresponding second-order MFG system.

Significance. If correct, this is the first existence theory for deterministic MFGs with acceleration control and a non-coercive Hamiltonian, and the representation of the population density by the optimal flow is a genuinely useful structural result. The proof is a long but coherent chain: Pontryagin maximum principle and optimal synthesis (Section 3), semiconcavity, vanishing viscosity and Fokker-Planck estimates (Section 4), and a Schauder fixed point argument (Section 5). I checked the potentially delicate steps: the use of estimate (3.33) in Lemma 3.2 is not circular because Corollary 3.1 is proved independently from the Pontryagin conditions; the uniqueness step in Proposition 4.2 is supported by Lemma 3.5(1) together with Lemma 3.4(3); and the strong regularity of F and G in (H3) is an explicit scope restriction rather than a hidden flaw. The paper contains no fitted parameters or circular normalizations, and the claims are falsifiable through the stated assumptions. The main limitation is that local couplings such as F[m]=f(m) are excluded, but the authors state this clearly through (H3).

minor comments (5)

[Section 3, Lemma 3.2] The proof of Lemma 3.2 uses estimate (3.33), which is proved later in Corollary 3.1. This is not circular, since Corollary 3.1 follows from the Pontryagin maximum principle independently of Lemma 3.2, but the forward reference should be flagged explicitly so the reader does not perceive a circularity.
[Section 4.2, Proposition 4.2] The sentence 'since for all t, u(·,·,t) is Lipschitz continuous' overstates the regularity proved in Lemma 3.2: u is Lipschitz in x and only locally Lipschitz in v. For the conclusion that the set in (4.13) is a singleton for almost every initial condition, one should explicitly invoke the semiconcavity of u from Lemma 3.3 together with Rademacher's theorem to obtain a.e. differentiability of u(·,·,0).
[Section 5, Proposition 2.1] The claim that u=u_1-u_2 is an admissible test function for the continuity equation should be justified by a brief density argument: u is only locally Lipschitz, while the weak formulation in Definition 2.1 uses smooth compactly supported test functions. The compact support of m_i together with the local Lipschitz regularity of u makes this standard, but the step is currently implicit.
[Section 6, equation (6.3)] Equation (6.3) is hard to parse as displayed: the drift term in the first weak formulation should involve D_vu_i, the coefficient corresponding to the equation for m_i, rather than D_vu without an index; the passage to the second equality and the subsequent subtraction step should be written out with explicit indices.
[Section 3, Lemma 3.5] There is a typo in the display preceding (3.41): the term 'u(x(s), x(s), s)' should read 'u(x(s), v(s), s)'.

Circularity Check

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No significant circularity: the existence and density-representation theorems are derived from stated hypotheses, external optimal-control lemmas, and standard fixed-point/superposition tools, with no fitted parameter, forced normalization, or load-bearing self-citation.

full rationale

The derivation chain is self-contained. Theorem 2.1 is proven by a Schauder fixed point over the map T that sends m to the solution of the continuity equation driven by the value function u(m), with all uniform bounds (Lemmas 3.2, 3.3, 4.1, 4.3) obtained directly from Hypotheses (H) and not from the target theorem. The representation of m as the image of m0 by the optimal flow is obtained in Proposition 4.2 from the superposition principle [2, Theorem 8.2.1], the disintegration theorem, and the optimal-synthesis Lemma 3.5; this is a genuine derivation rather than a restatement of the definition of m. The only prior work by the same authors cited with load-bearing role is [29] (Mannucci–Marchi–Mariconda–Tchou), used in Lemma 3.1 to argue as in [29, Proposition 5.1] for the dynamic programming principle; that result is a parameter-free optimal-control lemma with stated assumptions and is not equivalent to the MFG existence claim, so it does not make the argument circular. The uniqueness result Proposition 2.1 is a standard Lasry–Lions monotonicity argument and does not depend on the construction. I find no fitted input called a prediction, no uniqueness theorem imported from the authors' prior work, no ansatz smuggled in by citation, and no renaming of a known result as a new one. The strong regularity of F and G in (H3) is an explicit scope restriction, not a hidden circular assumption.

