Approximate controllability and optimal controls of fractional evolution systems in abstract spaces
- PDF / 442,948 Bytes
- 22 Pages / 595.276 x 793.701 pts Page_size
- 32 Downloads / 211 Views
RESEARCH
Open Access
Approximate controllability and optimal controls of fractional evolution systems in abstract spaces Haiyong Qin1 , Jianwei Liu1* , Xin Zuo1 and Lishan Liu2 *
Correspondence: [email protected] Department of Automation, China University of Petroleum (Beijing), Changping, Beijing 102249, China Full list of author information is available at the end of the article 1
Abstract In this paper, under the assumption that the corresponding linear system is approximately controllable, we obtain the approximate controllability of semilinear fractional evolution systems in Hilbert spaces. The approximate controllability results are proved by means of the Hölder inequality, the Banach contraction mapping principle, and the Schauder fixed point theorem. We also discuss the existence of optimal controls for semilinear fractional controlled systems. Finally, an example is also given to illustrate the applications of the main results. MSC: 26A33; 49J15; 49K27; 93B05; 93C25 Keywords: fractional evolution systems; approximate controllability; optimal controls; semigroup theory; fixed point theorem
1 Introduction During the past few decades, fractional differential equations have proved to be valuable tools in the modeling of many phenomena in viscoelasticity, electrochemistry, control, porous media, and electromagnetism, etc. Due to its tremendous scopes and applications, several monographs have been devoted to the study of fractional differential equations; see the monographs [–]. Controllability is a mathematical problem. Since approximately controllable systems are considered to be more prevalent and very often approximate controllability is completely adequate in applications, a considerable interest has been shown in approximate controllability of control systems consisting of a linear and a nonlinear part [–]. In addition, the problems associated with optimal controls for fractional systems in abstract spaces have been widely discussed [–]. Wang and Wei [] obtained the existence and uniqueness of the PC-mild solution for one order nonlinear integrodifferential impulsive differential equations with nonlocal conditions. Bragdi [] established exact controllability results for a class of nonlocal quasilinear differential inclusions of fractional order in a Banach space. Machado et al. [] considered the exact controllability for one order abstract impulsive mixed point-type functional integro-differential equations with finite delay in a Banach space. Approximate controllability for one order nonlinear evolution equations with monotone operators was attained in []. By the wellknown monotone iterative technique, Mu and Li [] obtained existence and uniqueness results for fractional evolution equations without mixed type operators in nonlinearity. © 2014 Qin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any mediu
Data Loading...
