Journal Article

Controllability of linear difference equations in Hilbert spaces and applications

Hugo Leiva and Jahnett Uzcategui

in IMA Journal of Mathematical Control and Information

Published on behalf of Institute of Mathematics and its Applications

Volume 25, issue 3, pages 323-340
Published in print September 2008 | ISSN: 0265-0754
Published online February 2008 | e-ISSN: 1471-6887 | DOI: https://dx.doi.org/10.1093/imamci/dnm027
Controllability of linear difference equations in Hilbert spaces and applications

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In this paper, we present necessary and sufficient conditions for the exact and approximate controllability of the following linear difference equation:where Z, U are Hilbert spaces, A(·) ∈ l(, L(Z)), B(·) ∈ l(, L(U, Z)), ul2(, U) and * = ∪ {0}. Moreover, in the case of exact controllability, the control ul2(, U) steering an initial state z0 to a final state z1 in time n0 is given by the 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, zZ, uU, t > 0, where BL (U, Z), uL2(0, τ;U) and A is the infinitesimal generator of a strongly continuous semigroup {T(t)}t ≥ 0 in Z, given byaccording 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

Journal Article.  0 words. 

Subjects: Mathematics

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