A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian

Abstract

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized -the so-called variational discretization approach -and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests.

Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

“…In the past decades lots of works [14][15][16][17][18][19] are devoted to develop numerical methods or algorithms for fractional differential equations. In recent years optimal control problems governed by different types of fractional differential equations have attracted increasing attentions [20][21][22][23][24][25][26][27][28][29][30].…”

Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

“…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.…”

“…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. PDE constraints contain the integral fractional Laplacian.…”

Finite element approximation of fractional Neumann problems

“…In recent years, finite element methods have been proposed and studied for a variety of equations involving the fractional Laplacian (1.2), such as Dirichlet [2,4,5,6,12,13], time-fractional evolution [3], phase field [1,7,31], optimal control [8,9,11,23,29], and obstacle [14,18,19,28] problems. Most of these references consider either Dirichlet or periodic boundary conditions; reference [8] deals with Neumann and Robin conditions, but does not address the convergence of finite element discretizations of such problems.…”