List of issues > Series «Mathematics». 2015. Vol. 13
Limiting Differential Inclusions and the Method of Lyapunov’s Functions
In article the method of research of asymptotic behaviour for solutions of the nonautonomous systems submitted in the form of differential inclusions develops. The received results carry the form of generalizations of the LaSalle’s principle of invariance.
Principle of invariance usually call as LaSalle’s theorem for the autonomous differential equations in which (in the frame of Lyapunov’s direct method) it is supposed, that derivative of Lyapunov’s function is nonpositivity. The conclusion which this implies, will be, that the right limiting sets of solutions belong to the greatest invariant subset from set of zero of derivative function of Lyapunov. Before Lyapunov’s functions with constant signs were used in Barbashin – Krasovsky’s known theorem about asymptotic stability of positions of balance of autonomous systems. This theorem (together with LaSalle’s theorem) also sometimes characterize, as a principle of invariancy.
For the nonautonomous equations on this way there are the difficulties connected to absence of properties such as invariancy of the right limiting sets of solutions, and also with the description of set of zero of a derivative of Lyapunov’s functions. Attempts of overcoming of these difficulties have led to concept of the limiting differential equations. Method of the limiting equations in a combination to Lyapunov’s direct method allows to investigate effectively asymptotic behaviour of solution of nonautonomous systems. These researches go back to works of G.R. Selll and Z. Artstein on topological dynamics of the nonautonomous differential equations. Distribution of a method of the limiting equations on wider classes of systems brings an attention to the question about structure and methods of construction of the limiting equations. We this question is solved with reference to differential inclusions.
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