Abstract: This work investigates an inverse problem of reconstructing the convection coefficient in a Kolmogorov equation from the final measurement data, which has important application in many fields of applied science. Compared with ordinary inverse problems, the unknown coefficient in this work depends on both the space variable x and the time variable t. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem and the existence of the minimizer for the cost functional is established. Then necessary condition of the minimizer is obtained. After the necessary condition of the minimizer satisfied, the stability and uniqueness of the minimizer are deduced.

Key words: optimal control, existence, uniqueness, inverse problem, stability, Kolmogorov equation