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2009 | Buch

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Observation and Control for Operator Semigroups

verfasst von: Marius Tucsnak, George Weiss

Verlag: Birkhäuser Basel

Buchreihe : Birkhäuser Advanced Texts / Basler Lehrbücher

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Observability and Controllability for Finite-dimensional Systems

Abstract

In this section we recall some basic concepts and results concerning normed vector spaces. Our aim is very modest: to list those facts which are needed in Chapter 1 (the treatment of controllability and observability for finite-dimensional systems). We do not give proofs — our aim is only to clarify our terminology and notation. A proper treatment of this material can be found in many books, of which we mention Brown and Pearcy [23], Halmos [86] and Rudin [194]. Introductions to functional analysis that stress the connections with, and applications in, systems theory are Nikol’skii [178], Partington [181] and Young [240].

Chapter 2. Operator Semigroups

Abstract

In this chapter and the following one, we introduce the basics about strongly continuous semigroups of operators on Hilbert spaces, which are also called operator semigroups for short. We concentrate on those aspects which are useful for the later chapters. As a result, there will be many glaring omissions of subjects normally found in the literature about semigroups. For example, we shall ignore analytic semigroups, compact semigroups, spectral mapping theorems and stability theory.

Chapter 3. Semigroups of Contractions

Abstract

A strongly continuous semigroup $ \mathbb{T} $ is called a contraction semigroup if % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacY % hatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-nj8 % ujab-bW90naaBaaaleaacGaxOniDaaqabaGccaGG8bGaaiiFaGqaai % aa+bcacqWFKjcHcaGFXaGae8ha3hdaaa!4E6D! \[ ||\mathbb{T}_t || \leqslant 1 \] for all t ≥0. This chapter is a continuation of the previous one: we present basic facts about unbounded operators and strongly continuous semigroups on Hilbert spaces, but now the emphasis is on contraction semigroups and their generators, which are called m-dissipative operators. We also discuss other important classes of operators (self-adjoint, positive and skew-adjoint operators) that arise as generators or as ingredients of generators of contraction semigroups.We also investigate some classes of self-adjoint differential operators: Sturm-Liouville operators and the Dirichlet Laplacian on various domains in ℝⁿ.

Chapter 4. Control and Observation Operators

Abstract

Notation. Throughout this chapter, U,X and Y are complex Hilbert spaces which are identified with their duals. $ \mathbb{T} $ is a strongly continuous semigroup on X, with generator A: D(A) → X and growth bound ω₀($ \mathbb{T} $). Recall from Section 2.10 that X₁ is D(A) with the norm ‖z‖₁ = ‖(βI - A)z‖, where β∋ρ(A) is fixed, while X₋₁ is the completion of X with respect to the norm ‖z‖₋₁ = ‖ (βI - A)⁻¹z‖. Remember that we use the notation A and $ \mathbb{T}_t $ also for the extension of the original generator to X and for the extension of the original semigroup to X₋₁. Recall also that X _d ¹ is D(A*) with the norm $ ||z||_1^d = ||(\bar \beta I - A*)z|| $ and X _d ⁻¹ is the completion of X with respect to the norm $ ||z||_{ - 1}^d = ||(\bar \beta I - A*)^{ - 1} z|| $ . Recall that X₋₁ is the dual of X _d ¹ with respect to the pivot space X.

Chapter 5. Testing Admissibility

Abstract

This chapter is devoted to results which can help to determine if an observation operator or a control operator is admissible for an operator semigroup. We use the same notation as listed at the beginning of Chapter 4.

Chapter 6. Observability

Abstract

Notation. Throughout this chapter, X and Y are complex Hilbert spaces which are identified with their duals. $ \mathbb{T} $ is a strongly continuous semigroup on X, with generator A:D(A)→X and growth bound ω₀($ \mathbb{T} $ ). Recall from Section 2.10 that X₁ is D(A) with the norm …z‖₁ = ‖(βI - A)z…, where β ∋ ρ(A) is fixed.

Chapter 7. Observation for the Wave Equation

Abstract

Notation. Throughout this chapter, Ω denotes a bounded open connected set in ℝⁿ, where n ∋ ℕ. We assume that either the boundary ∂Ω is of class C₂ or Ω is a rectangular domain. The remaining part of the notation described below is used in the whole chapter, with the exception of Section 7.6, where some notation (like X and A) will have a different meaning.

