Abstract
We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of ℝn. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder’s fixed point theorem.
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This work was supported by the Natural Science Foundation of China (No.10371136, 10771222).
Yuqing YAN was born in 1981. He received his B.S. degree in 2003 from Zhejiang Normal University. Now, he is a Ph.D. student in the Department of Mathematics of Sun Yat-Set University. His research interests are mainly in the areas of controllability of PDE and insesitizing controls.
Yi ZHAO was born in 1940. He received his B.S. degree in 1963 from Sun Yat-Set University. Now, he is currently a professor in Department of Mathematics, Sun Yat-Sen University. His research interests conclude infinite dimensional dynamical systems and control systems.
Yu HUANG received the B.S. degree in 1983, and M.S. degree in 1986, both in Mathematics Department from Zhongshan (Sun, Yat-Sen) University, Guangzhou, China. In 1995, he received his Ph.D. degree in Mathematics Department from the Chinese University of HongKong, Hongkong. Since July 1986, he has been a faculty member in Mathematics Department at Zhongshan university, where he is currently a professor. He is an associate editor of “Journal of Mathematical Analysis and Application”. His areas of current research interest include dynamical systems, distributed parameter control systems and switched linear systems.
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Yan, Y., Zhao, Y. & Huang, Y. The cost of approximate controllability for semilinear heat equations. J. Control Theory Appl. 7, 73–76 (2009). https://doi.org/10.1007/s11768-009-6153-3
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DOI: https://doi.org/10.1007/s11768-009-6153-3