Abstract : The article explores the applicability of the quadratic criterion of optimality to the control of distributed parameter systems or, alternatively, to differential equations in a Hilbert space. It is shown that such application is indeed possible and, perhaps more important from the engineering point of view, that the optimal controls thus found are limits of optimal controls corresponding to finite dimensional models of the distributed or infinite dimensional, system. The theory is first developed in an abstract setting and then applied to specific examples. (Author)