共查询到20条相似文献,搜索用时 0 毫秒
1.
Michel CHIPOT 《数学年刊B辑(英文版)》2018,39(3):597-606
The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution. 相似文献
2.
Maksim V. Dolgopolik 《Numerical Functional Analysis & Optimization》2018,39(4):467-490
In this paper, we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper, we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints. 相似文献
3.
Francesca Prinari 《Applied Mathematics and Optimization》2008,58(1):111-145
We study the weak* lower semicontinuity properties of functionals of the form
where Ω is a bounded open set of R
N
and u∈W
1,∞(Ω). Without a continuity assumption on f(⋅,ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if F is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in
general it is not possible to represent F through the level convex envelope of f. 相似文献
4.
Based on an earlier publication (Ref. 1), a coordinate transformation is proposed, which allows the direct global extremization of a class of integrals without the use of comparison methods such as variational or field techniques. This direct method is shown to be applicable to a class of unconstrained optimal control problems. A motivation for the proposed method as well as applications are presented. 相似文献
5.
6.
研究了非多项式增长的变分泛函,利用Orlicz空间理论,得到了其在Orlicz-Sobolev空间中弱序列下半连续的充要条件,推广了关于多项式增长的变分泛函的相应结论。 相似文献
7.
D. G. Hull 《Journal of Optimization Theory and Applications》1990,67(3):447-462
The variational process is established and applied to the development of the second variation for the free-final-time optimal control problem. First, it is shown that, given a change in the control (the independent variable), the change in the state (the dependent variable) consists of all orders of the change in the control. Hence, the change in the state is a total change. This implies that variations of dependent variations exist. Next, the variational relationship between time-constant and time-free variations is developed, and the formula for taking the variation of an integral is presented. The results are used to derive the second variation following three different approaches: taking the variation of the first variation after performing the integration by parts; taking the variation of the first variation before performing the integration by parts; and using the Taylor series approach. The ability to get the same result requires the existence of the total change in the state or of the variation of the state variation. Finally, if the nominal path is not an extremal, this process gives extra terms in the second variation. 相似文献
8.
In this paper, a variational problem is considered with differential equality constraints over a variable interval. It is stressed that the abnormality is a local character of the admissible set; consequently, a definition of regularity related to the constraints characterizing the admissible set is given. Then, for the local minimum necessary conditions, a compact form equivalent to the well-known Euler equation and transversality condition is given. By exploiting this result and the previous definition of regularity, it is proved that nonregularity is a necessary and sufficient condition for an admissible solution to be an abnormal extremal. Then, a necessary and sufficient condition is given for an abnormal extremal to be weakly abnormal. The analysis of the abnormality is completed by considering the particular case of affine constraints over a fixed interval: in this case, the abnormality turns out to have a global character, so that it is possible to define an abnormal problem or a normal problem. The last section is devoted to the study of an optimal control problem characterized by differential constraints corresponding to the dynamics of a controlled process. The above general results are particularized to this problem, yielding a necessary and sufficient condition for an admissible solution to be an abnormal extremal. From this, a previously known result is recovered concerning the linearized system controllability as a sufficient condition to exclude the abnormality. 相似文献
9.
G. Sorger 《Journal of Optimization Theory and Applications》1990,65(2):321-329
We show how to construct a Lagrange problem which has the property that its extremals are the solutions of a given differential equation and which satisfies certain structural assumptions. These assumptions require that the integrand is either a concave function or that it is additively separable. In the first case, which is relevant in economics, we present a continuous-time analogue of the indeterminacy result of Boldrin and Montrucchio. The second case is illustrated by a minimum principle for the logistic growth function. 相似文献
10.
We define, and discuss some properties of, a form which bears a relation to the Carathéodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincaré–Cartan form for one-dimensional variational problems. 相似文献
11.
Set-Valued and Variational Analysis - In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume... 相似文献
12.
Robert Černý 《Calculus of Variations and Partial Differential Equations》2007,28(2):203-216
We compute the relaxation
where
for sequences of functions from
converging strongly in the
-norm to
. 相似文献
13.
R. M. Bianchini 《Journal of Optimization Theory and Applications》1994,81(2):249-258
Concerning small-time local controllability at a point, we prove that it is possible to use a simple definition of variation of the control process at the initial point. The paper generalizes an analogous result by Frankowska.This research was supported by a grant from the Ministero dell'Università e della Ricerca Scientifica e Tecnologica, Rome, Italy. 相似文献
14.
The inverse problem of the calculus of variations for second-order nonlinear and linear systems of differential-difference equations is considered. The relationship between the formal potentiality of a linear system with constant coefficients and the parity of its characteristic function is established. 相似文献
15.
Stan Alama Lia Bronsard Etienne Sandier 《Transactions of the American Mathematical Society》2008,360(1):1-34
We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. To model experiments in which the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes, we study the structure of isolated vortices for a doubly periodic problem. We consider a singular limit which simulates certain experimental regimes in which isolated vortices have been observed, corresponding to letting the interlayer spacing of the superconducting planes tend to zero and the Ginzburg-Landau parameter simultaneously, but at a fixed relative rate.
16.
H. Wojewódka 《Statistics & probability letters》2013,83(10):2337-2347
The exponential rate of convergence for Markov operators is established. The operators correspond to continuous iterated function systems which are a very useful tool in some cell cycle models. 相似文献
17.
For problems in the calculus of variations with
isoperimetric side constraints, we provide in this paper a set of
points whose emptiness, independently of nonsingularity assumptions,
is equivalent to the nonnegativity of the second variation along
admissible variations. The main objective of introducing a
characterization of this condition should be, of course, to obtain a
simpler way of verifying it. There are two other sets of points
available in the literature, introduced by Loewen and Zheng (1994)
and Zeidan (1996), for which this necessary condition implies their
emptiness. However, we show that verifying membership of these sets
may be more difficult than checking directly if that condition
holds. Contrary to this behavior, we prove that the desired
objective of characterizing that condition is achieved by means of
the set introduced in this paper. 相似文献
18.
We consider a Ginzburg-Landau functional for a complex vector order parameter Ψ=(ψ+,ψ−), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R2 which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one component of the solution vanishes, but the other does not. We also prove the existence and uniqueness of equivariant entire solutions, and provide a second proof of uniqueness, valid for a large class of systems with variational structure. 相似文献
19.
We consider quasi-isometric mappings of domains in multidimensional Euclidean spaces. We establish that a mapping depends continuously in the sense of the topology of Sobolev classes on its metric tensor to within isometry of the space. In the space of metric tensors we take the topology determined by means of almost everywhere convergence. We show that if the metric tensor of a mapping is continuous then the length of the image of a rectifiable curve is determined by the same formula as in the case of mappings with continuous derivatives. (Continuity of the metric tensor of a mapping does not imply continuity of its derivatives.) 相似文献
20.
Holomorphic representation of Lie algebra can be realized through Kählerian symplectic formalism; underlying holomorphic convexity requires then the introduction of elliptic operators with complex coefficients. We construct the Stochastic Calculus of Variations for those elliptic operators; remote past vanishing of projections of the underlying process implies convergence in law; then limit laws lead to the unitarizing measure of the given representation; this general approach is developed in full details on Poincaré disk. 相似文献