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Some aspects of Variational Analysis and Applications

Organizer:
Vladimir Goncharov

Abstract:
The goal of the mini Symposium is to bring together various tools of Variational Analysis, modern discipline, which partially unifies such traditional fields of mathematics as Calculus of Variations, Optimal Control, Games Theory, Nonsmooth and Multivalued Analysis, etc. The crucial term in this discipline is variation, which serves mostly for optimization or for the choice of an optimal strategy, optimal values of some parameters needed, e.g., to improve the quality of technological processes, of economic behaviour and so on. Theoretical results in Variational Analysis are often obtained through so named variational as well as optimality principles. The First ones, such as Ekeland's Variational Principle, Strong Maximum Principle for elliptic systems, etc., being the fundamental mathematical properties, permit to prove existence, uniqueness and regularity of minimizers, or to study their qualitative characteristics; while others (e.g., the famous Pontryagin's Maximum Principle or Bellman's Dynamic Programming Principle) give necessary (sometimes sucient) conditions of minima (maxima) that is basis for the development of effective numerical methods and algorithms. On the other hand, various optimization problems appearing in practice (such as variational or optimal control problems with phase constraints) require detailed studying of new classes of sets and functions, not necessarily convex or smooth. The results obtained in this direction being of big importance themselves can be successfully applied in optimization and in various fields of Analysis. Thus, the mini Symposium focuses mainly on both variational and optimality principles emphasized above and recent achievements in studying of variational (geometric) properties of sets and functions. Among the concrete theoretical subjects covered by mini Symposium's talks let us point out the following:
 the weak convexity and proximal regularity of closed sets in Hilbert and Banach spaces;
 controllable sweeping processes;
 existence and qualitative properties (e.g., a priori estimates) of minimizers in variational problems;
 optimality conditions for the problems governed by differential inclusions.