Abstract: The necessary conditions of the optimal control for a class of distributed parameter systems are studied in the paper. The systems studied are described by 2nd order uniformly parabolic differential equations and the control is contained in the coefficients of the equations. For the first order cost functional the two necessary conditions are given. After proving the state which is a continuously Fréchet differentiable function of the control, the necessary conditions are also obtained for the quadratic cost functionals. The mentioned results can be used for searching the optimal control.