In this work we revisit the linear-quadratic optimal control problem for differential-algebraic systems on the infinite time horizon with zero terminal state. Based on the recently developed Lur'e equation for differential-algebraic equations we obtain new equivalent conditions for feasibility. These are related to the existence of a stabilizing solution of the Lur'e equation. This approach also allows us to determine optimal controls if they exist. In particular, we can characterize regularity of the optimal control problem. The latter refers to existence and uniqueness of optimal controls for any consistent initial condition.