Stabilizing solutions of the $H_\infty$ algebraic Riccati equation

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback $H_\infty$ control problem. It is parameterized by the $H_\infty$ norm bound $\gamma$ we want to achieve. The objective of this paper is to study the behaviour of the solution to the Riccati equation as a function of $\gamma$. It turns out that a stabilizing solution exists for all but finitely many values of $\gamma$ larger than some a priori determined boundary $\gamma_{-}$. On the other hand for values smaller than $\gamma_{-}$ there does not exist a stabilizing solution. The finite number of exception points turn out to be switching points where eigenvalues of the stabilizing solution can switch from negative to positive with increasing $\gamma$. After the final switching point the solution will be positive semi-definite. We obtain the following interpretation: the Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exist a static feedback such that the closed loop transfer matrix has no more than k unstable poles and an $L_\infty$ norm strictly less than $\gamma$. Keywords: The $H_\infty$ control problem, The Algebraic Riccati Equation, J-spectral factorization, Wiener-Hopf factorization.