The author discusses the linear quadratic optimal control problem for retarded functional differential systems in the Hilbert state space \(M^ 2\) on finite and infinite time intervals. After a review of the general theory in Hilbert spaces, he shows that solutions of the integral and algebraic Riccati equations are of trace class. Then strong and weak convergence of approximate Riccati equations to solutions of the infinite dimensional Riccati equation are discussed. These results are applied to delay equations and the so-called averaging approximation is treated in detail. For approximation on the infinite time interval a property is needed (conjecture 7.1) which ensures uniform exponential stability of the approximating semigroups provided that the system semigroup is uniformly exponentially stable. In the meantime, this conjecture has been confirmed by D. Salamon [Structure and stability for approximations of functional differential equations, to appear in: SIAM J. Control Optimization]. Finally, computational aspects are discussed and three numerical examples are presented.