Abstract
In this paper, we consider an optimal control problem for a linear second order elliptic system governed by a self-adjoint elliptic 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 elliptic equations are presented. The performance functional has the quadratic form. Finally, we impose some constraints on the control. Making use of the Lions scheme [18], necessary and sufficient conditions of optimality for the Dirichlet and Neumann problems with the quadratic performance functional and constrained control are derived.