Chapter 8. Non-harmonic Fourier Series and Exact Observability

Abstract

In this chapter we show how classical results on non-harmonic Fourier series imply exact observability for some systems governed by PDEs. The method of non-harmonic Fourier series for exact observability of PDEs is essentially limited to one space dimension or to rectangular domains in ℝⁿ, since it uses that the eigenfunctions of the operator can be expressed (or approximated) by complex exponentials. We shall see that, in some of the above-mentioned cases, this method yields sharp estimates on the observability time and on the observation region.

Chapter 9. Observability for Parabolic Equations

Abstract

In this section and the following one, we shall use the notation from Section 3.4: H is a Hilbert space with the inner product 〈 ·,· 〉 and the induced norm ‖·‖. The operator A₀ : D(A₀) → H is assumed to be strictly positive. The space D(A₀) endowed with the norm ‖z‖₁ = ‖A₀z‖ is denoted by H₁ and H_1/2 is the completion of D(A₀) with respect to the norm

$$ ||w||_{\tfrac{1} {2} } = \sqrt {\langle A_0 w,w\rangle } , $$

so that H_1/2 coincides with D(A ₀ ^1/2 ) with the norm $ ||w||_{\tfrac{1} {2} } = ||A_0^{\tfrac{1} {2}} w|| $. We have seen in Proposition 3.8.5 that -A₀ generates an exponentially stable semigroup $ \mathbb{S} $ on H.

Chapter 10. Boundary Control Systems

Abstract

Notation. We continue to use the notation listed at the beginning of Chapter 2. As in earlier chapters, if $ \mathbb{T} $ is a strongly continuous semigroup on the Hilbert space X, with generator A, then the spaces X₁ and X₋₁ are as in Section 2.10 and the extension of A to X is still denoted by A.

Chapter 11. Controllability

Abstract

Notation. Throughout this chapter, U, X and Y are complex Hilbert spaces which are identified with their duals. $ \mathbb{T} $ is a strongly continuous semigroup on X, with generator A : D(A)→X and growth bound ω₀ ($ \mathbb{T} $). Remember that we use the notation A and $ \mathbb{T}_t $ also for the extension of the original generator to X and for the extension of the original semigroup to X₋₁. Recall also that X ₁ ^d is D(A^*) with the norm $ ||z||_1^d = ||(\bar \beta I - A*)z|| $ and X ₋₁ ^d is the completion of X with respect to the norm $ ||z||_{ - 1}^d = ||(\bar \beta I - A*)^{ - 1} z|| $. Recall that X₋₁ is the dual of X ₁ ^d with respect to the pivot space X.

Chapter 12. Appendix I: Some Background on Functional Analysis

Abstract

In this section we state the closed-graph theorem without proof, and then we prove a few applications that are needed in the book.

Chapter 13. Appendix II: Some Background on Sobolev Spaces

Abstract

In this chapter we introduce some concepts about distributions, Sobolev spaces and differential operators acting on such spaces. For a more solid grounding the reader should consult Adams [1], Brezis [22], Dautray and Lions [42, 43], Grisvard [77], Hörmander [101], Lions and Magenes [157], Neças [176] and Zuily [246]. Starting from Section 13.5 we assume that the reader knows some basic concepts about differentiable manifolds, as can be found, for instance, in Spivak [208].

Chapter 14. Appendix III: Some Background on Differential Calculus

Abstract

The aim of this chapter is to provide an elementary proof of Theorem 9.4.3, after introducing the necessary tools from differential calculus. First we recall some basic concepts and prove a classical result of Sard. Then we give the detailed construction of n₀ from Theorem 9.4.3. Our method requires only a particular case of Sard’s theorem (which is proved below). We refer the reader to Coron [36, Lemma 2.68] and Fursikov and Imanuvilov [69, Lemma 1.1] for related proofs.

Chapter 15. Appendix IV: Unique Continuation for Elliptic Operators

Abstract

In this section we provide an elementary proof of a Carleman estimate for secondorder elliptic operators. As it has already been remarked by Carleman in [29], this kind of estimates provides a powerful tool for proving unique continuation results for linear elliptic PDEs. Our approach is essentially based on Burq and Gérard [26]. More sophisticated versions of Carleman estimates are currently applied to quite general linear partial differential operators (see, for instance, Hörmander [103], Fursikov and Imanuvilov [69], Tataru [214, 216], Imanuvilov and Puel [106] and Lebeau and Robbiano [151, 152]).

Backmatter

Titel: Observation and Control for Operator Semigroups
verfasst von: Marius Tucsnak
George Weiss
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8994-9
Print ISBN: 978-3-7643-8993-2
DOI: https://doi.org/10.1007/978-3-7643-8994-9