Publisher DOI: 10.1137/19m1266769
Title: Sufficient criteria and sharp geometric conditions for observability in banach spaces
Language: English
Authors: Gallaun, Dennis 
Seifert, Christian  
Tautenhahn, Martin 
Keywords: Banach space; C0-semigroups; Control costs; Elliptic operators; Null-controllability; Observability estimate
Issue Date: 2020
Source: SIAM Journal on Control and Optimization 58 (4): 2639-2657 (2020)
Abstract (english): 
Let X, Y be Banach spaces, (St)t≥ 0 a C0-semigroup on X, A the corresponding infinitesimal generator on X, C a bounded linear operator from X to Y , and T 0. We consider the system x(̇t) = Ax(t), y(t) = Cx(t), t ∈ (0, T], x(0) = x0 ∈ X. We provide sufficient conditions such that this system satisfies a final state observability estimate in Lr((0, T); Y ), r ∈ [1,∞ ]. These sufficient conditions are given by an uncertainty relation and a dissipation estimate. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider the example where A is an elliptic operator in Lp(Rd) for 1 p ∞ and where C = 1E is the restriction onto a thick set E ⊂ Rd. In this case, we show that the above system satisfies a final state observability estimate if and only if E ⊂ Rd is a thick set. Finally, we make use of the well-known relation between observability and null-controllability of the predual system and investigate bounds on the corresponding control costs.
ISSN: 1095-7138
Journal: SIAM journal on control and optimization 
Institute: Mathematik E-10 
Document Type: Article
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