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

Fangyuan Wang,Xiaodi Li,Zhaojie Zhou

doi:10.3390/fractalfract5030102

Fangyuan Wang, Xiaodi Li + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract5030102

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Journal: Fractal and Fractional	Publication Date: Aug 24, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Shandong Normal University

Abstract

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

Highlights

The aim of this paper is to develop a spectral Galerkin approximation of the following optimal control problem governed by fractional advection diffusion reaction equation: min
In this paper we mainly focus on the optimal control problem governed by a fractional advection diffusion equation
A spectral Galerkin approximation of optimal control problem governed by fractional equations with control constraint is firstly investigated in [34,35], where the weighted Jacobi polynomials are used to approximate the state variable and the adjoint state variable

Summary

Introduction

The aim of this paper is to develop a spectral Galerkin approximation of the following optimal control problem governed by fractional advection diffusion reaction equation: min. The stable density that solves the fractional diffusion equation can capture the super-diffusive spreading observed in the data Motivated by these facts, in this paper we mainly focus on the optimal control problem governed by a fractional advection diffusion equation. A spectral Galerkin approximation of optimal control problem governed by fractional equations with control constraint is firstly investigated in [34,35], where the weighted Jacobi polynomials are used to approximate the state variable and the adjoint state variable. As an extension in the present work we propose a spectral Galerkin approximation scheme for optimal control problem governed by fractional advection diffusion reaction equation under the constraints of state integration.

Weighed Sobolev Spaces and Jacobi Polynomials

Properties of the Fractional Laplacian Operations

First-Order Optimality Condition

Spectral Galerkin Approximation

Algorithm

Numerical Examples

Conclusions