On Characterization of Inverse Data in the Boundary Control Method

2016/02/15 by Mikhail Belishev, Belishev, Mikhail, Aleksei Vakulenko +1
Mathematics · Physics and Astronomy · #35L51 #35Qxx #35R30 #Analysis of PDEs (math.AP) #F.2.2 #FOS: Mathematics #FOS: Physical sciences #I.2.7 #Mathematical Physics (math-ph) #acm:35L51 #acm:35Qxx #acm:35R30 #math-ph #math.AP #math.MP #msc:35L51 #msc:35Qxx #msc:35R30

paper · pdf · doi:10.48550/arxiv.1602.05066

33 pages, 1 figure

arxiv created 2016/02/15 · arxiv updated 2016/02/17

Abstract

We deal with a dynamical system u_tt-Δu+qu=0 & \rm in Ω× (0,T)
u|_t=0=u_t|_t=0=0 & \rm in Ω
∂_νu = f & \rm in ∂Ω× [0,T] , where Ω⊂ \mathbb Rⁿ is a bounded domain, q ∈ L_∞(Ω) a real-valued function, ν the outward normal to ∂ Ω, u=u^f(x,t) a solution. The input/output correspondence is realized by a response operator R^T: f ↦ u^f|_{∂Ω× [0,T]} and its relevant extension by hyperbolicity R^2T. Ope\-rator R^2T is determined by q|_Ω^T, where Ω^T:=\x ∈ Ω | \rm dist (x,∂ Ω)<T\. The inverse problem is: Given R^2T to recover q in Ω^T. We solve this problem by the boundary control method and describe the \it ne\-ces\-sary and sufficient conditions on R^2T, which provide its solvability.

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