# Erratum to: The Controllability of the Gurtin-Pipkin Equation: A Cosine Operator Approach

Erratum to: The Controllability of the Gurtin-Pipkin Equation: A Cosine Operator Approach 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 ⊥ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Erratum to: The Controllability of the Gurtin-Pipkin Equation: A Cosine Operator Approach

, Volume 64 (3) – Dec 1, 2011
2 pages

/lp/springer_journal/erratum-to-the-controllability-of-the-gurtin-pipkin-equation-a-cosine-NPsowO8FFK
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics; Numerical and Computational Physics; Mathematical Methods in Physics; Systems Theory, Control
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-011-9149-6
Publisher site
See Article on Publisher Site

### Abstract

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 ⊥

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2011

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