Abstract and Applied Analysis
Volume 2012 (2012), Article ID 850529, 20 pages
http://dx.doi.org/10.1155/2012/850529
Research Article

Control Systems Described by a Class of Fractional Semilinear Evolution Equations and Their Relaxation Property

Xiaoyou Liu1 and Xi Fu2

1School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075, China
2Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China

Received 29 May 2012; Accepted 16 December 2012

Academic Editor: Irena Lasiecka

Copyright © 2012 Xiaoyou Liu and Xi Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a control system described by a class of fractional semilinear evolution equations in a separable reflexive Banach space. The constraint on the control is a multivalued map with nonconvex values which is lower semicontinuous with respect to the state variable. Along with the original system we also consider the system in which the constraint on the control is the upper semicontinuous convex-valued regularization of the original constraint. We obtain the existence results for the control systems and the relaxation property between the solution sets of these systems.