A FEM for an optimal control problem subject to the fractional Laplace equation

• Published in: Calcolo Vol. 56; no. 4; pp. 1 - 21
• Main Authors: Dohr, Stefan, Kahle, Christian, Rogovs, Sergejs, Swierczynski, Piotr
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2019; Springer Nature B.V
• Subjects: Approximation; Discretization; Equations of state; Finite element method; Laplace equation; Linear systems; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical Analysis; Optimal control; Quadratures; Theory of Computation; Finite element method; Dunford–Taylor integral; 49K20; A priori error estimates; 35J15; Fractional Laplacian; 65N30; Linear–quadratic optimal control problem
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-019-0334-3

We study the numerical approximation of linear–quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.