Open Access
Issue
ESAIM: COCV
Volume 28, 2022
Article Number 77
Number of page(s) 30
DOI https://doi.org/10.1051/cocv/2022066
Published online 22 December 2022
  1. R.A. Adams and J.J. Fournier, Sobolev Spaces. Elsevier (2003). [Google Scholar]
  2. M. Annunziato and A. Borzi, A Fokker-Planck control framework for multidimensional stochastic processes. J. Comput. Appi. Math. 237 (2013) 487–507. [CrossRef] [Google Scholar]
  3. M. Annunziato and A. Borzi, A Fokker-Planck control framework for stochastic systems. EMS Surv. Math. Sci. 5 (2018) 65–98. [CrossRef] [MathSciNet] [Google Scholar]
  4. M.S. Aronna and F. Tröltzsch, First and second order optimality conditions for the control of Fokker-Planck equations. ESAIM: COCV 27 (2021) 15. [CrossRef] [EDP Sciences] [Google Scholar]
  5. J. Bartsch, A. Borzi, F. Fanelli and S. Roy, A theoretical investigation of Brockett’s ensemble optimal control problems. Calc. Variat. Partial Differ. Equ. 58 (2019) 1–34. [CrossRef] [Google Scholar]
  6. J. Bartsch, A. Borzi, F. Fanelli and S. Roy, A numerical investigation of Brockett’s ensemble optimal control problems. Numer. Math. 149 (2021) 1–42. [Google Scholar]
  7. J. Bartsch, G. Nastasi and A. Borzi, Optimal control of the Keilson-Storer caster equation in a Monte Carlo framework. J. Comput. Theor. Transport 50 (2021) 454–482. [CrossRef] [MathSciNet] [Google Scholar]
  8. M. Bengfort, H. Malchow and F.M. Hilker, The Fokker-Planck law of diffusion and pattern formation in heterogeneous environments. J. Math. Biol. 73 (2016) 683–704. [Google Scholar]
  9. A. Bensoussan, Stochastic Control by Functional Analysis Methods. Elsevier (2011). [Google Scholar]
  10. A. Borzi, The Fokker-Planck framework in the modeling of pedestrians’ motion, Chapter 6 in Crowd Dynamics, Volume 2, Modeling and Simulation in Science, Engineering and Technology, edited by L. Gibelli. Birkhäuser, Cham, Switzerland (2020) 111–131. [CrossRef] [Google Scholar]
  11. 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]
  12. T. Breitenbach and A. Borzi, The Pontryagin maximum principle for solving Fokker-Planck optimal control problems. Comput. Optim. Appi. 76 (2020) 499–533. [CrossRef] [Google Scholar]
  13. T. Breitenbach and A. Borzi, A sequential quadratic Hamiltonian method for solving parabolic optimal control problems with discontinuous cost functionals. J. Dyn. Control Syst. 25 (2019) 403–435. [CrossRef] [MathSciNet] [Google Scholar]
  14. R.W. Brockett, Minimum attention control, in Proceedings of the 36th IEEE Conference on Decision and Control 3 (1997) 2628–2632. [Google Scholar]
  15. R.W. Brockett, Optimal control of the Liouville equation. Proceedings of the International Conference on Complex Geometry and Related Fields. AMS/IP Stud. Adv. Math. 39 (2007) 23–35. [CrossRef] [Google Scholar]
  16. R.W. Brockett, Notes on the control of the Liouville equation. Control of partial differential equations, Lecture Notes in Math. 2048. Springer, Heidelberg (2012) 101–129. [Google Scholar]
  17. B. Buonomo, D. Lacitignola and C. Vargas-De-Leon, Qualitative analysis and optimal control of an epidemic model with vaccination and treatment. Math. Comput. Simul. 100 (2014) 88–102. [Google Scholar]
  18. B. Buonomo, On the optimal vaccination strategies for horizontally and vertically transmitted infectious diseases. J. Biolog. Syst. 19 (2011) 263–279. [CrossRef] [Google Scholar]
  19. E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [Google Scholar]
  20. E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math. Ver. 117 (2015) 3–44. [CrossRef] [MathSciNet] [Google Scholar]
  21. M. Chipot, Elements of Nonlinear Analysis. Springer Science & Business Media (2000). [Google Scholar]
  22. L.C. Evans, Partial differential equations. Graduate Stud. Math. 19 (1998) 7. [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. D.A. Gomes and J. Saude, Mean field games models - a brief survey. Dyn. Games Appl. 4 (2014) 110–154. [CrossRef] [MathSciNet] [Google Scholar]
  25. P. Grandits, R.M. Kovacevic and V.M. Veliov, Optimal control and the value of information for a stochastic epidemiological SIS-model. J. Math. Anal. Appl. 476 (2019) 665–695. [Google Scholar]
  26. P. Han, Z. Chang and X. Meng, Asymptotic dynamics of a stochastic SIR epidemic system affected by mixed nonlinear incidence rates. Complexity (2020) 2020. [Google Scholar]
  27. J.M. Lasry and P.L. Lions, Mean field games. Jpn. J. Math. 2 (2007) 229–260. [Google Scholar]
  28. J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Science & Business Media, Berlin Heidelberg, (1972). [Google Scholar]
  29. H. Lou and J. Yong, Optimality conditions for semilinear elliptic equations with leading term containing controls. SIAM J. Control Optim. 48 (2009) 2366–2387. [Google Scholar]
  30. H. Lou and J. Yong, Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Math. Control Related Fields 8 (2018) 57–88. [Google Scholar]
  31. X. Mao, Stochastic Differential Equations and Applications. Elsevier (2007). [Google Scholar]
  32. A. Röosch and D. Wachsmuth, Numerical verification of optimality conditions. SIAM J. Control Optim. 47 (2008) 2557–2581. [Google Scholar]
  33. H. Risken, Fokker-Planck Equation. Springer, Berlin, Heidelberg (1996). [Google Scholar]
  34. S. Roy, M. Annunziato, A. Borzi and C. Klingenberg, A Fokker-Planck approach to control collective motion. Comput. Optim. Appl. 69 (2018) 423–459. [CrossRef] [MathSciNet] [Google Scholar]
  35. S. Roy, M. Annunziato and A. Borzi, A Fokker-Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transport 45 (2016) 442–458. [CrossRef] [MathSciNet] [Google Scholar]
  36. S. Roy, A. Borzi and A. Habbal, Pedestrian motion modelled by Fokker-Planck Nash games. Royal Soc. Open Sci. 4 (2017) 170648. [Google Scholar]
  37. M. Schienbein and H. Gruler, Langevin equation, Fokker-Planck equation and cell migration. Bull. Math. Biol. 55 (1993) 585–608. [Google Scholar]
  38. R.K. Tagiyev, Optimal control by the coefficients of a parabolic equation. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 24 (2004) 247–256. [Google Scholar]
  39. R.K. Tagiyev, On optimal control of the hyperbolic equation coefficients. Autom. Remote Control 73 (2012) 1145–1155. [CrossRef] [MathSciNet] [Google Scholar]
  40. D. Wachsmuth, The regularity of the positive part of functions in L2 (I; H1 (Q)) U H1(I; H1 (Q)*) with applications to parabolic equations. Comment. Math. Univers. Carolinae 57 (2016) 327–332. [Google Scholar]
  41. Z. Wu, J. Yin and C. Wang, Elliptic & Parabolic Equations. World Scientific (2006). [Google Scholar]
  42. Y. Zhang, Y. Li, Q. Zhang and A. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules. Physica A 501 (2018) 178–187. [Google Scholar]

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