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

Abstract

In this article, the limiting behavior of the solution $\bar u_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-\Delta)^s u_s(x)=f_s(x), \quad x\in \Omega$$ defined on domain $\Omega$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0, \quad x \in \Omega^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 $-\Delta u(x)=f(x), \quad x \in \Omega$ and $u(x)=0, \quad x\in \partial \Omega.$