Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

C1;1-smoothness of constrained solutions in the calculus of variations withapplication to mean field games

1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italy
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italy, & Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France
3 Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France

We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies in the fact that the presence of state constraints enters the Euler-Lagrange equations as a local feedback, which allows to derive the C1;1-smoothness of solutions.As an application, we discuss a constrained Mean Field Games problem, for which our optimality conditions allow to construct Lipschitz relaxed solutions, thus improving an existence result due to the first two authors.
  Article Metrics


1. Adams RA (1975) Sobolev Spaces. Academic Press, New York.

2. Ambrosio L, Gigli N, Savare G (2008) Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag.

3. Arutyanov AV, Aseev SM (1997) Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J Control Optim 35: 930–952.    

4. Benamou JD, Brenier Y (2000) A computational fluid mechanics solution to the Monge- Kantorovich mass transfer problem. Numer Math 84: 375–393.    

5. Benamou JD, Carlier G (2015) Augmented Lagrangian Methods for Trasport Optimization, Mean Field Games and Degenerate Elliptic Equations. J Optimiz Theory App 167: 1–26.    

6. Benamou JD, Carlier G, Santambrogio F (2017) Variational Mean Field Games, In: Bellomo N, Degond P, Tadmor E (eds) Active Particles, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 1: 141–171.

7. Brenier Y (1999) Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm Pure Appl Math 52: 411–452.    

8. Bettiol P, Frankowska H (2007) Normality of the maximum principle for nonconvex constrained bolza problems. J Differ Equations 243: 256–269.    

9. Bettiol P, Frankowska H (2008) Hölder continuity of adjoint states and optimal controls for state constrained problems. Appl Math Opt 57: 125–147.    

10. Bettiol P, Khalil N and Vinter RB (2016) Normality of generalized euler-lagrange conditions for state constrained optimal control problems. J Convex Anal 23: 291–311.

11. Cannarsa P, Capuani R (2017) Existence and uniqueness for Mean Field Games with state constraints. Available from: http://arxiv.org/abs/1711.01063.

12. Cannarsa P, Castelpietra M and Cardaliaguet P (2008) Regularity properties of a attainable sets under state constraints. Series on Advances in Mathematics for Applied Sciences 76: 120–135.    

13. Cardaliaguet P (2015)Weak solutions for first order mean field games with local coupling. Analysis and geometry in control theory and its applications 11: 111–158.

14. Cardaliaguet P, Mészáros AR, Santambrogio F (2016) First order mean field games with density constraints:pressure equals price. SIAM J Control Optim 54: 2672–2709.    

15. Cesari L (1983) Optimization–Theory and Applications: Problems with Ordinary Differential Equations, Vol 17, Springer-Verlag, New York.

16. Clarke FH (1983) Optimization and Nonsmooth Analisis, John Wiley & Sons, New York.

17. Dubovitskii AY and Milyutin AA (1964) Extremum problems with certain constraints. Dokl Akad Nauk SSSR 149: 759–762.

18. Frankowska H (2006) Regularity of minimizers and of adjoint states in optimal control under state constraints. J Convex Anal 13: 299.

19. Frankowska H (2009) Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern 38: 1327–1340.

20. Frankowska H (2010) Optimal control under state constraints. Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, 2915–2942.

21. Galbraith GN and Vinter RB (2003) Lipschitz continuity of optimal controls for state constrained problems. SIAM J Control Optim 42: 1727–1744.    

22. HagerWW(1979) Lipschitz continuity for constrained processes. SIAM J Control Optim 17: 321– 338.

23. Huang M, Caines PE and Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$ - Nash equilibria. IEEE T Automat Contr 52: 1560–1571.    

24. Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closedloop McKean-Vlasov systems and the Nash certainly equivalence principle. Communication in information and systems 6: 221–252.    

25. Lasry JM, Lions PL (2006) Jeux à champ moyen. I – Le cas stationnaire. CR Math 343: 619–625.

26. Lasry JM, Lions PL (2006) Jeux à champ moyen. II – Horizon fini et contrôle optimal. CR Math 343: 679–684.

27. Lasry JM, Lions PL (2007) Mean field games. Jpn J Math 2: 229–260.    

28. Lions PL (1985) Optimal control and viscosity solutions. Recent mathematical methods in dynamic programming, Springer, Berlin, Heidelberg, 94–112.

29. Loewen P, Rockafellar RT (1991) The adjoint arc in nonsmooth optimization. T Am Math Soc 325: 39–72.    

30. Kakutani S (1941) A generalization of Brouwer's fixed point theorem. Duke Math J 8: 457–459.    

31. Malanowski K (1978) On regularity of solutions to optimal control problems for systems with control appearing linearly. Archiwum Automatyki i Telemechaniki 23: 227–242.

32. Milyutin AA (2000) On a certain family of optimal control problems with phase constraint. Journal of Mathematical Sciences 100: 2564–2571.    

33. Rampazzo F, Vinter RB (2000) Degenerate optimal control problems with state constraints. SIAM J Control Optim 39: 989–1007.    

34. Soner HM (1986) Optimal control with state-space constraint I. SIAM J Control Optim 24: 552– 561.    

35. Vinter RB (2000) Optimal control. Birkhäuser Boston, Basel, Berlin.

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved