A result on output feedback linear quadratic control

In this note we consider the static output feedback linear quadratic control problem. We present both necessary and sufficient conditions under which this problem has a solution in case the involved cost depends only on the output and control variables.This result is used to present both necessary and sufficient conditions under which the corresponding linear quadratic differential game has a Nash equilibrium in case the players use static output feedback control.Another consequence of this result is that the conditions also provide sufficient conditions for the static output stabilizability problem. Of course, in case these conditions are not met this does not mean that the system is not stabilizable via static output feedback.