Infinite dimensional systems with unbounded control and observation operators : an introduction

01/10/2017
Auteurs : Marius Tucsnak
Publication e-STA e-STA 2004-2
OAI : oai:www.see.asso.fr:545:2004-2:20062
DOI :

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Infinite dimensional systems with unbounded control and observation operators : an introduction

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Infinite dimensional systems with unbounded control and observation operators : an introduction Marius TUCSNAK June 24, 2004 2 Contents 3 4 CONTENTS Chapter 1 Semigroups of linear operators in Hilbert spaces 1.1 Dissipative and m-dissipative linear operators in Hilbert spaces In this section we recall some basic facts concerning unbounded linear operators acting in a Hilbert space and in particular dissipative and m-dissipative operators. If H is a Hilbert space we denote (·, ·) the inner product in H, by | · | the corre- sponding norm in H and by k·k the associated norm in the space of linear bounded operators from H to H, denoted by L(H). Definition 1.1.1. Let H be a Hilbert space and D(A) ⊂ H be a vector space. A linear operator A : D(A) → H is called an unbounded linear operator in H. Definition 1.1.2. The unbounded operator A : D(A) → H is closed if its graph G(A) = {(f, Af)|f ∈ D(A)} is closed in H × H. Remark 1.1.3 It can be easily checked that A is closed iff for any sequence (fn) ⊂ D(A) such that fn → f and Afn → g we have f ∈ D(A) and Af = g. Definition 1.1.4. The unbounded operator A : D(A) → H is called dissipative if (Av, v) ≤ 0, ∀ v ∈ D(A). The dissipative operator A is called m-dissipative if R(I − A) = H, i.e ∀f ∈ H, ∃u ∈ D(A) such that u − Au = f. Notice first that dissipative operators can be characterized as follows 5 6CHAPTER 1. SEMIGROUPS OF LINEAR OPERATORS IN HILBERT SPACES Proposition 1.1.5. The operator A : D(A) → H is dissipative if and only if |z − µAz| ≥ |z|, ∀ (z, µ) ∈ D(A) × (0, ∞). (1.1.1) Proof. Suppose first that A is dissipative. Then |z − µAz|2 = |z|2 − 2µ(Az, z) + µ2 |Az|2 ≥ |z|2 , i.e. (??) holds true. On the other hand suppose that (??) holds true. Then, for all µ > 0 we have (Az, z) − µ 2 |Az|2 = |z|2 − |z − µAz|2 2µ ≤ 0. For µ → 0+ inequality above implies that A is dissipative. The following simple property of dissipative operators will be used in Section ??. Proposition 1.1.6. Let A be a dissipative operator in the Hilbert space H. More- over, suppose that A is closed. Then the range of I − A is a closed subspace of H. Proof. Let (fn) ⊂ (I − A) [D(A)], with fn → f in H. Then there exists a sequence (zn) ⊂ D(A) such that zn − Azn = fn, ∀ n ≥ 1. (1.1.2) The convergence of (fn) in H and the dissipativity of A imply, by (??), that zn → z in H. Moreover, (??) implies that Azn → z − f in H. Since A is closed it follows that z ∈ D(A) and that Az = z − f, which clearly implies that f is in the range of I − A. The first properties of m-dissipative operators are given in the result below. Proposition 1.1.7. Let A be an m-dissipative operator. Then the following asser- tions hold true (a) D(A) is dense in H. (b) A is closed (c) For all λ > 0 the operator λI − A is an isomorphism from D(A) onto H. Moreover (λI − A)−1 is a linear bounded operator such that k(λI − A)−1 k ≤ 1 λ . 