Behavior of the misfit functional for a one-dimensional hyperbolic inverse problem

In this paper we investigate the behavior of the misfit functional for a one-dimensional hyperbolic inverse problem when an unknown coefficient stands by a lowest term of a differential equation. Assuming an existence of an inverse problem solution we prove a uniqueness of a stationary point of the functional. If the minimization sequence belongs to a bounded set, we show that the following estimates of the convergence rate for the suggested method of the descent
$$J[q_k]\le J[q_0]\exp\{-c(k-1)\},\quad\|q_k-q_*\|^2_{L_2[-T,T]}\le CJ[q_0]\exp\{-c(k-1)\}$$
takes place.