pith. sign in

arxiv: 2605.05490 · v1 · submitted 2026-05-06 · 🧮 math.AP

H\"older continuity for non-coercive Hamilton-Jacobi equations associated to linear control systems

Megan Griffin-Pickering , Alp\'ar R. M\'esz\'aros This is my paper

Pith reviewed 2026-05-08 15:52 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hölder continuityviscosity solutionsHamilton-Jacobi equationslinear control systemsKalman rank conditionnon-coercive Hamiltoniansanisotropic estimates
0
0 comments X

The pith

Non-coercive Hamilton-Jacobi equations for linear control systems are Hölder continuous.

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

This paper establishes Hölder continuity estimates for viscosity solutions to first-order Hamilton-Jacobi equations that arise from linear control systems satisfying the Kalman rank condition. The model Hamiltonians are non-convex and lack coercivity in certain directions, rendering prior results inapplicable. The authors introduce a geometric argument driven by the control system to derive anisotropic Hölder estimates. These estimates remain valid for unbounded source terms, incorporating elements from De Giorgi-type methods for hypoelliptic operators. Establishing this regularity extends the reach of viscosity solution theory to degenerate control problems.

Core claim

Viscosity solutions to Hamilton-Jacobi equations linked to linear control systems under the Kalman rank condition are Hölder continuous. The estimates are anisotropic and follow from a geometric argument that compensates for the Hamiltonians' non-coercivity and non-convexity in the momentum variable. The result applies even when source terms are unbounded.

What carries the argument

The geometric argument dictated by the linear control system, which quantifies the Hölder estimates anisotropically within the control geometry.

If this is right

  • The estimates apply to unbounded source terms.
  • Hölder continuity is obtained in an anisotropic manner according to the system's geometry.
  • The approach succeeds where coercivity-based techniques fail.
  • Adaptation of De Giorgi methods supports the analysis for these degenerate equations.

Where Pith is reading between the lines

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

  • Numerical methods for solving such control problems may converge without extra smoothing.
  • The technique could apply to other degenerate PDEs arising in control theory.
  • Further regularity, such as differentiability, might follow from similar geometric ideas.

Load-bearing premise

The linear control systems satisfy the Kalman rank condition to enable the geometric argument to bypass the Hamiltonians' missing coercivity.

What would settle it

A concrete viscosity solution to one such equation, for a system satisfying the Kalman condition, that is not Hölder continuous with the stated anisotropic exponent.

Figures

Figures reproduced from arXiv: 2605.05490 by Alp\'ar R. M\'esz\'aros, Megan Griffin-Pickering.

Figure 1
Figure 1. Figure 1: Estimation of the values of 𝑢 ∗ 𝑔 on the boundary subset 𝜕 −𝑄 𝛾 1 ∩ {𝑡 > −1}. (i) If a controlled trajectory connects (𝑡, 𝑥) to a point (−1, 𝑦1) where 𝑦1 ∉ 𝑒 −𝐴ℎ (2Ω𝛿), then the associated cost is at least 𝑔(−1, 𝑦1) = 1. (ii) If a controlled trajectory connects (𝑡, 𝑥) to a point (−1, 𝑦2) where 𝑦2 ∈ 𝑒 −𝐴ℎ (2Ω𝛿), then the associated control cost can be bounded from below, since the trajectory must cross from… view at source ↗
Figure 2
Figure 2. Figure 2: A point (−1, 𝑦) with 𝑦 ∈ 𝑒 −𝐴ℎ (2Ω 𝛾 𝛿 ) can be connected to every point (𝑡, 𝑥) ∈ 𝑄 ℎ,𝛾 𝛿 (shaded region) by a controlled trajectory. If 𝛿 is sufficiently small, then trajectories with minimal control cost remain within the larger cylinder 𝑄ℎ 1 throughout the time interval (−1, 𝑡]. We will use the shorthand 𝜂 ∗ := 𝜂 𝛽 ∗ : [−1, 𝑡] → R 𝑁 to denote the controlled trajectory satisfying 𝜂¤ ∗ = 𝐴ℎ𝜂 ∗ + 𝑃0𝛽 ∗ 𝜂 ∗… view at source ↗
Figure 3
Figure 3. Figure 3: The constant zero path from (−1, 0) to (𝑡, 0) is perturbed to pass through 𝑤 ∈ 𝒲𝜖,𝑡 at the midpoint 1 2 (𝑡 − 1). The union of all such trajectories Ψ(·, 𝑤) creates a set of positive measure in R × R 𝑁 [region enclosed by dotted lines]. satisfies Φˇ ( 𝑡−1 2 , 𝑤) = 𝑤 and Φˇ (𝜏, ·) defines a diffeomorphism R 𝑁 → R 𝑁 for 𝑡−1 2 ≤ 𝜏 < 𝑡. We concatenate these two flows to define the controls 𝛽 𝑤 𝜏 = (Í𝜅 𝑖=0 (𝜏 + … view at source ↗
Figure 4
Figure 4. Figure 4: Optimising paths hit the boundary at points view at source ↗
Figure 5
Figure 5. Figure 5: We consider optimising paths that either reach the spatial boundary of view at source ↗
read the original abstract

