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Sparse Optimal Control for Fractional Diffusion

Enrique Otárola and Abner J. Salgado EMAIL logo


We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables.

Award Identifier / Grant number: DMS-1418784

Award Identifier / Grant number: 3160201

Funding statement: The first author has been supported in part by CONICYT through FONDECYT project 3160201. The second author has been supported in part by NSF grant DMS-1418784.


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Received: 2017-4-4
Revised: 2017-8-3
Accepted: 2017-8-5
Published Online: 2017-9-2
Published in Print: 2018-1-1

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