In this paper we consider the heat equation with memory in a bounded region $\Omega \subset\mathbb{R}^d$, $d\geq 1$, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class $C^1$. We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of $\Omega$. We prove approximate controllability of the system and, in contrast with this, we prove the existence of initial conditions which cannot be steered to hit the target $0$ in a certain time $T$, of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the heat equation.

Approximate controllability and lack of controllability to zero of the heat equation with memory / A., Halanay; Pandolfi, Luciano. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - STAMPA. - 425:(2015), pp. 194-211. [10.1016/j.jmaa.2014.12.021]

Approximate controllability and lack of controllability to zero of the heat equation with memory

PANDOLFI, LUCIANO
2015

Abstract

In this paper we consider the heat equation with memory in a bounded region $\Omega \subset\mathbb{R}^d$, $d\geq 1$, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class $C^1$. We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of $\Omega$. We prove approximate controllability of the system and, in contrast with this, we prove the existence of initial conditions which cannot be steered to hit the target $0$ in a certain time $T$, of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the heat equation.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2585568
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