Boundary Control and Observation to Inverse Coefficient Problem for Heat Equation With Unknown Source and Initial Value

This article investigates an inverse problem of determining the spatially variable diffusion coefficient of a one-dimensional heat equation with an unknown spatial varying source term and initial value. The big challenge of the problem comes from the multiple unknowns and very limited available data that are only boundary control and boundary observation at one end, in addition to the ill-posed nature of the inverse problem. We first design a switch ON/OFF boundary control and show that the diffusion coefficient can be uniquely determined by the boundary observation. Next, a stable numerical algorithm for reconstruction of the diffusion coefficient is proposed by means of the matrix pencil method and optimal perturbation regularization technique. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed identification algorithm.