On Frech`{e}t differentiability of cost functional in optimal control of coefficients of elliptic equations

In the work we consider nonlinear optimal control problems for semi-linear elliptic equations with discontinuous data and solutions (states) with controls in the boundary conditions of conjugation of heterogeneous media and in the right hand side of the state equation. We prove the differentiability and Lipshitz continuity for the grid analogue of the cost functional for extremum problems.