Abstract: We established an optimization model for the refractive index inversion problem in the terrain parabolic equation based on  the framework of optimal control theory. Firstly, through rigorous mathematical analysis, we proved the existence of minimum value  for the objective functional and its necessary conditions. Secondly, we proved  the local uniqueness and stability of the minimum value to the problem.

Key words: inverse problem, refractive index, terrain parabolic equation, optimal control