Open Access
Volume 25, 2019
Article Number 52
Number of page(s) 38
Published online 18 October 2019
  1. Y. Achdou and M. Laurière, On the system of partial differential equations arising in mean field type control. Discrete Contin. Dyn. Syst. 35 (2015) 3879–3900. [CrossRef] [Google Scholar]
  2. Y. Achdou and M. Laurière, Mean field type control with congestion. Appl. Math. Optim. 73 (2016) 393–418. [Google Scholar]
  3. A. Agrachev and Y. Sachkov, in Control Theory from the Geometric Viewpoint, Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2004). [CrossRef] [Google Scholar]
  4. G. Albi, M. Bongini, E. Cristiani and D. Kalise. Invisible control of self-organizing agents leaving unknown environments. SIAM J. Appl. Math. 76 (2016) 1683–1710. [Google Scholar]
  5. G. Albi, L. Pareschi and M. Zanella, Boltzmann type control of opinion consensus through leaders, Proc. of the Roy. Soc. A., 372 (2014) 20140138. [Google Scholar]
  6. L. Ambrosio. Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004) 227–260. [Google Scholar]
  7. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Oxford Mathematical Monographs. Clarendon Press, Oxford (2000). [Google Scholar]
  8. L. Ambrosio and W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures. Comm. Pure Appl. Math. 61 (2008) 18–53. [CrossRef] [Google Scholar]
  9. L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd. edn. Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag, Basel (2008). [Google Scholar]
  10. A.V. Arutyunov, D.Y. Karamzin and F.L. Pereira. The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: Revisited. J. Optim. Theor. Appl. 149 (2011) 474–493. [CrossRef] [Google Scholar]
  11. A.V. Arutyunov and R. Vinter, A simple ”finite approximations” proof of the Pontryagin Maximum Principle under reduced differentiability hypotheses. Set-Valued Anal. 12 (2004) 5–24. [CrossRef] [Google Scholar]
  12. N. Bellomo, M.A. Herrero and A. Tosin, On the dynamics of social conflicts: looking for the black swan. Kinet. Relat. Models 6 (2013) 459–479. [CrossRef] [Google Scholar]
  13. M. Bongini, M. Fornasier, F. Rossi and F. Solombrino, Mean field Pontryagin maximum principle. J. Optim. Theor. Appl. 175 (2017) 1–38. [CrossRef] [Google Scholar]
  14. B. Bonnet and F. Rossi. The Pontryagin Maximum Principle in the Wasserstein Space. Calc. Var. Partial Differ. Equ. 58 (2019) 11. [Google Scholar]
  15. A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, Vol. 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). [Google Scholar]
  16. H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differential Equations, In Universitext. Springer, Berlin (2010). [Google Scholar]
  17. M. Burger, R. Pinnau, O. Totzeck and O. Tse, Mean-field optimal control and optimality conditions in the space of probability measures. arXiv:1902.05339. [Google Scholar]
  18. P Cardaliaguet, F. Delarue, J-M. Lasry, and P.-L. Lions, The master equation and the convergence problem in mean field games. Preprint (2015). [Google Scholar]
  19. P. Cardaliaguet and L. Silvester, Hölder continuity to Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded right-hand side. Commun. Partial Differ. Equ. 37 (2012) 1668–1688. [CrossRef] [Google Scholar]
  20. R Carmona and F. Delarue, Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43 (2015) 2647–2700. [Google Scholar]
  21. R. Carmona, F. Delarue and A. Lachapelle, Control of McKean–Vlasov dynamics versus mean field games. Math. Financial Econ. 7 (2013) 131–166. [Google Scholar]
  22. J.A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42 (2010) 218–236. [CrossRef] [Google Scholar]
  23. G. Cavagnari, A. Marigonda, K.T. Nguyen and F.S Priuli, Generalized control systems in the space of probability measures. Set-Valued Var. Anal. 26 (2018) 663–691. [CrossRef] [Google Scholar]
  24. G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures. ESAIM COCV 24 (2018) 721–740. [CrossRef] [EDP Sciences] [Google Scholar]
  25. F Clarke., Functional Analysis, Calculus of Variations and Optimal Control. Springer, Berlin (2013). [CrossRef] [Google Scholar]
