Derivatives with respect to diffeomorphisms and their applications in control theory and geometrical optics

Abstract

We study nonsmooth problems of optimal control theory and geometrical optics that can be formalized as Dirichlet boundary value problems for first-order partial differential equations (including equations of Hamiltonian type). A methodology is elaborated for the identification and construction of singular sets with the use of multipoint derivatives. Four types of derivatives with respect to diffeomorphisms are introduced; they generalize the notions of classical derivative and one-sided derivative. Formulas are given for the calculation of derivatives with respect to diffeomorphisms for some classes of functions. The efficiency of the developed method of analysis is illustrated by the example of solving a time-optimal problem in the case of a circular velocity vectogram and a nonconvex target with nonsmooth boundary.