We consider the problems for identifying the parameters a(11)(x, t),...,a(mm)(x, t) and c(x, t) involved in a second-order, linear, uniformly parabolic equation partial derivative(t)u - partial derivative(i)(a(ij) (x, t)partial derivative(j)u) + bi(x, t)partial derivative(i)u + c(x, t)u = f(x, t) in Omega x (0, T), u(partial derivative Omega) = g, u(t = 0) = u(0)(x), x is an element of Omega. on the basis of noisy measurement data z(x) = u(x, T) + w(x), x is an element of Omega with equality and inequality constraints on the parameters and the state variable. The cost functionals are (one-sided) Gateaux-differentiable with respect to the state variables and the parameters. Using the Duboviskii-Miljutin lemma we get the two maximum principles for the two identification problems, respectively, i.e., the necessary conditions for the existence of optimal parameters.