Dynamical Inverse Problem for the Equation 𝒰ᵼᵼ − Δ𝒰 − ∇ln𝜌 · ∇𝒰 = 0 (the BC Method)

Belishev, M.I.

CUBO, A Mathematical Journal, Tome 10 (2008), / Harvested from Cubo, A Mathematical Journal

Résumé

A dynamical system of the form

𝑢tt − Δ𝑢 − ∇ln𝜌 · ∇𝑢 = 0,                   in   ℝ𝑛+ × (0, 𝑇)

𝑢|t=0 = 𝑢t|t=0|= 0,                            in   ℝ𝑛+

𝑢x𝑛 = f                                               on   ϑℝ𝑛+ × (0, 𝑇),

is considered, where ℝ𝑛+ := {x = {x1, . . . , x𝑛}| x𝑛 > 0} ; 𝜌 = 𝜌(x) is a smooth positive function (density) such that 𝜌, 1/𝜌 are bounded in ℝ𝑛+; f is a (Neumann) boundary control of the class L2(ϑℝ𝑛+ × [0, 𝑇]); 𝑢 = 𝑢f (x, t) is a solution (wave). With the system one associates a response operator RT : f ⟼ 𝑢f|ϑℝ𝑛+ × [0, 𝑇]. A dynamical inverse problem is to determine the density from the given response operator.

Fix an open subset 𝜎 ⊂ ϑℝ𝑛+; let L2(𝜎 × [0, 𝑇]) be the subspace of controls supported on 𝜎. A partial response operator RT𝜎 acts in this subspace by the rule RT𝜎 f = 𝑢f|𝜎×[0,T]; let R2T𝜎 be the operator corresponding to the same system considered on the doubled time interval [0, 2T]. Denote BT𝜎 := {x ∈  ℝ𝑛+|{x1, . . . , x𝑛-1,0} ∈ 𝜎, 0 < x𝑛 < T} and assume 𝜌|𝜎 to be known. We show that R2T𝜎 determines 𝜌|BT𝜎 and propose an efficient procedure recovering the density. The procedure is available for constructing numerical algorithms.

The instrument for solving the problem is the boundary control method which is an approach to inverse problems based on their relations with control theory (Belishev, 1986). Our presentation is elementary and can serve as introduction to the BC method.

Détail

Publié le : 2008-07-01