A new approach to the theory of the stability of linear canonical systems of differential equations with periodic coefficients

Abstract

The main element of the proposed approach to constructing a theory of the stability of canonical systems is an index function (defined below), which contains all the necessary information on the system. The fundamental results of existing theory, in particular, the necessary and sufficient condition for strong stability, are expressed in new terms. The corresponding proofs only use simple mathematical means; moreover, they are much shorter than existing proofs. A number of new assertions are established, in particular, a simple sufficient condition for strong stability is obtained, which essentially generalizes the well-known Yakubovich theorem [1] of the directed width of the stability regions, and the necessary and sufficient condition for their directed convexity is obtained. Using them, certain non-local qualitative results on the regions of stability of parametric oscillations of canonical systems are established (which enable, in particular, the existing practice of constructing stability regions in accordance with their boundaries to be justified), and the conditions for high-frequency parametric stabilization of unstable systems are obtained.