On the problem of stability in the critical case

Abstract

A critical case of m zero roots, n pairs of pure imaginary roots and q roots with negative real parts in which m groups of solutions correspond to the m-tuple zero root and no integral relations exist between the pure imaginary roots is considered, and a case in which asymptotic stability is impossible is singled out. The Kamenkov theorem on stability is extended to an essentially singular case. A theorem on stability for the nonessentially singular cases is established for the critical case of a single zero root, arbitrary number of pairs of pure imaginary roots and roots with negative real parts.