G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

Abstract

517.9To study problems of homogenization of one class of control objects described by nonlinear operator equations, the concept of G * -convergence of nonlinear operators is introduced. The sufficient conditions of identifiability of homogenized objects are obtained.Keywords: homogenization, elliptic operator, convergence in the sense of Kuratowski, control object, G-limit, operator graph.The objective of this work is the study of limit behavior of solutions of nonlinear operator equations of the formand problems of homogenization of control objects connected with these equations. To date, the problem of homogenization of differential partial differential equations and operator equations in Banach spaces is considered in many publications (see, for example, [1-10]). However, a distinctive feature of the majority of the results obtained is that, as a rule, a sequence of elements { } f e is assumed to be convergent in the strong topology of the space Y * . In this case, a homogenized object exists and can be identified in terms of the so-called G-limit operators. At the same time, a basic premise of the present work is that the property of compactness of the sequence { } f e being considered takes place only with respect to the weak topology of Y * . In this case, the fact is essential that the sequence { } f e does not contain any strongly convergent subsequence. As is well known, the practical significance of the solution of the problem of homogenization of such objects follows from the needs of mechanics of composite materials and the theory of filtration in porous media. In particular, as is shown in works on problems of limit analysis of boundary-value problems in perforated domains (see, for example, [4, 11-17]), their solutions converge to the solution of a boundary-value problem whose associated operator is the sum of a G-limit operator and some additional lowest term that characterizes the perforation structure of the initial domain.In this connection, we note that the apparatus of G-convergence of elliptic operators becomes unsuitable for the study of limit properties of the sequence of solutions of boundary-value problems (1) under the above-mentioned assumptions on functions { } f e . Therefore, to identify the mathematical model of the problem whose solution is the weak limit of the sequence of solutions of problems (1), the concept of G * -convergence of nonlinear operators is introduced in this paper that is a natural generalization of the well-known conception of G-limit analysis. At the same time, it is important to note that, for the same sequence of operators { } A e , their G-and G * -limits coincide in the case when the family of functions { } f e is compact in the strong topology of the space Y * . This is explained by the fact that, in the definition of G * -convergence, not only properties of operators { } A e but also structural properties of functions { } f e are used. We introduce the following agreements. Let Y be a separable reflexive Banach space with a norm || || × over the real numbe...