Approximate controllability and optimal controls of fractional dynamical systems of order 1 &lt; q &lt; 2 in Banach spaces

Haiyong Qin,Lishan Liu,Xin Zuo,Jianwei Liu

doi:10.1186/s13662-015-0399-5

Abstract

This paper investigates the approximate controllability and optimal controls of fractional dynamical systems of order $1< q<2$ in Banach spaces. We research a class of fractional dynamical systems governed by fractional integrodifferential equations with nonlocal initial conditions. Using the Krasnosel’skii fixed point theorem and the Schauder fixed point theorem, the approximate controllability results are obtained under two cases of the nonlinear term. We also present the existence results of optimal pairs of the corresponding fractional control systems with a Bolza cost function. Finally, an application is given to illustrate the effectiveness of our main results.

Highlights

During the past two decades, fractional differential equations have been proved to be one of the most effective tools in the modeling of numerous fields of science, physics, engineering and so on
Since fractional differential equations efficiently describe many practical dynamical phenomena, they have attracted the attention of many researchers in the past years
Many authors investigated the existence of mild solutions of fractional differential equations by using semigroup theory and fixed point theorems

Summary

Introduction

During the past two decades, fractional differential equations have been proved to be one of the most effective tools in the modeling of numerous fields of science, physics, engineering and so on. Many authors investigated the existence of mild solutions of fractional differential equations by using semigroup theory and fixed point theorems (see [ – ]). Shu and Wang [ ] considered the existence of mild solutions for a class of fractional integrodifferential equations of order < q < in a Banach space: CDqt x(t) = Ax(t) + f (t, x(t)) +. Sakthivel et al [ ] established the controllability results for a class of nonlinear fractional differential equations of order < q < with nonlocal conditions. They extended the main results to approximate controllability results for nonlocal fractional control systems with infinite delay. If Q(E) ⊂ E for a convex, closed, and bounded set E of X, Q has a fixed point in E, where ∂(·) denotes the Kuratowski measure of noncompactness

Approximate controllability

Existence of optimal controls

Applications

Results