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