Structural identification of an unknown source term in a heat equation

The identification of an unknown state-dependent source term in a reaction-diffusion equation is considered. Integral identities are derived which relate changes in the source term to corresponding changes in the measured output. The identities are used to show that the measured boundary output determines the source term uniquely in an appropriate function class and to show that a source term that minimizes an output least squares functional based on this measured output must also solve the inverse problem. The set of outputs generated by polygonal source functions is shown to be dense in the set of all admissible outputs. Results from some numerical experiments are discussed.