In this paper, we consider an optimal control problem for a linear second order parabolic system governed by a self-adjoint parabolic operator with an infinite number of variables. One from the initial conditions is given by control function. Sufficient conditions for      the existence of a unique solution of such parabolic equations are presented. The performance functional has the quadratic form. Finally, we impose some constraints on the control. Making use of the Lions scheme (Lions [19]), necessary and sufficient conditions of optimality for the Dirichlet and Neumann problems with the quadratic performance functional and constrained control are derived.

parabolic systems, Dirichlet and Neumann conditions, existence and uniqueness of solutions, operators of second order with an infinite number of variables, optimal control problems.