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

The central theorem rests on strong regularity assumptions on the couplings F and G, on the C^2-boundedness of the running cost, and on the compact support of the initial density. It also invokes standard results from optimal control, parabolic PDEs, and measure-valued transport. There are no free parameters fitted to data and no invented entities; the velocity state is part of the model, not a new postulate.

assumptions (5)

domain assumption Assumptions (H1)-(H4): F and G are continuous on P1 x R^{2N}, l is C^2 with bounded derivatives, F and G map P1 to C^2 with uniformly bounded C^2 norms, and m0 is a compactly supported density in C^{0,delta}.
These standing hypotheses define the class of problems studied. The theorem is conditional on them, and the C^2 regularity of F and G is used throughout to obtain semiconcavity and uniform estimates. Section 2, Assumptions (H).
standard math Comparison principle for quadratic-growth Hamilton-Jacobi equations (Da Lio and Ley, Theorem 2.1).
Used in Proposition 3.1 to identify the value function as the unique viscosity solution among functions with quadratic growth, and in Lemma 4.1 to obtain sigma-uniform bounds for the viscous problem.
standard math Pontryagin maximum principle for control problems with unbounded control (Clarke, Theorem 22.17).
Gives the necessary optimality conditions and the feedback formula alpha = p_v in Proposition 3.2 and Corollary 3.1, which are then used to construct the optimal synthesis.
standard math Superposition principle for continuity equations (Ambrosio, Gigli and Savare, Theorem 8.2.1).
Core tool in Proposition 4.2 for representing the solution of the continuity equation as the pushforward of m0 by the flow associated with the optimal control.
standard math Parabolic regularity and well-posedness results for Fokker-Planck equations with unbounded coefficients (Ikeda, Ladyzhenskaya et al., Karatzas and Shreve).
Used in Lemma 4.2 and Lemma 4.3 to obtain existence, uniqueness, positivity, and sigma-uniform estimates for the viscous Fokker-Planck equation.

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Cite this review

Pith. "Pith review of Deterministic mean field games with control on the acceleration." pith.science (2026). https://pith.science/paper/QQRYYKSR

@misc{pith2026190803330,
  author       = {Pith},
  title        = {Pith review of: Deterministic mean field games with control on the acceleration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQRYYKSR}},
  note         = {Machine review of arXiv:1908.03330}
}

abstract

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x, v)\in R^N\times R^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

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Pith tools

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

[1] [29]

Mannucci, C

P. Mannucci, C. Marchi, C. Mariconda, N. Tchou, Non-coercive ﬁrst order Mean Field Games , J. Diﬀerential Equations 269 (2020), no. 5, 4503-4543

work page 2020

[2] [1]

Achdou, F

Y. Achdou, F. J. Buera, J-M. Lasry, P-L. Lions, B. Moll, Partial diﬀerential equation models in macroeconomics , Fill’s. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014) 2028, 20130397, 19 pp

work page 2014

[3] [2]

Ambrosio, N

L. Ambrosio, N. Gigli, G. Savaré, Gradient ﬂows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhaüser Verlag, Ba sel 2005

work page 2005

[4] [3]

Benamou, G

J-D. Benamou, G. Carlier, F. Santambrogio, Variational mean ﬁeld games , Active particles. Vol. 1, Advances in theory, models, and ap plications, 141–171, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 201 7. 27

work page

[5] [4]

Bardi, I

M. Bardi, I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi Bel lman equations , Systems and Control: Foundations and Ap- plications. Birkhauser, Boston 1997

work page 1997

[6] [5]

V. I. Bogachev, N.V. Krylov, M. Röckner, S. V. Shaposhnikov, Fokker- Planck-Kolmogorov equations, Mathematical Surveys and Monographs, 207. American Mathematical Society, Providence, RI, 2015

work page 2015

[7] [6]

Buckdahn, P

R. Buckdahn, P. Cannarsa, M. Quincampoix, Lipschitz continuity and semi- concavity properties of the value function of a stochastic c ontrol problem , NoDEA Nonlinear Diﬀerential Equations Appl. 17 (2010), no. 6, 715 –728

work page 2010

[8] [7]

Cannarsa, R

P. Cannarsa, R. Capuani, Existence and uniqueness for Mean Field Games with state constraints, PDE models for multi-agent phenomena, 49–71, Springer INd AM Ser., 28, Springer, Cham, 2018

work page 2018