Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. In practice we often meet problems, leading to the problem of finding an “optimal” shape of systems, the behaviour of which is described by elliptic equations. Problems of this type have been studied by many authors from mathematical as well as from computation point of view. When surfaces of a specimen which have been damaged by a corrosion aggressive attack are not accessible to direct inspection, one is forced to rely on over-determined measurements performed on the accessible part of the boundary. In this study, we consider such a non-destructive inspection technique modelled as a shape optimization problem, which consists of determining an unknown part of the boundary of a simply-connected bounded domain. The solution of Laplace equation $u$ satisfies nonlinear Robin condition on an unknown part of boundaty. We have Dirichlet and Neumann boundary conditions on the rest of boundary. We look for a curve, that minimizes the cost functional represented as a sum of regularizing term and $L_2$-norm of the difference of the state variable $u$ and some additional measurement on the part of admissible boundary. The existence of at least one curve is proved for an appropriate choice of the class of admissible curves. This will lead us to study properties of continuity of J and compactness of the set of controls. We shall characterize the shape derivative of the cost functional with respect to perturbations of the domain defined by a sufficiently smooth function. In order obtain necessary optimality condition we use the mapping method. This method transforms the unknown domain to a fixed one. The problem under consideration is essentially nonlinear. We prove Frechet differentiability of the solution operator.