In this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed: given a linear, time-invariant, and complete system, nd the set of trajectories of the system that optimize a quadratic-type cost function and that satisfy some linear static constraints. This formulation provides a unifying framework, where several generalized versions of the classical LQ optimal control can be stated and solved. The existence of solutions is rst discussed. It is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by-product of a reduction procedure translating the problem into an equivalent one of minimum complexity. Such a procedure is based on the theory of l(2)-systems in the behavioral setting. Once the complexity is reduced, a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two-step optimization procedure.