Lack of controllability of thermal systems with memory

Andrei Halanay; Luciano Pandolfi

doi:10.3934/eect.2014.3.485

Article Contents

2014, Volume 3, Issue 3: 485-497. Doi: 10.3934/eect.2014.3.485

This issue Previous Article Constructing free energies for materials with memory Next Article A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$

Lack of controllability of thermal systems with memory

Andrei Halanay^1, and
Luciano Pandolfi^2,

1.
Department of Mathematics and Informatics, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest
2.
Dipartimento di Scienze Matematiche "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: April 30, 2013

Revised: December 31, 2013

Published: August 2014

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Abstract

Heat equations with memory of Gurtin-Pipkin type (i.e. Eq. (1) with $ \alpha=0 $) have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when $ \alpha>0 $ the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are square integrable initial conditions which cannot be controlled to zero. The proof of this fact, in previous papers, consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every smooth memory kernel $ M(t) $.

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Mathematics Subject Classification: Primary: 35Q93, 45K05; Secondary: 93B03.

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