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

Abstract

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.

Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

“…This means that all assumptions of Theorem 4 are satisfied, so the pair (z * , u * ) given by (21) and (19) is the optimal solution to problem (15)- (16).…”

“…Recently, optimal control problems containing control systems described by fractional Laplacians have received a lot of attention. We refer [1,[20][21][22]29], where linearquadratic optimal control problems involving fractional partial differential equations are studied. In [21] the numerical aproximation of such a type of problem, where the linear state equation involves a fractional Laplace operator with its spectral definition, is investigated.…”

“…We refer [1,[20][21][22]29], where linearquadratic optimal control problems involving fractional partial differential equations are studied. In [21] the numerical aproximation of such a type of problem, where the linear state equation involves a fractional Laplace operator with its spectral definition, is investigated. In [20,22] first order necessary and sufficient optimality conditions as well as a priori error estimates are derived.…”

Numerical approximations for fractional elliptic equationsviathe method of semigroups

“…Similarly to what we do in this work with the heat semigroup e t∆ B f , in [52] the fast decay of the solution of (1.11) in the extension variable y ∈ [0, ∞) is exploited to obtain a convenient truncation of the extended unbounded domain. Numerical methods for optimal control problems related to (1.1)-(1.2) have also been developed using characterization (1.11) in [4] and (1.10) in [33]. We refer to the very recent work by Bonito et al [13] for a general review of numerical methods for fractional diffusion.…”

Fractional semilinear optimal control: optimality conditions, convergence, and error analysis

“…In contrast to these advances, the numerical analysis of PDE-constrained optimization problems involving (−∆) s has been less explored. Restricting ourselves to problems considering the spectral definition, we mention [3,16,25] within the linear-quadratic scenario, [4] for optimization with respect to order, and [26] for bilinear optimal control. We also mention [6], where the authors analyze, at the continuous level, a semilinear optimal control problem for the spectral fractional Laplacian.…”