qChapter 12 studies the stabilizing solution to an *Algebraic **Riccati Equation* (ARE):

Suppose *H* has no *j**w*-axis eigenvalues and *X**-**(H)* is the stable invariant subspace
of *H *and suppose *X**1 *is nonsingular, then *X=Ric(H):=X**2 **X**1**-1 *is the stabilizing solution. A key result of
this chapter is the so-called Bounded Real Lemma: G(s)Î*R**H**¥* with *||G(s)||**¥** **<1* if and only if there exists an *X* such that *A+BB*X/**g**2 *is stable and