About A.M.Lyapunov reduction principle for singularly perturbed Systems
by
Lyudmila K.Kuzmina
Kazan Aviation Institute, Adamuck, 4-6, Kazan-12, 420012, Russia

The research is aimed to development of concepts and methods of the classical theory of A.M.Lyapunov’s stability with reference to dynamics problems for complex systems of a singularly perturbed class, including a reduction principle. This work is connected to various aspects and problems of systems dynamics of such type, with development of methods and estimations in modelling and analysis on the basis of the generalized methodology, that is synthesizing idea and methods of stability theory and asymptotic methods. The extended approach formed on methods A.M.Lyapunov and N.G.Chetayev, on postulates of stability and singularity, gives the universal tool allowing to obtain the solving of fundamental problems in the general stability theory for complex systems of a examined class, including problem of a decomposition. According to ideas of classical works on the stability theory (A.M.Lyapunov, N.G.Chetayev) some statements and problems (K.P.Persidsky, P.A.Kuzmin) with reference to systems features of singularly perturbed class generated by concrete examples of the physical nature are discussed. For these systems the initial mathematical model, having the large and small parameters, may be presented in the standard form of singularly perturbed system with parametric perturbations of non-regular type, peculiar to systems with multiple time scales. The solving of stability problems in critical cases (A.M.Lyapunov), inherent to applied problems of mechanics, is discussed. Singularly perturbed systems with specific peculiarities, corresponding to mechanical systems, are investigated in cases, when the property of uniform asymptotic stability is absent; the unperturbed subsystems are on boundary of stability domain; the generating systems are not limit systems; nominal systems - quasi-Tikhonov's (N.N.Moiseev). The conditions of reduction-decomposition are determined, at which the problem about stability for original system is reduced to investigation of approximate system of the less order (s-shortened system), with obtaining of estimations of N.G.Chetayev type. And shortened system is nonlinear, singularly perturbed one (S.Cambell). The solving of singularly perturbed problem about stability for systems, for which spectra of the appropriate matrixes are critical (both for slow, and for fast variables), is received. The regular algorithms for estimations of domains of system parameters values, allowing the reduction in stability problems, are constructed. The results are discussed which give a strict substantiation of reduction principle for considered systems. The constructed methods and obtained results are generalizing and supplementing ones, known in stability theory (including the classical A.M.Lyapunov theorem “Some generalization”) with reference to systems of singularly perturbed class, with multiple time scales. This development is lighting new approaches and novel trends in stability investigations both from theoretical view point and from engineering applications point.

The author is grateful to the Russian Foundation of Fundamental Investigations for support of research.

Date received: September 14, 2006