Fractional Integro-Dierential Equations of Mixed Type with Solution Operator and Optimal Controls

Wichai Wittayakiattilerd,A Chonwerayuth

doi:10.5539/jmr.v3n3p140

Abstract

Local and global existence and uniqueness of mild solution for the fractional integro-differential equations of mixed type with delay are proved by using a family of solution operators and the contraction mapping principle on Banach space. TheBolza optimal control problem of a corresponding controlled system is solved. The Gronwall lemma with singular and time lag is derived to be tool for obtaining a priori estimate. In addition, the application to the fractional nonlinear heat equation is shown.

Highlights

In this paper, we consider fractional integro-differential equations of mixed type with delay;⎧⎪⎪⎨⎪⎪⎩Dx(αtt)x(=t)φ=(t)A, x(t) + f (t, x(t), t ∈ [−r, 0], G x(t), S x(t)) +B(t)u(t), t∈I (1)on infinite dimensional Banach space X, where I = [0, T ], 0 < α ≤ 1, Dαt denote the fractional derivative in the sense of Riemann-Liouville, f : I × X × X × X → X and φ ∈ C([−r, 0], X) are given, A is a linear operator corresponding to a solution operator {Tα(t)}t≥0 in the Banach space X and G, S are nonlinear integral operators given by t TGx(t) = k(t, s)g(s, x(s))ds, S x(t) = h(t, s)q(s, x(s))ds. (2) −rMany research groups have studied and reported on integro-differential systems and fractional differential systems
On infinite dimensional Banach space X, where I = [0, T ], 0 < α ≤ 1, Dαt denote the fractional derivative in the sense of Riemann-Liouville, f : I × X × X × X → X and φ ∈ C([−r, 0], X) are given, A is a linear operator corresponding to a solution operator {Tα(t)}t≥0 in the Banach space X and G, S are nonlinear integral operators given by t
These reports include the proof of the existence and uniqueness of a classical solution of an integro-differential equation by Chonwerayuth and a portion of work on the nonlinear impulsive integro-differential equations of mixed type by Wei.W

Summary

Introduction

We consider fractional integro-differential equations of mixed type with delay;. Many research groups have studied and reported on integro-differential systems and fractional differential systems. These reports include the proof of the existence and uniqueness of a classical solution of an integro-differential equation by Chonwerayuth and a portion of work on the nonlinear impulsive integro-differential equations of mixed type by Wei.W. in 2009, Gisele M.Mophou proved existence and uniqueness of mild solution to impulsive fractional differential equations. We apply our result to fractional nonlinear heat equation

Preliminaries

Existence of Optimal Controls

Application to Fractional Nonlinear Heat Equation