Variational formulation and optimal control of fractional diffusion equations with Caputo derivatives

Qing Tang,Qingxia Ma

doi:10.1186/s13662-015-0593-5

Abstract

In this paper we start by giving a new definition of weak Caputo derivative in the sense of distributions, and we give a variational formulation to a fractional diffusion equation with Caputo derivative. We prove the existence and uniqueness of the solution to this weak formulation and use it to obtain a result on optimal control.

Highlights

1 Introduction In this paper we study fractional diffusion equations with controls by the method of an abstract variational formulation
For a comprehensive treatment of the subject of fractional calculus and fractional differential equations we refer to Kilba et al [ ]
An important obstacle to study solutions in fractional Sobolev spaces is that the Caputo derivative was not clearly defined when the first order derivative does not exist in the strong sense

Summary

Introduction

In this paper we study fractional diffusion equations with controls by the method of an abstract variational formulation. In this paper we attempt to give a new definition of the weak fractional Caputo derivative via distribution theory and an integration by parts formula. This definition makes it very natural to adopt the theory of operational differential equations (Lions [ ]) and gives an abstract variational formulation of the fractional diffusion equation. In order to use the theory of operational differential equations, we need to interpret the weak Caputo derivative in the sense of distributions, through fractional integration by parts in the formula (( ) in [ ]).

Hence we have

By the condition on the unbounded linear operator we obtain

Finally we have to verify the initial condition for α

Proof We introduce the spaces