In this paper we study the optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. We adopt a weight coefficient in the main part of elliptic operator as control in BV(Ω). Since the equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that this optimal control problem is regular. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the class of weak admissible solutions.