Optimal control of fractional Poisson equation from non-local to local

TL;DR

This work analyzes the limiting behavior of an optimal control problem constrained by the fractional Poisson equation as the fractional order $s$ approaches 1 from below. It formulates the energy $J_s(u_s,f_s)=\tfrac12 [u_s]_{H^s(\Omega)}^2+\tfrac{\mu}{2}\|f_s\|_{L^2(\Omega)}^2$ under $(-\Delta)^s u_s=f_s$ with zero exterior conditions and a bound on the control norm, proving existence and uniqueness of minimizers and establishing a Gamma-convergence framework to compare nonlocal and local problems. The main result shows that optimal controls and states converge ($f_k^*\rightharpoonup f^*$ in $L^2$, $u_k^*\to u^*$ in $L^2$, and $J_k(f_k^*)\to J(f^*)$) to the local Poisson problem as $s\to 1^{-}$, with $(-\Delta)u=f$ and $u=0$ on $\partial\Omega$. This demonstrates a rigorous nonlocal-to-local transition in PDE-constrained optimization and provides a solid foundation for approximating local models via fractional formulations.

Abstract

In this article, the limiting behavior of the solution $\bar u_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-Δ)^s u_s(x)=f_s(x), \quad x\in Ω$$ defined on domain $Ω$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0, \quad x \in Ω^c $ is established. We will prove that $\lim_{s\to 1^-} \bar u_s= \bar u$, where $\bar u$ is a solution of the optimal control problem subjected to classical Poisson equation $-Δu(x)=f(x), \quad x \in Ω$ and $u(x)=0, \quad x\in \partial Ω.$

Theorems & Definitions (2)

• proof
• proof