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October  2021, 8(4): 381-402. doi: 10.3934/jdg.2021013

## Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

 1 Dip. di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via Scarpa 16, 00161 Roma, Italy 2 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia 3 China University of Geosciences, Wuhan, China

* Corresponding author: Fabio Camilli

Received  June 2020 Revised  February 2020 Published  October 2021 Early access  March 2021

In this paper, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem.

Citation: Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013
##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.  doi: 10.1137/120882421. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477. [3] Y. Achdou and A. Porretta, Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186.  doi: 10.1137/15M1015455. [4] N. Almulla, R. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.  doi: 10.1007/s13235-016-0203-5. [5] M. Annunziato, A. Borzì, M. Magdziarz and A. Weron, A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.  doi: 10.1002/oca.2168. [6] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [7] L. M. Briceño-Arias, D. Kalise and F. J. Silva, Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., 56 (2018), 801-836.  doi: 10.1137/16M1095615. [8] F. Camilli and R. De Maio, A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554. [9] F. Camilli and A. Goffi, Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 37pp. doi: 10.1007/s00030-020-0624-0. [10] E. Carlini and F. J. Silva, A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete Contin. Dyn. Syst., 35 (2015), 4269-4292.  doi: 10.3934/dcds.2015.35.4269. [11] R. Carmona and M. Lauriére, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: Ⅱ – The finite horizon case, preprint, arXiv: 1908.01613v1. [12] Q. Du, An invitation to nonlocal modeling, analysis and computation, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. Ⅳ, World Sci. Publ., Hackensack, NJ, 2018, 3541–3569. doi: 10.1142/9789813272880_0191. [13] Q. Du, J. Yang and Z. Zhou, Time-fractional Allen-Cahn equations: Analysis and numerical methods, J. Sci. Comput., 85 (2020), 30pp. doi: 10.1007/s10915-020-01351-5. [14] Y. Giga, Q. Liu and H. Mitake, On a discrete scheme for time fractional fully nonlinear evolution equations, Asymptot. Anal., 120 (2020), 151-162.  doi: 10.3233/ASY-191583. [15] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [16] B. Jin, R. Lazarov, J. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960. [17] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.  doi: 10.1093/imanum/dru063. [18] B. Jin, R. Lazarov and Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview, Comput. Methods Appl. Mech. Engrg., 346 (2019), 332-358.  doi: 10.1016/j.cma.2018.12.011. [19] V. N. Kolokoltsov and M. A. Veretennikova, A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), 18pp. doi: 10.1685/journal.caim.484. [20] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [21] D. Li, J. Wang and J. Zhang, Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088. doi: 10.1137/16M1105700. [22] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001. [23] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, De Gruyter, Berlin, 2019. doi: 10.1515/9783110560244. [24] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208. doi: 10.1088/0305-4470/37/31/R01. [25] T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp. doi: 10.1007/s00030-018-0513-y. [26] G. Pang, L. Lu and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), A2603–A2626. doi: 10.1137/18M1229845. [27] J. Shen and C.-T. Sheng, An efficient space–time method for time fractional diffusion equation, J. Sci. Comput., 81 (2019), 1088-1110.  doi: 10.1007/s10915-019-01052-8. [28] Q. Tang, On an optimal control problem of time-fractional advection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 761-779.  doi: 10.3934/dcdsb.2019266. [29] T. Tang, H. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757–A3778. doi: 10.1137/18M1203560. [30] E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024. [31] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

show all references

##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.  doi: 10.1137/120882421. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477. [3] Y. Achdou and A. Porretta, Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186.  doi: 10.1137/15M1015455. [4] N. Almulla, R. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.  doi: 10.1007/s13235-016-0203-5. [5] M. Annunziato, A. Borzì, M. Magdziarz and A. Weron, A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.  doi: 10.1002/oca.2168. [6] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [7] L. M. Briceño-Arias, D. Kalise and F. J. Silva, Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., 56 (2018), 801-836.  doi: 10.1137/16M1095615. [8] F. Camilli and R. De Maio, A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554. [9] F. Camilli and A. Goffi, Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 37pp. doi: 10.1007/s00030-020-0624-0. [10] E. Carlini and F. J. Silva, A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete Contin. Dyn. Syst., 35 (2015), 4269-4292.  doi: 10.3934/dcds.2015.35.4269. [11] R. Carmona and M. Lauriére, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: Ⅱ – The finite horizon case, preprint, arXiv: 1908.01613v1. [12] Q. Du, An invitation to nonlocal modeling, analysis and computation, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. Ⅳ, World Sci. Publ., Hackensack, NJ, 2018, 3541–3569. doi: 10.1142/9789813272880_0191. [13] Q. Du, J. Yang and Z. Zhou, Time-fractional Allen-Cahn equations: Analysis and numerical methods, J. Sci. Comput., 85 (2020), 30pp. doi: 10.1007/s10915-020-01351-5. [14] Y. Giga, Q. Liu and H. Mitake, On a discrete scheme for time fractional fully nonlinear evolution equations, Asymptot. Anal., 120 (2020), 151-162.  doi: 10.3233/ASY-191583. [15] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [16] B. Jin, R. Lazarov, J. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960. [17] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.  doi: 10.1093/imanum/dru063. [18] B. Jin, R. Lazarov and Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview, Comput. Methods Appl. Mech. Engrg., 346 (2019), 332-358.  doi: 10.1016/j.cma.2018.12.011. [19] V. N. Kolokoltsov and M. A. Veretennikova, A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), 18pp. doi: 10.1685/journal.caim.484. [20] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [21] D. Li, J. Wang and J. Zhang, Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088. doi: 10.1137/16M1105700. [22] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001. [23] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, De Gruyter, Berlin, 2019. doi: 10.1515/9783110560244. [24] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208. doi: 10.1088/0305-4470/37/31/R01. [25] T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp. doi: 10.1007/s00030-018-0513-y. [26] G. Pang, L. Lu and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), A2603–A2626. doi: 10.1137/18M1229845. [27] J. Shen and C.-T. Sheng, An efficient space–time method for time fractional diffusion equation, J. Sci. Comput., 81 (2019), 1088-1110.  doi: 10.1007/s10915-019-01052-8. [28] Q. Tang, On an optimal control problem of time-fractional advection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 761-779.  doi: 10.3934/dcdsb.2019266. [29] T. Tang, H. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757–A3778. doi: 10.1137/18M1203560. [30] E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024. [31] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.
Mass evolution (A) for $\alpha = 1$ (B) for $\alpha = 0.85$ (C) for $\alpha = 0.7$
Level sets of the mass evolution (A) for $\alpha = 1$ (B) for $\alpha = 0.85$ (C) for $\alpha = 0.7$
Level sets of the mass evolution (A) for $\alpha = 1$ (B) for $\alpha = 0.85$ (C) for $\alpha = 0.7$
Level sets of the mass evolution (A) for $\alpha = 1$ (B) for $\alpha = 0.85$ (C) for $\alpha = 0.7$
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