On the Klimushchev–Krasovskii theorem

The Klimushchev–Krasovskii theorem describes a sufficient condition for the asymptotic stability of singularly perturbed systems. Motivated by some subtle examples, this note presents a new necessary and sufficient condition by imposing an additional assumption on the statement of the Klimushchev–Krasovskii theorem. This note also extends the main inertia theorem to singularly perturbed systems.