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Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

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Abstract

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.

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Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments. We also acknowledge the valuable comments and suggestions from the editors. Finally, J. Wang acknowledges the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169); M. Fec̆kan acknowledges the support by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0124/12 and APVV-0414-07 and Y. Zhou acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

    JinRong Wang

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48, Slovakia

    Michal Fec̆kan

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

    Michal Fec̆kan

  4. Department of Mathematics, Xiangtan University, Xiangtan, Hunan, 411105, P.R. China

    Yong Zhou

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  1. JinRong Wang
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  2. Michal Fec̆kan
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  3. Yong Zhou
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Correspondence to Michal Fec̆kan.

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Communicated by Franco Giannessi.

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Wang, J., Fec̆kan, M. & Zhou, Y. Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations. J Optim Theory Appl 156, 13–32 (2013). https://doi.org/10.1007/s10957-012-0170-y

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  • DOI: https://doi.org/10.1007/s10957-012-0170-y

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