Открытый доступ
Доступ предоставлен
Только для подписчиков
Том 217, № 6 (2016)
- Год: 2016
- Статей: 8
- URL: https://ogarev-online.ru/1072-3374/issue/view/14768
Article
To the 80th Birthday of Professor V. M. Tikhomirov
671-671
Mix of Controls and the Pontryagin Maximum Principle
Аннотация
In this paper, necessary conditions for a minimum (the Pontryagin maximum principle) for an optimal control problem are proved on the basis of the concept of a mix, which enables one to reduce the study of the original problem to some approximation thereof, which is linear in the control. The study of the latter problem proves more simple.
672-682
Connectedness and Other Geometric Properties of Suns and Chebyshev Sets
Аннотация
This survey is concerned with structural characteristics of “suns” in normed linear spaces. Special attention is paid to connectedness and monotone path-connectedness of suns. We address both direct theorems of the geometric approximation theory, in which approximative properties of sets are derived from their structural characteristics, and inverse theorems, in which from approximative characteristics of sets one derives their structural properties.
683-730
On the Continuity of Inverse Mappings for Lipschitz Perturbations of Covering Mappings
Аннотация
In this paper, we study the question of the existence of a continuous right inverse mapping for a covering mapping. To describe covering mappings that have a continuous right inverse, a concept of strong covering is introduced. It is shown that the property of strong covering is stable under small Lipschitz perturbations.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 4, pp. 93–99, 2014.
731-735
On Integral Representation of Γ-Limit Functionals
Аннотация
We consider the Γ-convergence of a sequence of integral functionals Fn(u), defined on the functions u from the Sobolev space W1,α(Ω) (α > 1); Ω is a bounded Lipschitz domain, where the integrand fn(x, u,∇u) depends on a function u and its gradient ∇u. As functions of ξ, the integrands fn(x, s, ξ) are convex and satisfy a two-sided power estimate on the coercivity and growth with different exponents α < β. Moreover, the integrands fn(x, s, ξ) are equi-continuous over s in some sense with respect to n. We prove that for the functions from L ∞ (Ω) ∩ W1,β(Ω) the Γ-limit functional coincides with an integral functional F(u) for which the integrand f(x, s, ξ) is of the same class as fn(x, s, ξ).
736-750
On Necessary Conditions for a Minimum
Аннотация
We discuss a general approach to necessary optimality conditions based on the so-called “optimality alternative,” which reduces a problem with constraints to an unconstrained problem or a sequence of unconstrained problems. The power of the approach is demonstrated by a proof of a necessary optimality condition in an abstract problem with mixed (convex vs. nonconvex) structure and a new proof of Clarke’s “stratified” maximum principle for optimal control of differential inclusions.
751-772
773-802
On an Estimate Connected with the Stabilization of a Normal Parabolic Equation by Start Control
Аннотация
After a brief revision of facts concerning semilinear parabolic equations of normal type and their nonlocal stabilization by start control, we provide a simplification of the proof of the lower bound for a functional of the solution to the heat equation with initial condition of a special type. This bound is essential to prove the nonlocal stabilization of equations of normal type. The simplification presented is required for further development of the nonlocal stabilization theory.
803-826