The purpose of this paper is to close the remaining gaps in the understandingof the role that the constrained generalized continuous algebraic Riccatiequation plays in singular linear-quadratic (LQ) optimal control. Indeed, inspite of the vast literature on LQ problems, it is only in a recent paper thata sufficient condition for the existence of a non-impulsive optimal control hasfor the first time connected this equation with the singular LQ optimal controlproblem. In this paper, we establish four equivalent conditions providing acomplete picture that connects the singular LQ problem with the generalizedcontinuous algebraic Riccati equation and with the geometric properties of theunderlying system.
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