The optimal control of linear distributed parameter systems, which are represent able by a linear vector integral equation, is investigated. Restricting the control action to be discrete in time, the problem of minimizing the mean-square error, between specified desired final state functions and the actual state functions at a prescribed final time, subject to an energy constraint on the controlling functions, is treated. A necessary and sufficient condition from functional analysis is used to derive an equation whose solution yields the optimal control vector. Two convergence properties for the discrete problem are established which can be used to determine a good approximate solution to the corresponding measurable optimal control problem. An illustrative example is given.