Perturbations of extremal problems with constraints and necessary optimality conditions

The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons

“…Earlier, a similar approximation approach was used to derive necessary optimality conditions for various nonclassical optimal control problems (see, e.g., [3], [4], [5], [7], [32], and also survey [6]). Based on relevant approximation techniques and the methodology presented here, one can extend the results of this paper to more complex infinite-horizon problems of optimal control (e.g., problems with nonsmooth data).…”

Гладкие Анормальные Задачи Теории Экстремума И Анализа

“…В задачах оптимального управления с фазовыми ограничениями и с нефик-сированным временем необходимые условия второго порядка, заключающиеся в непустоте множества множителей Лагранжа Λ s , при соответствующем вы-боре числа s были получены в [9]- [11]. В указанных работах необходимые условия первого порядка брались в форме принципа максимума Понтрягина, а не уравнения Эйлера-Лагранжа.…”

Second-order conditions in extremal problems. The abnormal points

“…It follows from the example given below that the assumption on the finitedimensionality of X is fundamental in Theorem 3.2-it cannot be omitted without being replaced by another one (for example, by the assumption that the form is of the Legendre type, as was done in [5,11]). …”

“…For the first time second-order necessary conditions that require the set Λ a (x 0 ) to be nonempty were obtained for the time optimality problem in [13]. Further, those conditions were generalized to a broad class of extremal and optimal control problems [5,9,11].…”

“…In [5] the author obtained estimates from below for Ls{x : A n x ∈ K} where K is a cone with a finite number of faces in a Banach space.…”