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

U. Biccari, V. Hernández-Santamaría , IMA J. Math. Control Inf., Vol. 36, No. 4 (2019), pp. 1199-1235. 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.

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