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

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.

Well-posedness and simulation of weak solutions to the time-fractional Fokker–Planck equation with general forcing

“…We mention that weak solutions to other nonlinear time-fractional PDEs have been previously studied using the Galerkin method in the published works [16,15,17]. In addition, preliminary steps have been taken in the optimal control [10] and analysis [34,27,28,32,40,33] of the time-fractional Fokker-Planck system. Nonetheless, mild, strong, and classical solutions have been investigated.…”

Optimal control problems in transport dynamics with additive noise

“…While, there, the decentralized control rules are embedded inside the dynamics of µ, in our setting a control mass ν interacts with the original population with the aim of influencing its behavior. For mean-field games in the context of Fokker-Planck-type equations, we refer the reader to [22,23,49].…”