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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 385324, 16

Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 385324, 16 pages doi:10.1155/2011/385324 Research Article Study of an Approximation Process of Time Optimal Control for Fractional Evolution Systems in Banach Spaces JinRong Wang1 and Yong Zhou2 1 2 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China Correspondence should be addressed to Yong Zhou, yzhou@xtu.edu.cn Received 1 October 2010; Accepted 9 December 2010 Academic Editor: J. J. Trujillo Copyright q 2011 J. Wang and Y. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,. | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 385324 16 pages doi 10.1155 2011 385324 Research Article Study of an Approximation Process of Time Optimal Control for Fractional Evolution Systems in Banach Spaces JinRong Wang1 and Yong Zhou2 1 Department of Mathematics Guizhou University Guiyang Guizhou 550025 China 2 Department of Mathematics Xiangtan University Xiangtan Hunan 411105 China Correspondence should be addressed to Yong Zhou yzhou@xtu.edu.cn Received 1 October 2010 Accepted 9 December 2010 Academic Editor J. J. Trujillo Copyright 2011 J. Wang and Y. Zhou. 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. The paper is devoted to the study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. We firstly convert time optimal control problem into Meyer problem. By virtue of the properties of the family of solution operators given by us the existence of optimal controls for Meyer problem is proved. Secondly we construct a sequence of Meyer problems to successive approximation of the original time optimal control problem. Finally a new approximation process is established to find the solution of time optimal control problem. Our method is different from the standard method. 1. Introduction It has been shown that the accurate modelling in dynamics of many engineering physics and economy systems can be obtained by using fractional differential equations. Numerous applications can be found in viscoelasticity electrochemistry control porous media electromagnetic and so forth. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical sense. One can see the monographs of Kilbas et al. 1 Miller and Ross 2 Podlubny 3 and Lakshmikantham etal. 4 . The .