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IMA Journal of Mathematical Control and Information Advance Access originally published online on February 8, 2008
IMA Journal of Mathematical Control and Information 2008 25(3):323-340; doi:10.1093/imamci/dnm027
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Controllability of linear difference equations in Hilbert spaces and applications

Hugo Leiva{dagger} and Jahnett Uzcategui

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

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

Received on June 13, 2007; Accepted on September 28, 2007

In this paper, we present necessary and sufficient conditions for the exact and approximate controllability of the following linear difference equation:

Formula
where Z, U are Hilbert spaces, A(·) isin l{infty}(Formula , L(Z)), B(·) isin l{infty}(Formula , L(U, Z)), u isin l2(Formula , U) and Formula * = Formula {cup} {0}. Moreover, in the case of exact controllability, the control u isin l2(Formula , U) steering an initial state z0 to a final state z1 in time n0 is given by the formula Formula according to Lemma 2.1. As a particular case, we consider the discretization on flow of the following controlled evolution equation z' = Az + Bu, z isin Z, u isin U, t > 0, where B isin L (U, Z), u isin L2(0, {tau};U) and A is the infinitesimal generator of a strongly continuous semigroup {T(t)}t ≥ 0 in Z, given by

Formula
according to Lemma 1.1. These results are applicable to a broad class of reaction–diffusion systems such as the heat equation, the wave equation, the equation modelling the damped flexible beam, the strongly damped wave equation, the thermoelastic plate equation, etc. In Section 4, these results are applied to a discrete version of the n-dimensional heat and n-dimensional wave equation.

Keywords: difference equations; exact controllability; approximate controllability; heat and wave equation.


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