Conclusion

Abstract

This monograph continues a series of the authors’ works devoted to the problem of constructing extensions and relaxations. It is also connected with questions related to both applied and ‘pure’ mathematics. In the first case it is a matter mainly of problems of control theory, including control under uncertainty. In the second case, questions of asymptotic (in essence) realization of FAM are investigated; i.e., questions of universal integrability and measurability. The idea of extension of either space of solutions in an appropriate class of FAM turned out to be useful in these questions. Sometimes, more general extension schemes, in which only a topological equipment of the space of generalized elements is essential, are discussed. The authors think that quite general constructions of Chapter 3 were rendered concrete either in a direct form or in the form of some natural analogues. In this monograph questions of extremum and related regularizations are not considered (corresponding modifications of basic constructions for this case are given in [32, 35] and many journal articles). The point is that it is not difficult to change the basic constructions for variants of perturbation of conditions of the problem which are examined in this book. Such a change allows us to obtain representations of corresponding analogues of the ‘usual’ extremum and to investigate the constructions of optimal generalized elements. The authors think that the nature of correct extensions is revealed, in the best way, in problems of attainability.