(v) ||*D*||< *g** *and there exists an *X*³*0* such that

and *A+BR**-1**D*C+BR**-1**B*X* has no eigenvalues on the imaginary
axis.

(vi) ||*D*||< *g** *and there exists an *X *> 0 such that

(vii) there exists and *X *> 0 such that

q**Proof:** We have already known: (i) Û (ii). (iii) Þ (ii) is obvious. To show that (ii) Þ (iii), we need to
show that (*A+BR**-1**D*C, BR**-1**B*) *is stabilizable (Theorem 12.2). In fact, we will show that *A+BR**-1**D*C * is stable for all those *g** *such that ||*G*||¥ < *g** .*