Optimal control of a parabolic fractional PDE: analysis and discretization

Abstract

We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega ))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions and derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

arXiv comment24 pages, 4 figures

References in corpus (5)

• Ten equivalent definitions of the fractional Laplace operator
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains
• Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
• Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

Cited by in corpus (2)

• Numerical methods for nonlocal and fractional models
• An adaptive finite element method for the sparse optimal control of fractional diffusion