ALGEBRAIC RICCATI EQUATION AND INEQUALITY FOR SYSTEMS WITH UNCONTROLLABLE MODES ON THE IMAGINARY AXIS

If (A, B) is stabilizable, one pretty well knows algebraic conditions for the solvability and for the existence of largest solutions of the algebraic Riccati equation and inequality AX+XA-XBB*X+Q=0 and A*X+XA- XBBX+Q greater than or equal to 0, which leads to immediate existence results for positive definite solutions. In this paper we work out ho w far these properties may be generalized if (A, B) could have uncontr ollable modes on the imaginary axis. Since the relations of the equati on and inequality are not as tight any more, we provide separate condi tions for the existence of Hermitian or positive definite solutions an d give a detailed discussion how to verify them. As auxiliary steps we discuss various new aspects for the corresponding Lyapunov equation/i nequality and a complete solvability test for the quadratic equation X RX+SX+(SX)*+T=0 with Hermitian R and T. Finally, we briefly point out the consequences of our results for the general state-feedback H infi nity-control problem at optimality.