Stabilizing solutions of the H∞ algebraic Riccati equation

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H_∞ control problem. It is parameterized by the H_∞ norm bound γ 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 γ. It turns out that a stabilizing solution exists for all but finitely many values of γ larger than some a priori determined boundary γ*. On the other hand for values smaller than γ* 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 γ. 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_∞ norm strictly less than γ.