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6.2. Generalization of Lemma 4.13 see [22, Proposition 3.3.5, pag 76]. Now we introduce ℓ = k + α. − � ∞ 0 2 α−1 e −t/2 L α n(t)e At/2 x0dt = n+α � ℓ=α 2 ℓ (n + α)! ℓ!(n + α − ℓ)! (A − I)α−1−ℓ x0 � n+α � = (A − I) α−1 α−1 � − = (A − I) α−1 ℓ=0 (n + α)! ℓ!(n + α − ℓ)! 2ℓ (A − I) −ℓ (n + α)! ℓ!(n + α − ℓ)! ℓ=0 2ℓ (A − I) −ℓ � α−1 � − ℓ=0 (I + 2 (A − I) −1 ) n+α = A n+α d (A − I) α−1 α−1 � x0 − ℓ=0 � x0 (n + α)! ℓ!(n + α − ℓ)! 2ℓ (A − I) −ℓ � x0 (n + α)! ℓ!(n + α − ℓ)! 2ℓ (A − I) α−1−ℓ x0. where we used the binomial theorem in the second last step and equation (1.17) in the last step. Thus equation 6.14 holds. � Remark 6.8 To give some insight into equation (6.9), we specify the equation for some values of α. First the case α = 0: − The case α = 1: � ∞ 0 − 1 2 e−t/2 L 0 n(t)e At/2 x0dt = A n d (A − I) −1 x0. (6.11) � ∞ 0 e −t/2 L 1 n(t)e At/2 x0dt = A n+1 d x0 − x0. (6.12) This is the same equation as equation (4.19). Hence Lemma 4.13 is a special case of Lemma 6.7. The case α = 2: − � ∞ 0 2e −t/2 L 2 n(t)e At/2 x0dt =A n+2 d (A − I) x0 − (A − I) x0 + 2(n + 2)x0. (6.13) Note that for every step in α, on the right-hand side an extra term is added. 79
Chapter 6. Norm relations using Laguerre polynomials In Chapter 4 we derived equation (6.12) in Lemma 4.13. In Theorem 4.14 we used this equation to obtain a norm relation between semigroups and their cogenerators. To deal with the −x0 term on the right-hand side of equation (6.12), we compared the difference of two semigroups. If such a norm relation can be obtain using equation (6.13) is not a easy question, since there is no easy trick to deal with the two extra terms on the right-hand side. The same holds for α ≥ 3. On the other hand, with equation (6.11) it is possible to obtain a norm equality between the semigroup and the cogenerator. In Section 6.3 this norm equality is derived as a special case of Theorem 6.11. 6.3 Norm equality In Section 6.2 we derived a relation between the semigroup and the cogenerator in Lemma 6.7. In the case α = 1 we used equation 6.9 to obtain a norm relation, see Theorem 4.14. In this section we derive a relation between the semigroup and the cogenerator as well. However, now we add a t α term to the integral on the left-hand side of equation (6.14). In this way we can obtain a norm relation for general α. Note in this section α ∈ R. As in Section 6.2, we use the Laguerre polynomials to transform the semigroup into the cogenerator. Lemma 6.9 Let n ≥ 0, let α > −1 and let A generate a semigroup with growth bound ω < 1. Further let L α n(t) be the Laguerre polynomials, see (6.1), and let {bn}n≥0 be given by (6.4). Then for every x0 ∈ X the following relation holds, 80 � ∞ bne 0 −t/2 t α L α n(t)e At/2 x0dt = A n d (I − A) −(α+1) x0, x0 ∈ X. (6.14)
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Stability analysis in continuous an
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STABILITY ANALYSIS IN CONTINUOUS AN
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This thesis is dedicated to my pare
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Preface side(s) of the North Sea an
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Contents 5 Extension Bergman distan
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Chapter 1. Introduction This family
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Chapter 1. Introduction In this the
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Chapter 1. Introduction 600 500 400
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Chapter 1. Introduction Lemma 1.3 L
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Chapter 1. Introduction A contracti
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Chapter 1. Introduction 1.5 General
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Chapter 1. Introduction we find tha
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Chapter 2 Stability In this chapter
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is strongly stable. However its adj
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2.1. Continuous-time case For furth
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2.2 Discrete-time case Next, we def
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2.2. Discrete-time case Remark 2.16
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2.2. Discrete-time case To prove th
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Page 111 and 112: Bibliography [10] R. F. Curtain and
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Page 115 and 116: Summary of the power sequences of t
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