The aim of this work is to study the optimal control problem associated to a linear parabolic equation with homogeneous Dirichlet boundary condition. The control variable is the matrix of L1-coefficients in the main part of the parabolic operator. The precise answer about existence or none-existence of an L1-optimal solution heavily depends on the class of admissible controls. The main questions concern the right setting of the optimal control problem with L1-controls in the coefficients, and the right class of admissible solutions to the above problem. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problem in the class of H-admissible solutions.