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Controllability and observability of a heat equation with hyperbolic memory kernel |
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| Authors: | Xiaoyu Fu Xu Zhang |
| Institution: | a School of Mathematics, Sichuan University, Chengdu 610064, China b Department of Mathematics, University of Central Florida, FL 32816, USA c School of Mathematical Sciences, Fudan University, Shanghai 200433, China d Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China e Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China |
| Abstract: | The exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered. Due to the appearance of such a kind of memory, the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations, and is significantly different from that of the usual parabolic equations. By means of Carleman estimate, we establish a positive controllability and observability result under some geometric condition. On the other hand, by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory, we present a negative controllability and observability result when the geometric condition is not satisfied. |
| Keywords: | Heat equation with memory Controllability Observability estimate Carleman estimate Highly concentrated approximate solution |
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