Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

摘要

A dynamical Maxwell system, is e_t = curl h, h_t = -curie in Q × (0, T), e=0 = 0, h=0 = 0 in Ω, e_θ = f in ?Ω × [0, T], where Ω is a smooth compact oriented 3-dimensional Riemannian manifold with boundary, (·)_θ is a tangent component of a vector at the boundary, e = e~f (x, t) and h = h~f (x, t) are the electric and magnetic components of the solution. One associates with this system a response operator R~T: f → v ? h~f ?Ω×(0,T), where v is an outward, normal to ?Ω. The time-optimal setup of the inverse problem, which is relevant to the finiteness of the wave speed propagation, is as follows: given R~(2T) to recover the part Q~T:= {x ε Ω dist(x, ?Ω) 0 and provide a procedure that recovers Ω~T from R~(2T). Our approach is a version of the boundary control method (Belishev, 1986).