On the observability of abstract time discrete linear parabolic equations

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Niveau: Supérieur, Doctorat, Bac+8
On the observability of abstract time-discrete linear parabolic equations Sylvain Ervedoza? and Julie Valein† March 26, 2009 Abstract This article aims at analyzing the observability properties of time- discrete approximation schemes of abstract parabolic equations z˙+Az = 0, where A is a self-adjoint positive definite operator with dense domain and compact resolvent. We analyze the observability properties of these diffusive systems for an observation operator B ? L(D(A?), Y ) with ? < 1/2. Assuming that the continuous system is observable, we prove uni- form observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type al- gorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly ob- servable. In particular, our results can be combined with the existing ones on the observability of space semi-discrete systems, yielding observability properties for fully discrete approximation schemes. Keywords: Time discretization, Observability, Controllability, Parabolic equa- tions, Filtering techniques. Mathematics Subject Classifications: 35K05, 93B05, 93B07, 93B40, 93C55. 1 Introduction Let X be a Hilbert space endowed with the norm ?·?X and let A : D(A) ? X be a positive definite self-adjoint operator with dense domain and compact resolvent.

• observability properties


• k? ?


• such assumptions


• time discretization schemes


• discrete approximation


• c?


• observation operator


• efficient computational technique


• self-adjoint positive

Sujets

• United States
• Uniqueness quantification
• Inverse problem

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 11
• Langue: English

Extrait

On

the

observability of abstract time-discrete linear parabolic equations Sylvain Ervedoza ∗ and Julie Valein †

March 26, 2009

Abstract This article aims at analyzing the observability properties of time- discrete approximation schemes of abstract parabolic equationsz˙ +Az= 0, whereAis a self-adjoint positive definite operator with dense domain and compact resolvent. We analyze the observability properties of these diffusive systems for an observation operatorB∈L(D(A ν ), Y)withν < 1/2. Assuming that the continuous system is observable, we prove uni- form observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type al- gorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly ob- servable. In particular, our results can be combined with the existing ones on the observability of space semi-discrete systems, yielding observability properties for fully discrete approximation schemes.

Keywords:Time discretization, Observability, Controllability, Parabolic equa- tions, Filtering techniques.

Mathematics Subject Classifications:35K05, 93B05, 93B07, 93B40, 93C55.

1 Introduction

LetXbe a Hilbert space endowed with the normk∙k X and letA:D(A)→ Xbe a positive definite self-adjoint operator with dense domain and compact resolvent. We consider the following abstract system:

z˙(t) +Az(t) = 0, t∈[0, T], z(0) =z 0 .(1.1) ∗ Mathématiques de Versailles, Université de Versailles Saint-Quentin enLaboratoire de Yvelines, 45 avenue des États Unis, 78035 Versailles Cedex, [e-mail]. † Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Insti- tut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France, [e-mail].

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Here and henceforth, a dot(˙)denotes differentiation with respect to the time t. The elementz 0 ∈Xis called theinitial state, andz=z(t)is thestateof the system. Such equations are often used as models for diffusive systems and especially heat equations. Assume thatYis another Hilbert space equipped with the normk∙k Y . We denote byL(X, Y)the space of bounded linear operators fromXtoY, endowed with the classical operator norm. LetB∈L(D(A ν ), Y), withν≤1/2, be an observation operator, and define the output function

y(t) =Bz(t).

(1.2)

To give a sense to (1.2), we will assume thatBis an admissible observation operator, i.e. for everyT >0there exists a constantK T >0such that any solution of system (1.1) with initial dataz 0 ∈ D(A)satisfies T Z kBz(t)k 2 Y dt≤K T kz 0 k 2 X .(1.3) 0

Under this assumption, the output functionyin (1.2) is well-defined as a func- tion inL 2 ((0, T);Y)any solution of (1.1) with initial datafor z 0 ∈X. Actually, this property is automatically satisfied whenB∈L(D(A ν ), Y) withν≤1/2(see, e.g., [25] and Theorem 2.2 below), which we will always assume in the following. The exact observability property for system (1.1)-(1.2) can be formulated as follows:

Definition 1.1.System(1.1)-(1.2)is exactly observable in timeT ∗ if there existsk ∗ >0such that any solution of system(1.1)with initial dataz 0 ∈X satisfies ∗ 2 k ∗ kz(T ∗ )k X ≤ Z 0 T kBz(t)k 2 Y dt.(1.4) Moreover, system(1.1)-(1.2)to be exactly observable if it is exactly ob-is said servable in some timeT ∗ >0.

Inequalities (1.3) and (1.4) are relevant in controllability theory due to the duality argument given by the Hilbert Uniqueness Method (HUM in short), see [18] and Section 4. In the following, we assume that (1.4) holds for the continuous system. Such results have been proved, often by means of Carleman estimates, for various models including the heat equation [10, 12, 16], the Stokes equations [9], and some other singular models such as [3, 7, 20, 24].

This article aims at studying the admissibility and observability properties for time-discrete approximation schemes of (1.1)-(1.2), and corresponding con- trollability properties.

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