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

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