# Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

U. Biccari, V. Hernández-Santamaría , IMA Journal of Mathematical Control and Information DOI: 10.1093/imamci/dny025

Abstract: We analyze the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d^2_x)^s$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega\subset(-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leq 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method (HUM) and a finite element (FE) scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.