Identification problem for damped sine-Gordon equation with point sources

We establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gateaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Frechet differentiable with respect to the parameters. An expression for the Frechet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered. (C) 2010 Elsevier Inc. All rights reserved.