References
- M. Annunziato and A. Borzi, Optimal control of probability density functions of stochastic processes, Math. Model. Anal. 15 (2010), pp. 393–407.
- M. Annunziato and A. Borzi, A Fokker–Planck control framework for multidimensional stochastic process, J. Comput. Appl. Math. 237 (2013), pp. 487–705.
- M. Annunziato, A. Borzi, F. Nobile, and R. Tempone, On the connection between the Hamilton–Jacobi–Bellman and the Fokker–Planck control frameworks, Appl. Math. 5 (2014), pp. 2476–2484.
- T. Breitenbach, A. Borzi, and T. Breitenbach, The Pontryagin maximum principle for solving Fokker–Planck optimal control problems, Comput. Optim. Appl. 76 (2020), pp. 499–533.
- M.M. Butt, Two-level method for a time-independent Fokker–Planck control problem, Int. J. Comput. Math. 98(8) (2020), pp. 1542–1560. 10.1080/00207160.2020.1825696
- M.M. Butt, Two-level difference scheme for the two-dimensional Fokker–Planck equation, Math. Comput. Simul. 180 (2021), pp. 276–288.
- E. Carlini and F.J. Silva, On the discretization of some nonlinear Fokker–Planck–Kolmogorov equations and applications, SIAM J. Numer. Anal. 56 (2018), pp. 2148–2177.
- L. Chen and J.-Q. Sun, The closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for nonlinear systems, Commun. Nonlinear Sci. Numer. Simul. 41 (2016), pp. 1–10.
- J.S. Chang and G. Cooper, A practical difference scheme for Fokker–Planck equations, J. Comput. Phys. 6 (1970), pp. 1–16.
- M. Dehghan and M. Tatari, The use of he's variational iteration method for solving a Fokker–Planck equation, Phys. Scr. 74 (2006), pp. 310–316.
- A.N. Drozdov and M. Morillo, Solution of nonlinear Fokker–Planck equation, Phys. Rev. E 54 (1996), p. 931.
- A. Fleig and R. Guglielm, Bilinear optimal control of the Fokker–Planck equation, IFAC Papers Online49(8) (2016), pp. 254–259.
- J.B. Krawczyk, On loss-avoiding payoff distribution in a dynamic portfolio management problem, J. Risk Finance 9 (2008), pp. 151–172.
- O.A. Ladyzhenskaya, R. Bellman, and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, 2016.
- J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
- N.G. Meyers, An
Lp-estimate for the gradient of solutions of second-order elliptic divergence equations, Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze
Série, Tome 17 (1963), pp. 189–206.
- M. Mohammadi and A. Borzi, Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations, J. Numer. Math. 23 (2015), pp. 271–288.
- B.K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Spinger, Berlin, 2003.
- V. Palleschi, F. Sarri, G. Marcozzi, and M.R. Torquati, Numerical solution of the Fokker–Planck equation: a fast and accurate algorithm, Phys. Lett. A 146 (1990), pp. 363–465.
- V. Palleschi and N. de Rosa, Numerical solution of the Fokker–Planck equation. II. Multidimensional case, Phys. Lett. A 163 (1992), pp. 381–391.
- H. Risken, The Fokker–Planck Equation Method of Solution and Applications, Springer Verlag, Berlin, 1989.
- S. Roy, A sparsity-based Fokker–Planck optimal control framework for modeling traffic flows, AIP Conf. Proc. 2302(1) (2020), p. 110007.
- S. Roy, A. Borzi, and A. Habbal, Pedestrian motion modelled by Fokker–Planck Nash games, R. Soc. Open Sci. 4(9) (2017), p. 170648.
- S. Roy, M. Annunziato, A. Borzi, and C. Klingenberg, A Fokker–Planck approach to control collective motion, Comput. Optim. Appl. 69 (2018), pp. 423–459.
- F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, AMS, Providence, RI, 2010.
- H. Yoshioka, T. Tanaka, F. Aranishi, M. Tsujimura, and Y. Yoshioka, Impulsive fishery resource transporting strategies based on an open-ended stochastic growth model having a latent variable (2021). 10.1002/mma.7982
- M.P. Zorzano, H. Mais, and L. Vazquez, Numerical solution of two dimensional Fokker–Planck equations, Appl. Math. Comput. 98 (1999), pp. 109–117.