Optimal control of parabolic variational inequalities with delays and state constraint

In this paper, an optimal control problem for parabolic variational inequalities with delays and state constraint is investigated and the necessary conditions for optimal controls are derived.     

Optimal control of parabolic variational inequalities with delays and state constraint

In this paper, an optimal control problem for parabolic variational inequalities with delays and state constraint is investigated and the necessary conditions for optimal controls are derived.     

Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs

Optimal stopping and impulse control problems with certain multiplicative functionals are considered. The stopping problems are solved by showing the unique existence of the solutions of relevant variational inequalities. However, since functions defining the multiplicative costs change the signs, some difficulties arise in solving the variational inequalities. Through gauge transformation we rewrite the variational inequalities in different forms with the obstacles which grow exponentially fast but with positive killing rates. Through the analysis of such variational inequalities we construct optimal stopping times for the problems. Then optimal strategies for impulse control problems on the infinite time horizon with multiplicative cost functionals are constructed from the solutions of the risk-sensitive variational inequalities of "ergodic type" as well. Application to optimal investment with fixed ratio transaction costs is also considered.     

Optimal shape design for systems governed by variational inequalities,part 3: Necessary conditions in the elliptic case

General optimality conditions are obtained for optimal shape design for systems governed by a class of elliptic variational inequalities. The conditions are established by making use of the necessary conditions for optimal control of systems governed by strongly monotone variational inequalities. These conditions are then applied to an electrochemical machining problem.     

Optimal Control of Obstacle Problems by H1-Obstacles

Optimal control of variational inequalities with the controls given by the obstacles is considered. Existence optimal solutions are proved for obstacles with H¹ regularity and first-order optimality conditions are derived which, under additional assumptions, are also sufficient. A numerical algorithm is proposed and its practical feasibility is investigated.     

Optimality conditions for strongly monotone variational inequalities

Necessary conditions for optimal controls have been obtained for strongly monotone variational inequalities by the penalty method, Ekeland's Variational Principle, and lower-semicontinuity of set-valued mappings. It has been shown that these conditions are easy to apply and can imply some known necessary conditions. They also yield new optimality conditions.     

On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind

The directional differentiability of the solution mapping for a class of variational inequalities of the second kind inspired by applications in fluid mechanics and moving free boundary problems is investigated. The result is particularly relevant for the model predictive control or optimal control of such variational inequalities in that it can be used to derive stationarity conditions and efficient numerical methods.     

Optimal control of multivalued quasi variational inequalities

Optimal control of various variational problems has been an area of active research. On the other hand, in recent years many important models in mechanics and economics have been formulated as multi-valued quasi variational inequalities. The primary objective of this work is to study optimal control of the general nonlinear problems of this type. Under suitable conditions, we ensure the existence of an optimal control for a quasi variational inequality with multivalued pseudo-monotone maps. Convergence behavior of the control is studied when the data for the state quasi variational inequality is contaminated by some noise. Some possible applications are discussed.     

直线上函数的Banach空间的Poincaré型不等式的变分公式

基于研究对数Sobolev,Nash和其它泛函不等式的需要,将Poincare不等式 的变分公式拓广到一大类直线上函数的Banach(Orlicz)空间.给出了这些不等式成立 与否的显式判准和显式估计. 作为典型应用,仔细考察了对数Sobolev常数.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Maximum Principles for Optimal Controls for Elliptic Variational Inequalities of the Second Kind

A maximum principle is obtained for optimal controls for someelliptic variational inequalities of the second kind by usingthe penalty method, Ekelans's variational principle, and lowersemicontinuity of some set-valued mappings. It has been shownthat this principle leads to some known optimality conditionsin many cases. It also yields new optimality conditions. Insome cases, it leads to an analogy of pontryagin's principle. Permanent address: Mathematics Department, Centre South Universityof Technology, China.     

Projection Method Approach for General Regularized Non-convex Variational Inequalities

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.     

Direct Numerical Algorithm for Constrained Variational Problems

We propose a direct treatment for the numerical simulation of optimal solutions for vector, one-dimensional variational problems under pointwise constraints in the form of several inequalities. It is an iterative procedure to approximate the optimal solutions of such variational problems that rely on our ability to e?ciently approximate the optimal solutions of variational problems without restrictions, except possibly for end point constraints. One main advantage is that there is no need to control the free boundary, or the contact set, during the iterative process where constraints are active. In addition to proving some convergence results, the scheme is illustrated through several typical situations.     

Optimal Control of Semilinear Elliptic Variational Bilateral Problem

This paper is concerned with an optimal control problem for semilinear elliptic variational inequalities associated with bilateral constraints. Existence and optimality conditions of the optimal pair are established. Received August 17, 1998, Accepted December 28, 1998     

Approximation of optimal distributed control problems governed by variational inequalities

Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.     

Optimization of the thickness of layers of sandwich conical shells

We deal with an optimal control problem for variational inequalities, where the linear operators as well as the convex sets of possible states depend on the control parameter. The existence theorem for optimal control is applied to optimal design problems for sandwich conical shells where a variable thickness of given layers appears as a control variable.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

半直线上L~p-Poincaré不等式最优常数的估计

运用分析方法给出半直线上测度有限时在Dirichlet边界条件下L~p-Poincaré不等式最优常数的变分公式,并运用迭代方法结合变分公式得到了最优常数的显式估计.     

A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C^1,1-regular away for the free boundaries of the action sets.     

模糊映射的完全广义混合型强变分不等式

研究关于模糊映射的一类新的变分不等式－模糊映射的完全广义混合型强变分不等式,得到此类变分不等式解的存在定理分解的一个逼近算法,推广了文[4]和文[8]的主要结果。