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

• Main Authors: Biccari, U. (Author), Hernández-Santamaría, V. (Author)
• Format: Article
• Language: English
• Published: Oxford University Press 2019
• Subjects: Approximate controllability; controllability; Controllability; Controllability problems; fractional heat equation; Fractional heat equation; fractional Laplacian; Fractional Laplacian; Heat equation; Heat transfer; Laplace transforms; Numerical aspects; Numerical methods; One-dimensional; One-dimensional heat; Partial differential equations; penalized HUM; Penalized HUM; Theoretical aspects
• Online Access: View Fulltext in Publisher

 We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian (-d 2x )s on the interval (-1, 1). Using classical results and techniques, we show that, acting from an open subset ω ⊂ (-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. © The Author(s) 2019.