An inverse problem of determining the shape of rotating body by temperature measurements

In this paper, we would like to give an indirect possible way to reconstruct the shape of rotating body by temperature measurements. The mathematical model is governed by a heat conduction equation in which the parameter to determine is denoted by an unknown coefficient of first-order derivative. Firstly, the uniqueness of the solution is proved by using the maximum principle. Then, on the basis of optimal control framework, the inverse problem is transformed into an optimization problem. To ensure the well-posedness of the considered problem, we prove the existence and derive the necessary condition of the minimizer. By carefully analysing the necessary condition, we obtain the local uniqueness and stability on the basis of some a-priori estimates of forward problem. Finally, some numerical implements are performed to show the validity of the proposed method.