In this paper we establish H\"older continuity estimates for viscosity solutions to first order Hamilton-Jacobi equations linked to linear control systems satisfying the Kalman rank condition. Our model Hamiltonians are non-convex in the generalised momentum variable and - more importantly - they lack coercivity in certain directions. Therefore, all previously available results from the literature cannot be applied to these degenerate settings. In order to overcome these obstructions, we design a geometric argument, dictated by the linear control system. As a result of this, the obtained H\"older estimates are quantified in an anisotropic way within this geometric framework. The estimates hold true for unbounded source terms, for which one part of our analysis is inspired by a recent result on De Giorgi type methods for hypoelliptic operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes anisotropic Hölder continuity estimates for viscosity solutions of first-order Hamilton-Jacobi equations associated to linear control systems satisfying the Kalman rank condition. The Hamiltonians are non-convex and lack coercivity in certain directions; the proof combines a control-theoretic geometric construction with a De Giorgi-type iteration adapted to handle unbounded source terms.

Significance. If the estimates hold, the result meaningfully extends regularity theory for HJ equations beyond coercive settings by exploiting the linear control structure and Kalman condition to obtain quantified anisotropic Hölder moduli. The geometric approach and the adaptation of hypoelliptic De Giorgi techniques constitute a concrete technical contribution that could apply to other degenerate control-related PDEs.

major comments (2)
  1. [proof of main theorem (geometric construction)] The central geometric construction (outlined after the statement of the main theorem) relies on the Kalman rank condition to produce an anisotropic distance that compensates for the missing coercivity; however, the manuscript does not provide an explicit verification that the resulting Hölder exponent is strictly positive and independent of the solution size when the source is unbounded. This step is load-bearing for the claim that the estimates are uniform.
  2. [De Giorgi iteration section] In the De Giorgi iteration for unbounded sources, the adaptation from the cited hypoelliptic result requires that the anisotropic norms close the Caccioppoli-type inequality and the subsequent oscillation decay; the manuscript sketches the argument but does not display the precise constants or the dependence on the control matrices that would confirm the iteration converges. This is central to extending the result beyond bounded sources.
minor comments (2)
  1. [introduction / notation] The notation for the anisotropic Hölder seminorm should be introduced with a precise definition (including the scaling weights induced by the control system) before it is used in the statement of the main theorem.
  2. [introduction] A short comparison table or paragraph contrasting the new anisotropic exponent with the isotropic exponents from the coercive literature would clarify the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications into the revised version.

read point-by-point responses

  1. Referee: [proof of main theorem (geometric construction)] The central geometric construction (outlined after the statement of the main theorem) relies on the Kalman rank condition to produce an anisotropic distance that compensates for the missing coercivity; however, the manuscript does not provide an explicit verification that the resulting Hölder exponent is strictly positive and independent of the solution size when the source is unbounded. This step is load-bearing for the claim that the estimates are uniform.

    Authors: We thank the referee for this observation. The anisotropic distance is constructed from the controllability Gramian of the linear system under the Kalman rank condition, and the Hölder exponent is determined by the minimal controllability time and the rank of the controllability matrix; this ensures the exponent is strictly positive and depends only on the system matrices and dimension, remaining independent of the solution size and the (possibly unbounded) source term. While this follows from the construction, we agree that an explicit verification was not displayed. We will add a short lemma immediately after the geometric construction that computes the exponent explicitly and confirms its uniformity with respect to the source. This will be included in the revised manuscript. revision: yes

  2. Referee: [De Giorgi iteration section] In the De Giorgi iteration for unbounded sources, the adaptation from the cited hypoelliptic result requires that the anisotropic norms close the Caccioppoli-type inequality and the subsequent oscillation decay; the manuscript sketches the argument but does not display the precise constants or the dependence on the control matrices that would confirm the iteration converges. This is central to extending the result beyond bounded sources.

