An inverse radiative coefficient problem arising in a two-dimensional heat conduction equation with a homogeneous Dirichlet boundary condition in a circular section

摘要

In this study, we investigate the well-posedness of the solution of an optimal control problem related to a nonlinear inverse coefficient problem. Problems of this type have important applications in several fields of applied science. Unlike other terminal control problems, the observation data are only given for a fixed direction rather than for the whole domain, which may make the conjugate theory for parabolic equations ineffective. Moreover, the coefficients in our model are singular, so we propose some weighted Sobolev spaces. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After deducing the necessary conditions that must be satisfied by the minimizer, we prove the uniqueness and stability of the minimizer. Following a minor modification of the cost functional and imposing some a priori regularity conditions on the forward operator, we obtain the convergence of the minimizer for the noisy input data considered in this study. The results obtained in this study are interesting and useful, and they can be extended to more general parabolic equations with singular coefficients. (C) 2015 Elsevier Inc. All rights reserved.