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Exact controllability of the heat equation with hyperbolic memory kernel. (English) Zbl 1085.35038

Imanuvilov, Oleg (ed.) et al., Control theory of partial differential equations. Papers of the conference, Washington, D. C., USA, May 30 – June 1, 2003. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-2546-8/pbk). Lecture Notes in Pure and Applied Mathematics 242, 387-401 (2005).
In this paper, the authors study the exact controllability of the following controlled system \[ y_t-\nabla\cdot \int^t_0a(t-s)\nabla y(s,x)\,ds= u\chi_w\text{ in }Q, \tag{1} \]
\[ y=0\text{ on }\Sigma,\quad y(0)=y_0\text{ in }\Omega. \] Because of its hyperbolic nature, they show the exact controllability of the (1) under suitable conditions on the waiting time \(T\) and the controller \(w\). On the other hand, because of its finite propagation speed, the exact controllability of (1) is impossible unless \(T\) is large enough. Also, a Gaussian beam construction of highly localized solutions for the dual system of (1) show that the exact controllability of (1) is impossible without geometric conditions on the controller \(w\). Consequently, their results show that the controllability proper of (1) is very similar to that of the usual wave equation.
For the entire collection see [Zbl 1066.93001].

MSC:

35B37 PDE in connection with control problems (MSC2000)
93B05 Controllability
45K05 Integro-partial differential equations
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