    Authors: We agree that additional explicit details would strengthen the presentation. The adaptation of the De Giorgi iteration relies on the anisotropic norms induced by the linear control system to close the Caccioppoli inequality, absorbing the unbounded source via the hypoelliptic structure; the oscillation decay then follows from a geometric series whose ratio is controlled by constants depending on the control matrices A and B. We will expand the relevant section to display these constants explicitly, including their dependence on the matrices and a verification that they remain independent of the source bound, thereby confirming convergence of the iteration for unbounded sources. This will be added in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives anisotropic Hölder continuity estimates for viscosity solutions of non-coercive Hamilton-Jacobi equations by constructing a geometric argument that directly exploits the linear control system structure and the external Kalman rank condition. This is combined with an adaptation of established De Giorgi-type techniques for hypoelliptic operators to accommodate unbounded sources. No load-bearing step reduces the target estimate to a self-definition, a fitted input renamed as a prediction, or a chain of self-citations whose validity depends on the present work. The assumptions and methods are independent of the final estimates, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Kalman rank condition as the key structural assumption and on standard notions of viscosity solutions; no free parameters or new postulated entities appear in the abstract.

axioms (2)
  • domain assumption Linear control systems satisfy the Kalman rank condition.
    Invoked to design the geometric argument that replaces coercivity.
  • standard math Viscosity solutions are the correct weak-solution framework for the first-order Hamilton-Jacobi equations.
    Standard background assumption in the field.

pith-pipeline@v0.9.0 · 5433 in / 1338 out tokens · 46796 ms · 2026-05-08T15:52:16.841349+00:00 · methodology

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

62 extracted references · 62 canonical work pages

  1. [1]

    Y. Achdou. State-constrained optimal control problems with control on the acceleration: applications to kinetic mean field games.SIAM J. Control Optim., 64(1):262–287, 2026

    work page 2026

  2. [2]

    Achdou, P

    Y. Achdou, P. Mannucci, C. Marchi, and N. Tchou. Deterministic mean field games with control on the acceleration.NoDEA Nonlinear Differential Equations Appl., 27(3):Paper No. 33, 32, 2020

    work page 2020

  3. [3]

    D. M. Ambrose, M. Griffin-Pickering, and A. R. Mészáros. Kinetic-type mean field games with non- separable local Hamiltonians.J. Lond. Math. Soc. (2), 111(6):Paper No. e70202, 36, 2025

    work page 2025

  4. [4]

    Poincar´ e inequality and quantitative de giorgi m ethod for hypoelliptic operators, 2025

    F. Anceschi, H. Dietert, J. Guerand, A. Loher, C. Mouhot, and A. Rebucci. Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators, 2024. arXiv:2401.12194

    work page arXiv 2024

  5. [5]

    Anceschi, M

    F. Anceschi, M. Piccinini, and A. Rebucci. New perspectives on recent trends for Kolmogorov operators. InKolmogorov operators and their applications, volume 56 ofSpringer INdAM Ser., pages 57–92. Springer, Singapore, [2024]©2024

    work page 2024

  6. [6]

    Anceschi and S

    F. Anceschi and S. Polidoro. A survey on the classical theory for Kolmogorov equation.Matematiche (Catania), 75(1):221–258, 2020

    work page 2020

  7. [7]

    Z. M. Balogh, A. Calogero, and R. Pini. The Hopf-Lax formula in Carnot groups: a control theoretic approach.Calc. Var. Partial Differential Equations, 49(3-4):1379–1414, 2014

    work page 2014

  8. [8]

    Bardi and I

    M. Bardi and I. Capuzzo-Dolcetta.Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equa- tions. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia

    work page 1997

  9. [9]

    Convergenceofsomemeanfieldgamessystemstoaggregationandflocking models.Nonlinear Anal., 204:Paper No

    M.BardiandP.Cardaliaguet. Convergenceofsomemeanfieldgamessystemstoaggregationandflocking models.Nonlinear Anal., 204:Paper No. 112199, 24, 2021

    work page 2021

  10. [10]

    Regularityoftheminimumtimeandofviscositysolutionsofdegenerate eikonal equations via generalized Lie brackets.Set-Valued Var

    M.Bardi,E.Feleqi,andP.Soravia. Regularityoftheminimumtimeandofviscositysolutionsofdegenerate eikonal equations via generalized Lie brackets.Set-Valued Var. Anal., 29(1):83–108, 2021

    work page 2021

  11. [11]

    G. Barles. A short proof of the𝐶 0,𝛼-regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications.Nonlinear Anal., 73(1):31–47, 2010

    work page 2010

  12. [12]

    Springer Monogr

    A.Bonfiglioli,E.Lanconelli,andF.Uguzzoni.StratifiedLiegroupsandpotentialtheoryfortheirsub-Laplacians. Springer Monogr. Math. New York, NY: Springer, 2007

    work page 2007

  13. [13]

    Bramanti.An invitation to hypoelliptic operators and Hörmander’s vector fields

    M. Bramanti.An invitation to hypoelliptic operators and Hörmander’s vector fields. SpringerBriefs in Mathe- matics. Springer, Cham, 2014

    work page 2014

  14. [14]

    Brigati and C

    G. Brigati and C. Mouhot. Introduction to quantitative De Giorgi methods.arXiv:2510.11481, 2025

    work page arXiv 2025

  15. [15]

    Hölderestimatesinspace-timeforviscositysolutionsofHamilton-Jacobi equations.Comm

    P.CannarsaandP.Cardaliaguet. Hölderestimatesinspace-timeforviscositysolutionsofHamilton-Jacobi equations.Comm. Pure Appl. Math., 63(5):590–629, 2010

    work page 2010

  16. [16]

    Cannarsa and C

    P. Cannarsa and C. Mendico. Mild and weak solutions of mean field game problems for linear control systems.Minimax Theory Appl., 5(2):221–250, 2020. 37

    work page 2020

  17. [17]

    Birkhäuser Boston, Inc., Boston, MA, 2004

    P.CannarsaandC.Sinestrari.Semiconcavefunctions,Hamilton-Jacobiequations,andoptimalcontrol,volume58 ofProgress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2004

    work page 2004

  18. [18]

    Capuzzo Dolcetta, F

    I. Capuzzo Dolcetta, F. Leoni, and A. Porretta. Hölder estimates for degenerate elliptic equations with coercive Hamiltonians.Trans. Amer. Math. Soc., 362(9):4511–4536, 2010

    work page 2010

  19. [19]

    Cardaliaguet

    P. Cardaliaguet. A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable.ESAIM Control Optim. Calc. Var., 15(2):367–376, 2009

    work page 2009

  20. [20]

    Cardaliaguet and P

    P. Cardaliaguet and P. J. Graber. Mean field games systems of first order.ESAIM Control Optim. Calc. Var., 21(3):690–722, 2015

    work page 2015

  21. [21]

    SobolevregularityforthefirstorderHamilton-Jacobiequation

    P.Cardaliaguet,A.Porretta,andD.Tonon. SobolevregularityforthefirstorderHamilton-Jacobiequation. Calc. Var. Partial Differential Equations, 54(3):3037–3065, 2015

    work page 2015

  22. [22]

    Hölderregularityforviscositysolutionsoffullynonlinear,localornonlocal, Hamilton-Jacobiequationswithsuperquadraticgrowthinthegradient.SIAMJ.ControlOptim.,49(2):555– 573, 2011

    P.CardaliaguetandC.Rainer. Hölderregularityforviscositysolutionsoffullynonlinear,localornonlocal, Hamilton-Jacobiequationswithsuperquadraticgrowthinthegradient.SIAMJ.ControlOptim.,49(2):555– 573, 2011

    work page 2011

  23. [23]

    Cardaliaguet and B

    P. Cardaliaguet and B. Seeger. Hölder regularity of Hamilton-Jacobi equations with stochastic forcing. Trans. Amer. Math. Soc., 374(10):7197–7233, 2021

    work page 2021

  24. [24]

    Cardaliaguet and L

    P. Cardaliaguet and L. Silvestre. Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side.Comm. Partial Differential Equations, 37(9):1668– 1688, 2012

    work page 2012

  25. [25]

    InFrom particle systems to partial differential equations, volume 209 ofSpringer Proc

