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Variational Analysis in Nonsmooth Optimization and Discrete Optimal Control

Department of Mathematics, Wayne State University, Detroit, Michigan 48202
boris{at}math.wayne.edu

This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the "lack of compactness" in infinite dimensions, which leads to imposing certain "normal compactness" properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions.

Key Words: variational analysis; nonsmooth optimization and optimal control; discrete-time inclusions; generalized differentiation; infinite dimensions; necessary optimality conditions
History: Received: March 19, 2006; revision received: December 7, 2006;