On Some Elliptic Problems in Unbounded Domains

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.     

New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

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.     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

On a Class of Direct Optimization Problems

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.     

二维无界连通域上反常积分敛散性的判别准则

讨论了二维无界连通域上反常积分的敛散性.从柯西积分判别法出发,考察了三类特定区域上二元函数的反常积分,利用列维定理,得到了一些新的判别准则.     

变分泛函的半连续性问题

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。     

On the variational process in optimal control theory

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.     

A Study on Abnormality in Variational and Optimal Control Problems

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.     

On the optimality of given feedback controls

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.     

The Hilbert–Carathéodory Form for Parametric Multiple Integral Problems in the Calculus of Variations

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.     

Infinite Horizon Problems in the Calculus of Variations

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...     

Relaxation of an area-like functional for the function $$\frac{x}{|x|}$$

We compute the relaxation

Variations of a control process at the initial point

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.     

Inverse Problem of the Calculus of Variations for Systems of Differential-Difference Equations of Second Order

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.     

On the shape of interlayer vortices in the Lawrence-Doniach model

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.

     

Exponential rate of convergence for some Markov operators

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.     

A New Notion of Conjugacy for Isoperimetric Problems

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.     

On the structure of fractional degree vortices in a spinor Ginzburg-Landau model

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.     

Mappings of Domains in Rn and Their Metric Tensors

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.)     

Stochastic Calculus of Variations on complex line bundle and construction of unitarizing measures for the Poincaré disk

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.