1.1. DISSIPATIVE OPERATORS 7 Proof. (a) Let f ∈ H be such that (f, v) = 0 for all v ∈ D(A). Since I − A is onto it follows that there exists v0 ∈ D(A) such that v0 − Av0 = f. Relations above imply that 0 = (f, v0) = |v0|2 − (Av0, v0) ≥ |v0|2 . This implies that v0 = 0 so f = 0 which yields the assertion(a) of the Proposition. (b) It can be easily checked that for all f ∈ H there exists a unique z ∈ D(A) such that z − Az = f. Moreover we have |z|2 − (Az, z) = (f, z), which implies that |z| ≤ |f|. This means that (I −A)−1 is a linear bounded operator in H satisfying k(I − A)−1 k ≤ 1. (1.1.3) Let now (zn) ⊂ D(A) be a sequence such that zn → z and Azn → f. Since (I −A)−1 is bounded we obtain zn = (I − A)−1 (zn − Azn) → (I − A)−1 (z − f). It follows that z = (I − A)−1 (z − f) i.e. z ∈ D(A) and z − Az = z − f. (c) Suppose that there exists µ0 > 0 such that R(I−µ0A) = H. We will show that R(I −µA) = H for all µ > µ0 2 . As in (b) we clearly have that k(I −µ0A)−1 kL(H) ≤ 1. Consider the equation z − µAz = f, with µ > 0. (1.1.4) Equation (??) can be rewritten as z − µ0Az = µ0 µ f + 1 − µ0 µ ! z, which is equivalent to z = (I − µ0A)−1 " µ0 µ f + 1 − µ0 µ ! z # , (1.1.5) For µ > µ0 2 we have 1 − µ0 µ < 1 so, by the Banach fixed point theorem, equation (??) has a unique solution z ∈ D(A). It follows that equation (??) admits a unique solution. We can now finish the proof of assertion (c). Indeed since I − A is onto, consid- erations above imply that I − µA is onto for µ > 1 2 and then for µ > 1 4 and so on. By induction we obtain that I − µA is onto for all µ > 0. If we denote λ = 1 µ we obtain that λI − A is onto, for all λ > 0. Moreover, since λI − A is clearly injective we conclude that, for all λ > 0, the operator λI − A is an isomorphism from D(A) onto H. Moreover, if we take the inner product of the equation λz − Az = f by z and we use the dissipativity of A we obtain that λ|z|2 ≤ (f, z). By applying now the Cauchy-Schwartz inequality we get the inequality |z| ≤ 1 λ |f|. This ends up the proof of the assertion (c). 8CHAPTER 1. SEMIGROUPS OF LINEAR OPERATORS IN HILBERT SPACES Let A be an m-dissipative operator, λ > 0 and denote R(λ; A) = (λI − A)−1 , Aλ = λAR(λ; A) = λ2 R(λ; A) − λI. (1.1.6) Definition 1.1.8. R(λ; A) is called the resolvent of A and Aλ is called the Yosida approximation of A. From Proposition ?? we know that kR(λ, A)k ≤ 1 λ . (1.1.7) Proposition 1.1.9. Let A be an m-dissipative operator. Then the following asser- tions hold true: a) Aλv = λR(λ; A)Av, ∀ (v, λ) ∈ D(A) × (0, ∞); b) limλ→∞ λR(λ; A)v = v, ∀ v ∈ H. c) limλ→∞ Aλv = Av, ∀ v ∈ D(A); d) (Aλv, v) ≤ 0, ∀ (v, λ) ∈ H × (0, ∞); e) |Aλv| ≤ λ|v|, ∀ (v, λ) ∈ H × (0, ∞); Proof. a) From the definition of R(λ; A) we obtain R(λ; A)Av = −v + λR(λ; A)v = 1 λ Aλv, ∀ v ∈ D(A), which clearly implies assertion a). b) Suppose first that v ∈ D(A). Then we have |λR(λ; A)v − v| = |AR(λ; A)v| =, = |R(λ; A)Av| ≤ 1 λ |Av| → 0 as λ → ∞. But D(A) is dense in H and kλR(λ; A)k ≤ 1, so we get the conclusion b). c) From a) and b) it follows that lim λ→∞ Aλv = lim λ→∞ λR(λ; A)(Av) = Av, for all v ∈ D(A). d) We have (Aλv, v) = (Aλv, v − λR(λ; A)v) + λ (Aλv, R(λ; A)v) = − 1 λ |Aλv|2 + λ2 (AR(λ; A)v, R(λ; A)v) . Relation above implies that (Aλv, v) ≤ − 1 λ |Aλv|2 , (1.1.8) which clearly implies that assertion d) holds true. e) The result clearly follows from (??) and the Cauchy Schwartz inequality. 1.2. DEFINITION AND FIRST PROPERTIES 9 1.2 Definition and first properties of linear semigroups Definition 1.2.1. Let H be a Hilbert space. A one parameter family S(t)t≥0, of bounded linear operators from H into H is a semigroup of bounded linear operators on H if • S(0) = I, • S(t + s) = S(t)S(s) for every t, s ≥ 0 (the semigroup property). The linear operator A defined by D(A) = ( z ∈ H| lim t→0+ S(t)z − z t exists ) and Az = lim t→0+ S(t)z − z t , ∀ z ∈ D(A), is called the infinitesimal generator of the semigroup S(t) and D(A) is called the domain of A. Definition 1.2.2. A semigroup S(t) of bounded linear operators is called uniformly continuous if lim t→0+ kS(t) − Ik = 0. Proposition 1.2.3. A linear operator A is the infinitesimal generator of a uni- formly continous semigroup if and only if A is a bounded linear operator. Proof. Let A ∈ L(H) and set S(t)