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

  • * Corresponding author: Fabio Camilli

    * Corresponding author: Fabio Camilli 
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  • 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.

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

    Citation:

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

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

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

    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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