Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions

He Yang,Qian Ren

doi:10.3390/sym14020248

Abstract

In this paper, under symmetric properties of multivalued operators, the existence of mild solutions as well as optimal control for the nonlocal problem of fractional semilinear evolution inclusions are investigated in abstract spaces. At first, the existence results are proved by applying the theory of operator semigroups and the fixed-point theorem of multivalued mapping. Then the existence theorem on the optimal state-control pair is proved by constructing the minimizing sequence twice. An example is given in the last section as an application of the obtained conclusions.

Highlights

As an important branch of the nonlinear analysis theory, fractional differential inclusions have gained a lot of attention in recent years, because it has wide applications in fluid mechanics, economics, control theory, and so forth
By constructing minimizing sequences twice, we delete the Lipschitz continuity of the nonlinear term f, which is extensively used as an essential assumption in existing papers, and without the uniqueness of mild solutions, we prove the existence of optimal state-control pair of (4)
In this work, we first proved the existence theorem on mild solutions of (4) by using the theory of operator semigroups and fixed-point theorems of multi-valued mapping

Summary

Introduction

As an important branch of the nonlinear analysis theory, fractional differential inclusions have gained a lot of attention in recent years, because it has wide applications in fluid mechanics, economics, control theory, and so forth (see [1,2,3,4] and the references therein). [9], Lian et al were concerned with the existence of mild solutions for the nonlinear fractional differential system in Banach space X They firstly proved the existence results of mild solutions of Equation (3) by using the Schauder fixed-point theorem and the semigroup theory. The main contributions of this work can be listed as follows: (i) Under the case that the nonlocal function g is Lipschitz-continuous or completely continuous, the existence of mild solutions of the fractional evolution inclusion (4) is proved by using a fixed-point theorem of multi-valued operators. (ii) Under the case that the nonlinearity f is not Lipschitz-continuous, the existence of an optimal state-control pair of (4) is obtained by utilizing an approach established in the Ref. Results obtained in this paper extend some existing research, such as that by the Refs. [6,7,8,9], and so forth

Preliminaries

Existence of Mild Solutions

Existence of Optimal Control

Conclusions