2015
DOI: 10.1137/140975061
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A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators

Abstract: Abstract. We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichletto-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical a… Show more

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Cited by 71 publications
(110 citation statements)
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“…for more detail, see (Antil and Otarola , 2015). As a result, the right Caputo fractional derivative can be written as a left Caputo fractional derivative.…”
Section: (6)mentioning
confidence: 99%
“…for more detail, see (Antil and Otarola , 2015). As a result, the right Caputo fractional derivative can be written as a left Caputo fractional derivative.…”
Section: (6)mentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…PDE-constrained optimization problems involving fractional and nonlocal equations are not new in the literature; we mention, e.g., the works by Antil and Otárola [4], Otárola [41], and D'Elia and Gunzburger [18,19]. In [4], the authors consider a linear-quadratic optimal control problem for the spectral definition of the fractional Laplacian; control constraints are also considered. The authors also propose and study solution techniques to approximate the underlying solution.…”
Section: Introductionmentioning
confidence: 99%