EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 27 (2021) ESAIM: COCV, 27 (2021) 15 References
Free Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 15
Number of page(s) 26
DOI https://doi.org/10.1051/cocv/2021014
Published online 22 March 2021
  1. G. Albi, Y.-P. Choi, M. Fornasier and D. Kalise, Mean field control hierarchy. Appl. Math. Optim. 76 (2017) 93–135. [Google Scholar]
  2. G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20140138. [Google Scholar]
  3. H. Amann, Linear and quasilinear parabolic problems. Abstract linear theory. Vol. I, Vol. 89 of Monographs in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1995). [Google Scholar]
  4. M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15 (2010) 393–407. [Google Scholar]
  5. M. Annunziato and A. Borzì, A Fokker–Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. 237 (2013) 487–507. [Google Scholar]
  6. M. Annunziato and A. Borzì, A Fokker–Planck control framework for stochastic systems. EMS Surv. Math. Sci. 5 (2018) 65–98. [Google Scholar]
  7. M.S. Aronna, J.F. Bonnans, and A. Kröner, Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations. Math. Program. 168 (2018) 717–757. [Google Scholar]
  8. M.S. Aronna and F. Tröltzsch, First and second order optimality conditions for the control of Fokker-Planck equations. Preprint (2020) arxiv.org/abs/2002.03988v1. [Google Scholar]
  9. M.S. Aronna and F. Tröltzsch, First and second order optimality conditions for the control of Fokker-Planck equations. Preprint (2021) arxiv.org/abs/2002.03988v3. [Google Scholar]
  10. A. Ashyralyev and P.E. Sobolevskiĭ, Well-posedness of parabolic difference equations. Vol. 69 of Operator Theory: Advances and Applications. Translated from the Russian by A. Iacob. Birkhäuser Verlag, Basel (1994). [Google Scholar]
  11. J.-P. Aubin, Un théorème de compacité. CR Acad. Sci. Paris 256 (1963) 5042–5044. [Google Scholar]
  12. T. Bose and S. Trimper, Stochastic model for tumor growth with immunization. Phys. Rev. E 79 (2009) 051903. [Google Scholar]
  13. T. Breiten, K. Kunisch and L. Pfeiffer, Control strategies for the Fokker- Planck equation. ESAIM: COCV 24 (2018) 741–763. [CrossRef] [EDP Sciences] [Google Scholar]
  14. T. Breiten, K. Kunisch and L. Pfeiffer, Infinite-horizon bilinear optimal control problems: sensitivity analysis and polynomial feedback laws. SIAM J. Control Optim. 56 (2018) 3184–3214. [Google Scholar]
  15. P. Cardaliaguet, Notes on mean field games. Technical report, (2010). [Google Scholar]
  16. E. Casasand F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [Google Scholar]
  17. E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems. SIAM J. Control Optim. 56 (2018) 4203–4227. [Google Scholar]
  18. P.-H. Chavanis, Nonlinear mean field Fokker-Planck equations. application to the chemotaxis of biological populations. Eur. Phys. J. B 62 (2008) 179–208. [CrossRef] [EDP Sciences] [Google Scholar]
  19. M. Chipot, Elements of nonlinear analysis. Birkhäuser (2012). [Google Scholar]
  20. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology: Evolution Problems I, volume 5. Springer Science & Business Media (1992). [Google Scholar]
  21. R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion. Commun. Math. Phys. 300 (2010) 95–145. [Google Scholar]
  22. L.C. Evans, Partial differential equations. American Mathematical Society (2010). [Google Scholar]
  23. A. Fleig and R. Guglielmi, Optimal control of the Fokker-Planck equation with space-dependent controls. J. Optim. Theory Appl. 174 (2017) 408–427. [Google Scholar]
  24. G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker–Planck equations in the modeling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27 (2017) 115–158. [Google Scholar]
  25. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations: theory and algorithms. In vol. 5. Springer Science & Business Media (2012). [Google Scholar]
  26. D.A. Gomes and J. Saúde, Mean field games models—a brief survey. Dyn. Games Appl. 4 (2014) 110–154. [Google Scholar]
  27. M. Herty, C. Jörres and A.N. Sandjo, Optimization of a model Fokker-Planck equation. Kinet. Relat. Models 5 (2012). [Google Scholar]
  28. A.D. Ioffe, Necessary and sufficient conditions for a local minimum 3: Second order conditions and augmented duality. SIAM J. Control Optim. 17 (1979) 266–288. [Google Scholar]
  29. T. Kato, Perturbation theory for linear operators, Reprint of the corr. print of the 2nd edition. Classics in Mathematics. Springer-Verlag New York, Inc., New York (1980). [Google Scholar]
  30. O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I. (1968). [CrossRef] [Google Scholar]
  31. J.-M. Lasry and P.-L. Lions, Mean field games. Jpn. J. Math. 2 (2007) 229–260. [CrossRef] [MathSciNet] [Google Scholar]
  32. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. Dunod, Paris (1968). [Google Scholar]
  33. E.M. Ouhabaz, Analysis of heat equations on domains. Vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2005). [Google Scholar]
  34. S. Roy, M. Annunziato and A. Borzì, A Fokker–Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transport 45 (2016) 442–458. [Google Scholar]
  35. L. Ryzhik, Lectures notes (on mean field games). Technicalreport (2018) available at: link. [Google Scholar]
  36. M. Schienbein and H. Gruler, Langevin equation, Fokker-Planck equation and cell migration. Bull. Math. Biol. 55 (1993) 585–608. [Google Scholar]
  37. H. Triebel, Interpolation theory, function spaces, differential operators. Vol. 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam-New York (1978). [Google Scholar]
  38. F. Tröltzsch, Optimal control of partial differential equations. Theory, methods and applications, Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.