    C.H.ChanandA.Vasseur.DeGiorgitechniquesappliedtotheHölderregularityofsolutionstoHamilton- Jacobi equations. InFrom particle systems to partial differential equations, volume 209 ofSpringer Proc. Math. Stat., pages 117–137. Springer, Cham, 2017

    work page 2017

  26. [26]

    W.-L. Chow. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung.Math. Ann., 117:98–105, 1939

    work page 1939

  27. [27]

    M. Cirant. On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations.J. Funct. Anal., 288(2):Paper No. 110692, 35, 2025

    work page 2025

  28. [28]

    Cirant and A

    M. Cirant and A. Goffi. Lipschitz regularity for viscous Hamilton-Jacobi equations with𝐿𝑝 terms.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 37(4):757–784, 2020

    work page 2020

  29. [29]

    Cirant and A

    M. Cirant and A. Goffi. Maximal𝐿𝑞-regularity for parabolic Hamilton-Jacobi equations and applications to mean field games.Ann. PDE, 7(2):Paper No. 19, 40, 2021

    work page 2021

  30. [30]

    M. G. Crandall, L. C. Evans, and P.-L. Lions. Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 282(2):487–502, 1984

    work page 1984

  31. [31]

    M. G. Crandall, H. Ishii, and P.-L. Lions. Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited.J. Math. Soc. Japan, 39(4):581–596, 1987

    work page 1987

  32. [32]

    M. G. Crandall and P.-L. Lions. Viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 277(1):1–42, 1983

    work page 1983

  33. [33]

    Cutrì, P

    A. Cutrì, P. Mannucci, C. Marchi, and N. Tchou. The continuity equation in the Heisenberg-periodic case: a representation formula and an application to mean field games.NoDEA Nonlinear Differential Equations Appl., 31(5):Paper No. 91, 25, 2024

    work page 2024

  34. [34]

    Di Francesco and S

    M. Di Francesco and S. Polidoro. Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form.Adv. Differ. Equ., 11(11):1261–1320, 2006

    work page 2006

  35. [35]

    116, 22, 2018

    F.DragoniandE.Feleqi.ErgodicmeanfieldgameswithHörmanderdiffusions.Calc.Var.PartialDifferential Equations, 57(5):Paper No. 116, 22, 2018

    work page 2018

  36. [36]

    Dragoni, Q

    F. Dragoni, Q. Liu, and Y. Zhang. Horizontal semiconcavity for the square of Carnot-Carathéodory distanceonstep2CarnotgroupsandapplicationstoHamilton-Jacobiequations.Nonlinearity,38(4):Paper No. 045009, 34, 2025

    work page 2025

  37. [37]

    Estimatesforthe ¯𝜕𝑏 complexandanalysisontheHeisenberggroup.Comm

    G.B.FollandandE.M.Stein. Estimatesforthe ¯𝜕𝑏 complexandanalysisontheHeisenberggroup.Comm. Pure Appl. Math., 27:429–522, 1974. 38

    work page 1974

  38. [38]

    Approximationofdeterministicmeanfieldgameswithcontrol-affinedynamics

    J.GianattiandF.J.Silva. Approximationofdeterministicmeanfieldgameswithcontrol-affinedynamics. Found. Comput. Math., 24(6):2017–2061, 2024

    work page 2017

  39. [39]

    RegularityofsolutionsofFokker-Planckequationswithroughcoefficients.Riv.Mat.Univ.Parma (N.S.), 15(1):143–173, 2024

    F.Golse. RegularityofsolutionsofFokker-Planckequationswithroughcoefficients.Riv.Mat.Univ.Parma (N.S.), 15(1):143–173, 2024

    work page 2024

  40. [40]

    HarnackinequalityforkineticFokker-Planckequations with rough coefficients and application to the Landau equation.Ann

    F.Golse,C.Imbert,C.Mouhot,andA.F.Vasseur. HarnackinequalityforkineticFokker-Planckequations with rough coefficients and application to the Landau equation.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(1):253–295, 2019

    work page 2019

  41. [41]

    Griffin-Pickering and A

    M. Griffin-Pickering and A. R. Mészáros. A variational approach to first order kinetic mean field games with local couplings.Comm. Partial Differential Equations, 47(10):1945–2022, 2022

    work page 1945

  42. [42]

    Hörmander

    L. Hörmander. Hypoelliptic second order differential equations.Acta Math., 119:147–171, 1967

    work page 1967

  43. [43]

