Abstract: For a linear time-invariant system, if its feed-back control law u=Kx incorporates an observer for Kx in its implementaion, then theresultant close-loop system forms a composite system with state feedback. It is a quadratic performance index optimal system if matrix K satisfys some conditions. The problem of investigating these conditions is called observer linear quadratic state feedback inverse problem (OLQSF inverse problem), which is an extension of the well-known inverse problem of optimal control. In this paper, some necessary conditions and a set of necessary and sufficient conditions for OLQSF inverse problem are discussed. Furthermore, the inverse problem of optimal control for some stabilizable systems is discussed too.