The quadratic cost control problem for a linear parabolic equation controlled by the constrained right-hand side of the equation is considered. Finite-dimensional approximation scheme for finding approximate solutions is proposed and its convergence rate is estimated. The set of optimal controls and its continuity properties with respect to the problem's data are studied. The convergence of the approximation scheme in the case when the control consists of both the constrained initial condition and the constrained right-hand side of the equation is proved.