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

Abstract

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 $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ 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.

Controllability of the one-dimensional fractional heat equation under positivity constraints

“…In the absence of constraints, the fractional heat equation (1.1) is null-controllable in any positive time T > 0, provided s > 1/2. This has been proved in [3] by using the gap condition on the eigenvalues, and it has been validated through numerical experiments. In space dimension N ≥ 2, the best possible controllability result available for the fractional heat equation is the approximate controllability recently obtained in [39].…”

“…Moreover, we focus on the case s > 1/2, in which we know that (2.1) is controllable. We recall that, if s ≤ 1/2, it has been shown in [3], both on a theoretical and numerical level, that the fractional heat equation (2.1) is not controllable, not even in the absence of constraints. 5.1.…”

“…. , N x , with N x = 20 (see [3]). Moreover, we use an explicit Euler scheme for the time integration on the time-grid t j = T j Nt , j = 0, .…”

Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method

“…However, to our knowledge, the question of the approximation of controls in the case of fractional wave equations has never been investigated so far, despite its interest from a theoretical and applicative point of view. To conclude, let us mention that a numerical study of the one-dimensional fractional heat equation with finite element method has been performed in [5].…”

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

“…In [5], the null controllability of the one-dimensional fractional heat equation has been treated by using the gap condition on the eigenvalues. This result has been extended to the case of controls acting from the exterior of the domain in [56].…”