An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.