  One often encounters numerical difficulties in solving linear matrix
inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss
the reason from the viewpoint of optimization, and provide necessary and
sufficient conditions for LMI problem and its dual not to be strongly feasible.
Moreover, we interpret them in terms of control system. In this analysis,
facial reduction, which was proposed by Borwein and Wolkowicz, plays an
important role. We show that a necessary and sufficient condition closely
related to the existence of invariant zeros in the closed left-half plane in
the system, and present a way to remove the numerical difficulty with the null
vectors associated with invariant zeros in the closed left-half plane.
Numerical results show that the numerical stability is improved by applying it.
