Positive Definiteness in Linear Matrix Inequality Problem for $H$ -Infinity Output Feedback Control Problem

We focus on the positive definiteness in the linear matrix inequality (LMI) problem obtained from H
_∞
output feedback control problem. In particular, we provide a necessary condition for an inequality constraint of the LMI problem, which is called the coupling constraint, to strictly hold at optimal solutions. It is reasonable to expect that the condition does not hold for most generalized plants. In other words, the strict inequality at the optimal solutions rarely happens in H
_∞
output feedback control for most generalized plants. Thus, we must apply remedies for constructing a controller after solving the LMI problem.