# American Institute of Mathematical Sciences

September  2017, 6(3): 471-486. doi: 10.3934/eect.2017024

## Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions

 1 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China 2 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 842 48, Bratislava, Slovakia 3 Mathematical Institute, Slovak Academy of Sciences, Śtefánikova 49,814 73 Bratislava, Slovakia 4 Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China 5 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

* Corresponding author: Michal Fečkan

Received  October 2015 Revised  April 2017 Published  July 2017

In this paper, we study the approximate controllability of Sobolev-type fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results.

Citation: Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024
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