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

Abstract

A dynamical system of the form

ð‘¢_tt − Δ𑢠− ∇ln𜌠· ∇𑢠= 0,                   in   â„^ð‘›₊ × (0, ð‘‡)

ð‘¢|_t=0 = ð‘¢_t|_t=0|= 0,                            in   â„^ð‘›₊

ð‘¢_x^ð‘› = f                                               on   Ï‘â„^ð‘›₊ × (0, ð‘‡),

is considered, where â„^ð‘›₊ := {x = {x¹, . . . , x^ð‘›}| x^ð‘› > 0} ; 𜌠= ðœŒ(x) is a smooth positive function (density) such that ðœŒ, 1/𜌠are bounded in â„^ð‘›₊; f is a (Neumann) boundary control of the class L₂(Ï‘â„^ð‘›₊ × [0, ð‘‡]); ð‘¢ = ð‘¢^f (x, t) is a solution (wave). With the system one associates a response operator R^T : f ⟼ ð‘¢^f|_{Ï‘â„^ð‘›+ × [0, ð‘‡]}. A dynamical inverse problem is to determine the density from the given response operator.

Fix an open subset 𜎠⊂ Ï‘â„^ð‘›₊; let L₂(ðœŽ × [0, ð‘‡]) be the subspace of controls supported on ðœŽ. A partial response operator R^T_𜎠acts in this subspace by the rule R^T_𜎠f = ð‘¢^f|_{ðœŽ×[0,T]}; let R^2T_𜎠be the operator corresponding to the same system considered on the doubled time interval [0, 2T]. Denote B^T_𜎠:= {x ∈  â„^ð‘›₊|{x¹, . . . , x^ð‘›-1,0} ∈ ðœŽ, 0 < x^ð‘› < T} and assume ðœŒ|_𜎠to be known. We show that R^2T_𜎠determines ðœŒ|_B^T𜎠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.