The problem of optimal control over the system governed by the Dirichlet boundary value problem for a linear parabolic equation is constructed. Point-wise constraints are imposed on both control and state functions. The right-hand side of the equation is a control function in the problem. The objective functional contains an observation which is distributed in the space-time domain. Finite-difference approximation is constructed for the optimal control problem based on the Euler forward scheme for the state parabolic equation. The existence of its unique solution is proved. Constrained saddle point problem corresponding to the mesh optimal control problem is constructed. The existence of a solution for this saddle point problem and the converg ence of the generalized Uzawa iterative method are proved. The results of numerical experiments are given.