Abstract

We address the existence of equilibria for mean field type control problems of first-order with non-convex action functional. Introducing a relaxed Lagrangian approach on the Wasserstein space to handle the lack of convexity, we prove the existence of new relaxed Nash equilibria and we show that our existence result encompasses the classical Mean Field Control problem's existence result under convex data conditions

Author information

Contact details are reproduced from the original publication and may be historical.

Kaizhi Wang

School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China

kzwang@sjtu.edu.cn

Yuchen Xu

School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China

math_rain@sjtu.edu.cn

Keywords

  • Mean field control
  • relaxed Lagrangian approach
  • relaxed controls
  • residual neural networks

Mathematics Subject Classification

49N80, 49N90, 68T07, 93C10

References

  1. L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser, Basel (2008).Google Scholar
  2. A. Bensoussan, J. Frehse, P. Yam: Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York (2013.https://doi.org/10.1007/978-1-4614-8508-7Google Scholar
  3. P. Billingsley: Convergence of Probability Measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York (1999).https://doi.org/10.1002/9780470316962Google Scholar
  4. G. Buttazzo: Some relaxation problems in optimal control theory, J. Math. Anal. Appl. 125/1 (1987) 272–287.https://doi.org/10.1016/0022-247X(87)90182-XGoogle Scholar
  5. G. Buttazzo: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series Vol. 207, Longman Scientific & Technical, Harlow; copublished with John Wiley & Sons, New York (1989).Google Scholar
  6. G. Buttazzo: Relaxed optimal control problems and applications to shape optimization, in: Nonlinear Analysis, Differential Equations and Control, Montreal 1998, NATO Sci. Ser. C Math. Phys. Sci. Vol. 528, Kluwer Academic Publishers, Dordrecht (1999) 159–206.https://doi.org/10.1007/978-94-011-4560-2_3Google Scholar
  7. P. Cannarsa, R. Capuani: Existence and uniqueness for mean field games with state constraints, in: PDE models for multi-agent phenomena, Springer INdAM Ser. 28, Springer, Cham (2018) 49–71.https://doi.org/10.1007/978-3-030-01947-1_3Google Scholar
  8. P. Cannarsa, T. D’Aprile: Introduction to Measure Theory and Functional Analysis, Unitext Vol. 89, Springer, Cham (2015).https://doi.org/10.1007/978-3-319-17019-0Google Scholar
  9. P. Cannarsa, C. Mendico: Mild and weak solutions of mean field game problems for linear control systems, Minimax Theory Appl. 5/2 (2020) 221–250.Google Scholar
  10. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications Vol. 58, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  11. P. Cardaliaguet: Notes on mean field game, unpublished manuscript (2013).Google Scholar
  12. R. Carmona, F. Delarue: Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51/4 (2013) 2705–2734.https://doi.org/10.1137/120883499Google Scholar
  13. R. Carmona, F. Delarue: Probabilistic Theory of Mean Field Games with Applications. I: Mean field FBSDEs, Control, and Games, Probability Theory and Stochastic Modelling Vol. 83, Springer, Cham (2018).https://doi.org/10.1007/978-3-319-56436-4Google Scholar
  14. R. Carmona, F. Delarue: Probabilistic Theory of Mean Field Games with Applications. II: Mean Field Games with Common Noise and Master Equations, Probability Theory and Stochastic Modelling Vol. 84, Springer, Cham (2018).https://doi.org/10.1007/978-3-319-56436-4Google Scholar
  15. R. Carmona, F. Delarue, A. Lachapelle: Control of McKean-Vlasov dynamics versus mean field games, Math. Finance Econ. 7/2 (2013) 131–166.https://doi.org/10.1007/s11579-012-0089-yGoogle Scholar
  16. F. H. Clarke, Y. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics Vol. 178, Springer, New York (1998).Google Scholar
  17. W. E: A proposal on machine learning via dynamical systems, Commun. Math. Stat. 5/1 (2017) 1–11.https://doi.org/10.1007/s40304-017-0103-zGoogle Scholar
  18. W. E, J. Han, Q. Li: A mean-field optimal control formulation of deep learning, Res. Math. Sci. 6/1 (2019), art.no. 10, 41p.https://doi.org/10.1007/s40687-018-0172-yGoogle Scholar
  19. M. Fischer, F. J. Silva: On the asymptotic nature of first order mean field games, Appl. Math. Optim. 84/2 (2021) 2327–2357.https://doi.org/10.1007/s00245-020-09711-1Google Scholar
  20. D. A. Gomes, J. Saúde: A mean-field game approach to price formation, Dyn. Games Appl. 11/1 (2021) 29–53.https://doi.org/10.1007/s13235-020-00348-xGoogle Scholar
  21. U. G. Haussmann, J. P. Lepeltier: On the existence of optimal controls, SIAM J. Control Optim. 28/4 (1990) 851–902.https://doi.org/10.1137/0328049Google Scholar
  22. W. Lee, S. Liu, H. Tembine, W. Li, S. Osher: Controlling propagation of epidemics via mean-field control, SIAM J. Appl. Math. 81/1 (2021) 190–207.https://doi.org/10.1137/20M1342690Google Scholar
  23. J. Maass, J. Fontbona: Symmetries in overparametrized neural networks: a mean-field view, arXiv: 2405.19995 (2024)Google Scholar
  24. G. Mazanti, F. Santambrogio: Minimal-time mean field games, Math. Models Methods Appl. Sci. 29/8 (2019) 1413–1464.https://doi.org/10.1142/S0218202519500258Google Scholar
  25. C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics Vol. 58, American Mathematical Society, Providence (2003).https://doi.org/10.1090/gsm/058Google Scholar
  26. M. A. Zaman, E. Miehling, T. Başar: Reinforcement learning for non-stationary discrete-time linear-quadratic mean-field games in multiple populations, Dyn. Games Appl. 13/1 (2023) 118–164.https://doi.org/10.1007/s13235-022-00448-wGoogle Scholar

Suggested citation

C. Mendico, K. Wang, Y. Xu. “Relaxed Lagrangian Approach to First-Order Non-Convex Mean Field Type Control Problems.” Journal of Convex Analysis 34 (2027), No. 1, 77–95. https://doi.org/10.68381/jca34006

Download BibTeXPublished by Banach Press LLC.

© 2026 Banach Press LLC. Published under CC BY 4.0.