2022/10/18 by Alessandro Alla, Alla, Alessandro, Marta D’Elia +5
Mathematics · #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2210.09827
openalex publication_date 2022/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.