Stabilization and regularization for solutions of an ill posed problem for the wave equation

In this paper we study the problem of finding u: [O, T] → D(A), D(A) ? H, H a Hibert space such that: Is a linear positive, self-adjoint operator with a compact inverse. The problem is well known to be illposed because uniqueness and existence generally fail. We restore the stability with an a priori bound on ∥ du(0)/dt ∥ for some particular values of T.