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

Fabio Camilli; Serikbolsyn Duisembay; Qing Tang

doi:10.3934/jdg.2021013

Article Contents

2021, Volume 8, Issue 4: 381-402. Doi: 10.3934/jdg.2021013

This issue Previous Article Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach Next Article Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization

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
^* Corresponding author: Fabio Camilli

Received: May 31, 2020

Revised: January 31, 2020

Early access: March 2021

Published: October 2021

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Abstract

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.

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Mathematics Subject Classification: 65N06, 91A16, 35R11.

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Figure 1. Mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $

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Figure 2. Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $

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Figure 3. Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $

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Figure 4. Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $

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