  26. F. Cucker and S. Smale, On the mathematics of emergence. Jpn. J. Math. 2 (2007) 197–227. [CrossRef] [Google Scholar]
  27. R.L. Di Perna and P.-L Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–548. [Google Scholar]
  28. J. Diestel and J.J. Jr Uhl. Vector Measures, Vol. 15. American Mathematical Society, Rhode Island (1977). [CrossRef] [Google Scholar]
  29. J. Dugundji. An extension of Tietze’s theorem. Pac. J. Math. 1 (1951) 353–367. [CrossRef] [Google Scholar]
  30. M. Duprez, M. Morancey and F. Rossi, Approximate and exact controllability of the continuity equation with a localized vector field. SIAM J. Control Optim. 57 (2019) 1284–1311. [Google Scholar]
  31. K. Elamvazhuthi and S. Berman, Optimal control of stochastic coverage strategies for robotic swarms. IEEE International Conference on Robotics and Automation (2015). [Google Scholar]
  32. A. Ferscha and K. Zia. Lifebelt: crowd evacuation based on vibro-tactile guidance. IEEE Pervasive Comput. 9 (2010) 33–42. [Google Scholar]
  33. M. Fornasier, S. Lisini, C. Orrieri and G. Savaré. Mean-field optimal control as gamma-limit of finite agent controls. Eur. J. Appl. Math. (2019) 1–34. [Google Scholar]
  34. M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control. Phil. Trans. R. Soc. A 372 (2014) 20130400. [CrossRef] [Google Scholar]
  35. M. Fornasier and F. Solombrino, Mean field optimal control. Esaim COCV 20 (2014) 1123–1152. [CrossRef] [EDP Sciences] [Google Scholar]
  36. W Gangbo, T Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space. Methods Appl. Anal. 15 (2008) 155–184. [Google Scholar]
  37. W. Gangbo and A. Tudorascu, On differentiability in Wasserstein spaces and well-posedness for Hamilton-Jacobi equations, Technical Report, 2017. [Google Scholar]
  38. S.Y. Ha and J.G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7 (2009) 297–325. [Google Scholar]
  39. R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5 (2002) 1–33. [Google Scholar]
  40. A.D. Ioffe, A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints. Math. Program. 58 (1993) 137–145. [Google Scholar]
  41. A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North Holland Publishing Company, Elsevier (1979). [Google Scholar]
  42. L.V. Kantorovich, On the translocation of mass. Dokl. Akad. Nauk. USSR 37 (1942) 199–201. [Google Scholar]
  43. J-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. 2 (2007) 229–260. [CrossRef] [MathSciNet] [Google Scholar]
  44. J.P. Penot and P. Michel, Calcul sous-différentiel pour les fonction Lipschitziennes et non-Lipschitziennes. C. R. Acad. Sci. Paris Sér. I 298 (1984) 269–272. [Google Scholar]
  45. B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes. Acta Appl. Math. 124 (2013) 73–105. [Google Scholar]
  46. B. Piccoli, F. Rossi, E. Trélat, Control of the kinetic Cucker-Smale model. SIAM J. Math. Anal. 47 (2015) 4685–4719. [CrossRef] [Google Scholar]
  47. N. Pogodaev, Numerical algorithm for optimal control of continuity equations. Preprint arXiv:1708.05516 (2017). [Google Scholar]
  48. N. Pogodaev, Optimal control of continuity equations. Nonlinear Differ. Equ. Appl. 23 (2016) 21. [CrossRef] [Google Scholar]
  49. W. Rudin, Real and Complex Analysis. Mathematical Series. McGraw-Hill International Editions (1987). [Google Scholar]
  50. F. Santambrogio, Optimal Transport for Applied Mathematicians, Vol. 87. Birkhauser, Basel (2015). [CrossRef] [Google Scholar]
  51. I.A. Shvartsman, New approximation method in the proof of the maximum principle for nonsmooth optimal control problems with state constraints. J. Math. Anal. Appl. 326 (2006) 974–1000. [Google Scholar]
  52. F Tröltzsch, Optimal Control of Partial Differential Equations. American Mathematical Society, Rhode Island (2010). [Google Scholar]
  53. C. Villani, Optimal Transport: Old and New. Springer-Verlag, Berlin (2009). [Google Scholar]
  54. R.B. Vinter, Optimal Control. Modern Birkhauser Classics. Birkhauser, Basel (2000). [Google Scholar]
  55. A.A. Vlasov, Many-Particle Theory and its Application to Plasma. Gordon and Breach, New York (1961). [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.