Stability of memory reconstruction from the Dirichlet-to-Neumann operator

On the indicated class of functions we define the so-called (nonstationary) Dirichlet-to-Neumann operator H that assigns to g(x, t) the normal derivative a~u of a solution to (1), (2) on the boundary aft x R. In the present article, we consider the inverse problem consisting in reconstructing an unknown memory k(x, t) given the Dirichlet-to-Neumann operator H. We prove conditional stability of a solution to the problem; i.e., we show that for some class of functions k(z, t) we may guarantee closeness of two different memories, provided that the corresponding Dirichlet-to-Neumann operators are close. We suppose that the function k(z, t) belongs to the well-posedness set