Method of limiting differential inclusions and asymptotic behavior of systems with relay controls

In this paper, problems of asymptotic behavior of non-autonomous controlled systems with a matrix of derivatives and the feedbacks of relay type are considered. The research is based on the method of limiting equations in combination with the direct method of Lyapunov functions with semidefinite derivatives. The method of the limiting equations has arisen in works G.R. Sell (1967) and Z. Artstein (1977, 1978) on topological dynamics of nonautonomous systems. Now this method is advanced for a wide class of systems, including the systems with delay. Nevertheless the method of the limiting equations till now has not received development with reference to nonautonomous differential inclusions and discontinuous systems for which it has fragmentary character. The main results are bound up with development of this method for discontinuous systems represented in the form of differential inclusions. In this case, specific methods of multivalued analysis and development of new methods for constructing limiting differential inclusions were required. The structure of the systems under scrutiny makes it possible, in particular, to study mechanical systems controlled on the decomposition principle of E.S. Pyatnitsky, and systems with dry friction submitted by equations Lagrange of 2-nd kind.