Optimal Control of Evolution Systems in Banach Spaces

Abstract

The next two chapters are on optimal control, which is among the most important motivations and fruitful applications of modern methods of variational analysis and generalized differentiation. It is not accidental that the very concepts of basic normals, subgradients, and coderivatives used in this book were introduced and applied by the author in connection with problems of optimal control. In fact, already the simplest and historically first problems of optimal control are intrinsically nonsmooth, even in the case of smooth functional data describing dynamics and constraints on feasible arcs. The crux of the matter is that a characteristic feature of optimal control problems, in contrast to the classical calculus of variations, is the presence of pointwise constraints on control functions, which may be (and often are) defined by highly irregular sets consisting, e.g., of finitely many points. In particular, this is the case of typical problems in automatic control that provided the primary motivation for developing optimal control theory.