Appl Math Optim (2011) 64:467–468 DOI 10.1007/s00245-011-9149-6 ERRATUM Erratum to: The Controllability of the Gurtin-Pipkin Equation: A Cosine Operator Approach Luciano Pandolﬁ Published online: 20 September 2011 © Springer Science+Business Media, LLC 2011 Erratum to: Appl Math Optim (2005) 52: 143–165 DOI 10.1007/s00245-005-0819-0 −1 ⊥ ⊥ Lemma 18 states that A [R ] ⊆[R ] . Its proof is based on Lemma 17 which ∞ ∞ is not correct since an integral in the (sketched) computations does not cancel out. A proof of Lemma 18 which does not use Lemma 17 is as follows. Using formula (7), the Laplace transform of θ(t) with θ(0) = 0is −1 θ(λ) =−A I − A Du(λ). ˆ (1) b(λ) −t Let u(t ) = u e . For every λ (in a right half-plane) and ξ ⊥ R we have 0 ∞ −1 1 λ 0 =−ξ, θ(λ)= ξ, A I − A Du , ∀u ∈ U. 0 0 1 + λ b(λ) The assumptions on b(t ) imply that this equality can be extended by continuity to λ = 0 and for λ = 0we have ξ, Du = 0 for every u ∈ U . Hence, if ξ ⊥
Applied Mathematics and Optimization – Springer Journals
Published: Dec 1, 2011
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