Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

Global and local variational derivatives and integral representations of Gâteaux differentials

Let J be a real-valued functional on the space of continuous functions with the supremum norm. Suppose J has a continuous linear Gâteaux variation δJ[y; h]. Then δJ[y; h] has an integral representation δJ[y; h] = ∝ hdμ_y, where μ_y is a regular Borel measure depending on y. This article establishes conditions under which μ_y is absolutely continuous and its Radon-Nikodým derivative is a continuous function. Under these conditions, the (local) Volterra variational derivative is shown to exist everywhere and to be equal to the (globally defined) Radon-Nikodým derivative. The conditions are imposed directly on the Gâteaux variation. In addition, the article clears up some ambiguities in the literature on variational derivatives and provides a strong linkage between the global and local approaches to the variational derivative. Several of the lemmas developed for the proofs of the main results also establish what seem to be new measure-theoretic results for functions.     

Derivatives of the energy functional for 2D‐problems with a crack under Signorini and friction conditions

We consider the two‐dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non‐penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non‐linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed. Copyright © 2000 John Wiley & Sons, Ltd.     

向量优化问题广义有效点的一个存在性定理

利用集值映射不动点定理及最优化问题与变分不等式的关系给出线性G^↑ateaux可微的锥凸映射的广义有效点的一个存在性定理。     

Application of an Extremization Method to a Linear Integral of a Statistical Decision Problem

We consider the variational inequality that represents the first-order optimality condition for the class of variational problems with the property that the integrand in the objective functional does not depend on the derivative of the unknown function. This allows the development of an iterative method for solving the statistical decision problem of testing simple hypotheses.     

Maximum principle for optimal control of neutral stochastic functional differential systems

This paper is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.     

Fractional variational problems with the Riesz-Caputo derivative

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.     

An application of the Hurwitz theorem to the root analysis of the characteristic equation

In this work we show that a classical result of A. Hurwitz is still very effective in studying the root analysis of the characteristic equation for a linear functional differential equation. A conjecture was made by Funakubo et al. (2006) [3] regarding the asymptotic stability condition of the zero solution of a linear integro-differential equation of Volterra type. We applied the Hurwitz theorem to the characteristic equation in question and showed the existence of a root with positive real part and solved the conjecture. The Hurwitz theorem is expected to work well for the root analysis in critical cases.     

A general adjoint relation for functional differential and Volterra integral equations with application to control

A general adjoint relation is developed between solutions of linear functional differential equations and linear Volterra integral equations. Several useful representations for solutions of such equations arise as a consequence of the adjoint relationship. These representations are then used to obtain directly several results for controlling systems described by either linear functional differential equations or linear Volterra integral equations.This work was supported by the National Science Foundation under Grant No. GK-5798.     

On the Extremization of Linear Integrals

By means of a suitable variational inequality, we consider an extremization method for a particular class of integrals with the integrand of the objective functional linear with respect to the derivative of the unknown function. This method is closely related to the one proposed by Miele (Refs. 1–3) and, based on an application of the Green theorem concerning the transformation of line integrals into surface integrals, it can be extended to vector extremum problems under suitable regularity assumptions.     

Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions

The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on Cⁿ[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cⁿ[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.     

Variational procedures for a class of exterior interface problems

The problem under consideration is that of determining a function which is a solution of the Helmholtz equation in a planar region exterior to a simple closed curve and of an inhomogeneous Helmholtz equation inside the curve. Jump conditions on the function and its normal derivative across the cruve are given. The problem is first transformed into one involving the inner region only with a boundary condition which is non-local. This means that the solution at a point on the boundary is a functional of its values elsewhere. This second problem is further transformed into a variational form with all boundary conditions natural. It is shown that the variational problem has a solution. Finite dimensional approximate problems are defined and they are shown to have solutions converging to the solution of the variational problem.     

弹性力学中的立兹法和屈列弗兹法的一般推导

本文采用一般的数学表示形式推导了线弹性力学中的立兹法和屈列弗兹法,证明了立兹法给出相应泛函极值的上限,屈列弗兹法则给出其下限.同时发现,特征值问题(例如自振频率问题)泛函变分法中的屈列弗兹法同求特征值的放松边界条件下限法是一致的.当然,此处的推导结果,也适用于一类泛函的变分法中,这类泛函的欧拉方程是线性正定的.     

Properties of the solutions of a certain class of variational inequalities

In a curvilinear quadrangle one considers an elliptic operator with linear principal terms and discontinuous leading coefficients. One investigates the solution of a variational inequality with a constraint on the derivatives, tangent to the boundary and to the discontinuity lines of the coefficients. On certain parts of the boundary one imposes the first boundary condition and on others a condition on a directional derivative. One proves the existence of a solution with square summable second derivatives at each point of the subdomains where the leading coefficients are smooth.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 83–92, 1986.     

Lp-solutions of singular integro-differential equations

We study a variety of scalar integro-differential equations with singular kernels including linear, nonlinear, and resolvent equations. The first result involves a type of existence theorem which uses a fixed point mapping defined by the integro-differential equation itself and produces a unique solution with a continuous derivative in a very simple way. We then construct a Liapunov functional yielding qualitative properties of solutions. The work answers questions raised by Volterra in 1928, by Levin in 1963, and by Grimmer and Seifert in 1975. Previous results had produced bounded solutions from bounded perturbations. Our results mainly concern integrable solutions from integrable perturbations.     

Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

时滞积-微分方程配置解的多重校正

本文讨论线性Volterra时滞积一微分方程的连续分片多项式样条配置方法．在适当条件下,获得了配置解的导数在结点集上成立的精细误差展开式．以此为基础,给出了配置解的多重校正格式．     

Differential properties and optimality conditions for generalized weak vector variational inequalities

In this paper, we study a generalized weak vector variational inequality, which is a generalization of a weak vector variational inequality and a Minty weak vector variational inequality. By virtue of a contingent derivative and a Φ-contingent cone, we investigate differential properties of a class of set-valued maps and obtain an explicit expression of its contingent derivative. We also establish some necessary optimality conditions for solutions of the generalized weak vector variational inequality, which generalize the corresponding results in the literature. Furthermore, we establish some unified necessary and sufficient optimality conditions for local optimal solutions of the generalized weak vector variational inequality. Simultaneously, we also show that there is no gap between the necessary and sufficient conditions under an appropriate condition.     

A parabolic variational inequality related to the perpetual American executive stock options

A parabolic variational inequality is investigated which comes from the study of the optimal exercise strategy for the perpetual American executive stock options in financial markets. It is a degenerate parabolic variational inequality and its obstacle condition depends on the derivative of the solution with respect to the time variable. The method of discrete time approximation is used and the existence and regularity of the solution are established.     

屈服条件由应力张量的二次齐次函数与一次齐次函数之和来表达的极限分析定理

本文建立了由应力张量σ_ij的二次齐次函数与一次齐次函数的和来表达其屈服条件的刚理想塑性体的极限分析变分原理,它可用于岩土力学的极限分析问题,并把屈服条件为应力张量σ_ij 的二次齐次函数或一次齐次函数来表达的情况作为其特例.