On the uniqueness of a solution to the multiplicative control problem for the electron drift–diffusion model

The multiplicative control problem for a stationary diffusion-drift model of charging a polar dielectric is studied. The role of control is played by a leading coefficient in the model equation, which has the meaning of the electron diffusion coefficient. The global solvability of the boundary value problem and the local uniqueness of its solution, as well as the solvability of the extremum problem under consideration, have been proved in the previous papers of the authors. In this paper, an optimality system is derived for the control problem and local regularity conditions for the Lagrange multiplier are established. Based on the analysis of this system, the local uniqueness of the multiplicative control problem's solution for specific cost functionals is proved.