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IMA Journal of Mathematical Control and Information Advance Access originally published online on May 15, 2007
IMA Journal of Mathematical Control and Information 2008 25(3):269-280; doi:10.1093/imamci/dnm017
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications

Edgar Iturriaga and Hugo Leiva{dagger}

Universidad de Los Andes, Departamento de Matemáticas, Mérida 5101-Venezuela

{dagger} Email: hleiva{at}ula.ve

Received on February 28, 2006; As we have announced in the title of this work, we show that a broad class of linear evolution equations is exactly controllable. This class is represented by the following infinite-dimensional linear control system:

Formula
where Z, U are Hilbert spaces, the control function u belong to L2(0, t1;U), t1 > 0, Formula and Formula generates a strongly continuous semigroup operator T(t) according to Pazy. We give a necessary and sufficient condition for the exact controllability of this system and apply this result to a linear controlled damped wave equation.

Keywords: linear evolution equations; exact controllability; damped wave equation.


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