Dual Problems of Lagrange for Arcs of Bounded Variation.

摘要

In Lagrange problems in optimal control and the calculus of variations, an integral functional of state and velocity is minimized over a class of arcs in R to the Nth satisfying an endpoint condition and other constraints. For problems with joint convexity properties in state and velocity, a theory of duality is available in the context of arcs which are absolutely continuous. However, for inherent reasons, a fully satisfactory treatment of state constraints is not possible without an extension of the basic foundations so as to admit arcs which are merely of bounded variation. Such an extension in symmetric form is carried out here for the first time. Results are obtained on the characterization of optimal arcs in terms of a generalized Hamiltonian equation, as well as on their existence and the possibility of identifying or approximating them by absolutely continuous arcs. (Author)