    Imbert and L

    C. Imbert and L. Silvestre. The weak Harnack inequality for the Boltzmann equation without cut-off.J. Eur. Math. Soc. (JEMS), 22(2):507–592, 2020

    work page 2020

  44. [44]

    StochastichomogenizationofviscoussuperquadraticHamilton- Jacobi equations in dynamic random environment.Res

    W.Jing,P.E.Souganidis,andH.V.Tran. StochastichomogenizationofviscoussuperquadraticHamilton- Jacobi equations in dynamic random environment.Res. Math. Sci., 4:Paper No. 6, 20, 2017

    work page 2017

  45. [45]

    R. E. Kalman. On the general theory of control systems. InProceedings first international conference on automatic control, Moscow, USSR, pages 481–492, 1960

    work page 1960

  46. [46]

    S. N. Kružkov. First order quasilinear equations with several independent variables.Mat. Sb. (N.S.), 81(123):228–255, 1970

    work page 1970

  47. [47]

    S. N. Kružkov. Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems;existence,uniquenessandstabilitytheorems;certainpropertiesofthesolutions.Mat.Sb.(N.S.), 98(140)(3(11)):450–493, 496, 1975

    work page 1975

  48. [48]

    Onaclassofhypoellipticevolutionoperators

    E.LanconelliandS.Polidoro. Onaclassofhypoellipticevolutionoperators. volume52,pages29–63.1994. Partial differential equations, II (Turin, 1993)

    work page 1994

  49. [49]

    Mannucci, C

    P. Mannucci, C. Marchi, and N. Tchou. Non coercive unbounded first order mean field games: the Heisenberg example.J. Differential Equations, 309:809–840, 2022

    work page 2022

  50. [50]

    Weakandrenormalizedsolutionstoahypoellipticmeanfieldgamessystem

    N.Mimikos-Stamatopoulos. Weakandrenormalizedsolutionstoahypoellipticmeanfieldgamessystem. SIAM J. Math. Anal., 56(2):2312–2356, 2024

    work page 2024

  51. [51]

    Polidoro

    S. Polidoro. On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type.Matematiche (Catania), 49(1):53–105, 1994

    work page 1994

  52. [52]

    P. K. Rashevski˘ı. About connecting two points of a completely nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknechta, 2:83–94, 1938

    work page 1938

  53. [53]

    L. P. Rothschild and E. M. Stein. Hypoelliptic differential operators and nilpotent groups.Acta Math., 137(3-4):247–320, 1976

    work page 1976

  54. [54]

    Sánchez Morgado

    H. Sánchez Morgado. Homogenization for sub-riemannian Lagrangians.Nonlinearity, 36(6):3043–3067, 2023

    work page 2023

  55. [55]

    R. W. Schwab. Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio- temporal media.Indiana Univ. Math. J., 58(2):537–581, 2009

    work page 2009

  56. [56]

    T. I. Seidman. How violent are fast controls?Math. Control Signals Systems, 1(1):89–95, 1988

    work page 1988

  57. [57]

    T. I. Seidman and J. Yong. How violent are fast controls? II.Math. Control Signals Systems, 9(4):327–340, 1996

    work page 1996

  58. [58]

    L.F.StokolsandA.F.Vasseur.DeGiorgitechniquesappliedtoHamilton-Jacobiequationswithunbounded right-hand side.Commun. Math. Sci., 16(6):1465–1487, 2018

    work page 2018

  59. [59]

    American Mathematical Society, Providence, RI, [2021]©2021

    H.V.Tran.Hamilton-Jacobiequations—theoryandapplications,volume213ofGraduateStudiesinMathematics. American Mathematical Society, Providence, RI, [2021]©2021

    work page 2021

  60. [60]

    Wang and L

    W. Wang and L. Zhang. The𝐶𝛼 regularity of a class of non-homogeneous ultraparabolic equations.Sci. China Ser. A, 52(8):1589–1606, 2009. 39

    work page 2009

  61. [61]

    Wang and L

    W. Wang and L. Zhang. The𝐶𝛼 regularity of weak solutions of ultraparabolic equations.Discrete Contin. Dyn. Syst., 29(3):1261–1275, 2011

    work page 2011

  62. [62]

    Zabczyk.Mathematical control theory—an introduction

    J. Zabczyk.Mathematical control theory—an introduction. Systems & Control: Foundations & Applications. Birkhäuser/Springer, Cham, second edition, [2020]©2020. 40

    work page 2020