Lipschitz stability in an inverse hyperbolic problem with impulsive forces

Let u = u(q) satisfy a hyperbolic equation with impulsive input: ∂ t u(x, t)−4u(x, t) + q(x)u(x, t) = δ(x1)δ(t) and let u|t<0 = 0. Then we consider an inverse problem of determining q(x), x ∈ Ω from data u(q)|ST and (∂u(q)/∂ν) |ST . Here Ω ⊂ {(x1, . . . , xn) ∈ R|x1 > 0}, n ≥ 2, is a bounded domain, ST = {(x, t); x ∈ ∂Ω, x1 < t < T + x1}, ν = ν(x) is the unit outward normal vector to ∂Ω at x ∈ ∂Ω, and T > 0. For suitable T > 0, we prove an estimate: ‖q1 − q2‖L2(Ω) ≤ C ( ‖u(q1)− u(q2)‖H1(ST ) + ‚‚‚‚ ∂u(q1) ∂ν − ∂u(q2) ∂ν ‚‚‚‚ L(ST ) ) , provided that q1 satisfies a boundedness condition and q2 satisfies a smallness condition in the Sobolev norm of order n + 2.