Control of fractional diffusion problems via dynamic programming equations

TL;DR

This paper develops a numerical method combining semi-Lagrangian schemes and Shepard approximation to efficiently compute feedback controls for high-dimensional fractional diffusion equations, addressing the curse of dimensionality.

Contribution

It introduces a novel coupling of semi-Lagrangian schemes with Shepard approximation for solving high-dimensional Hamilton-Jacobi-Bellman equations in fractional diffusion control problems.

Findings

Numerical convergence demonstrated for various examples.

Method robust against system disturbances.

Effective in high-dimensional settings.

Abstract

We explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation we use the Hamilton--Jacobi--Bellman equation. It is well-known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.