Determination of point wave sources by pointwiseobservations: stability and reconstruction

Abstract

We consider a wave equation with point source terms where λ∊C1[0,T] is a known function such thatλ(0)≠0, αk∊ℝ,δ(·-xk)is the Dirac delta function at xk, 1⩽k⩽N. Wediscuss the inverse problem of determining point sources {N,α1,...,αN,x1,...,xN} or {x1,...,xN} from observation data u(η,t), 0<t<T withgiven η∊(0,1) and T>0.We prove uniqueness and stability in determining point sourcesin terms of the norm in H1(0,T) of observations. Theuniqueness result requires that η is an irrational numberand T⩾1, and our stability result needs further a priori (but reasonable) information of unknown {x1,...,xN}. Moreover, we establish two schemes forreconstructing {x1,...,xN} which are stable againsterrors in L2(0,T).