Fundamental inertia conditions for the solution of H^∞-problems

We study the relation between the solutions of two minimization problems with indefinite quadratic forms. We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions. While these inertia conditions are automatically satisfied in a standard Hilbert space setting, they nevertheless turn out to mark the differences between the two optimization problems in indefinite metric spaces. They also include, as special cases, the well-known conditions for the existence of H^∞-filters and controllers.