最值原理及其应用

最值原理:设m是常数. (1)若变量y≥m恒成立,且其等号会成立,则y的最小值是m; (2)若变量y≤m恒成立,且其等号会成立,则y的最大值是m. 灵活运用最值原理给解题带来简便,由于课本没有明确描述此原理,因此许多人对它的认识很模糊,以至不能很好地运用它。

一阶线性积分微分方程的边值问题

研究一阶线性积分—微分方程的边值问题．首先，应用格林函数得到核在Ｌ∞中的估计，从而得出解的存在唯一性结果．然后，构造出两个函数。

一类时空的分数阶组合扩散方程的初边值问题

本文研究了一类时空的具有分数阶组合导数的分数阶扩散方程,并给出了这类方程的最值原理,并应用最值原理得到了该方程解的估计以及解的唯一性与连续依赖性.

裂解法求函数最值的探讨

介绍了求函数最值的一种特殊解法—裂解法 ,并作了详尽的阐述 ,给出了具体的实例 .

一类具有年龄结构的捕食-食饵反应扩散恒化器模型

为研究物种的死亡率对恒化器模型中物种生存状态的影响,提出一类具有年龄结构的捕食-食饵反应扩散恒化器模型。应用最值原理、一致持续性理论、分歧理论等方法,讨论了模型解的全局存在性,分析了模型半平凡解的稳定性以及解的一致持续性,给出模型稳态解以死亡率为分歧参数的全局分歧结构。结果表明:存在死亡率的临界值,当死亡率小于临界值时物种共存,超过临界值捕食者死亡。

费马点最值问题推广例析

费马点最值问题是几何旋转变换的经典问题,在近年全国中考中多次考查.文章对此举几个实例从系数及最值原理两个角度去推广应用.

THE MAXIMUM PRINCIPLE FOR PARTIALLY OBSERVED OPTIMAL CONTROL OF FORWARD-BACKWARD STOCHASTIC SYSTEMS WITH RANDOM JUMPS

This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.

OPTIMAL PRICING, PRODUCTION, AND SALES POLICIES FOR NEW PRODUCT UNDER SUPPLY CONSTRAINT

This paper considers a firm that sells a durable product with a given market potential.The purpose of the firm is to maximize its profit by determining how much capacity to install before the sales horizon, how many products to produce in accordance with the capacity, and how many products to sell by pricing. Appealing to Pontryagin maximum principle in control theory, the authors obtain the closed-forms of all optimal decisions the firm should make. Furthermore, the optimal production rate and optimal sales rate are both equal to the demand rate, which is caused by the optimal pricing policy during the whole horizon, and the optimal pricing path is increasing with the cost of installing a unit of capacity. Furthermore, numerical analysis reveals the visual impression of the relationship of the parameters.

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms.When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions,the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.

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.

STOCHASTIC MAXIMUM PRINCIPLE FOR MIXED REGULAR-SINGULAR CONTROL PROBLEMS OF FORWARD-BACKWARD SYSTEMS

This paper considers a stochastic optimal control problem of a forward-backward system with regular-singular controls where the set of regular controls is not necessarily convex and the regular control enters the diffusion coefficient.This control problem is difficult to solve with the classical method of spike variation.The authors use the approach of relaxed controls to establish maximum principle for this stochastic optimal control problem.Sufficient optimality conditions are also investigated.

GEOMETRY OF COMPLETE HYPERSURFACES EVOLVED BY MEAN CURVATURE FLOW

Some geometric behaviours of complete solutions to mean curvature flow before the singu-larities occur are studied. The author obtains the estimates of the rate of the distance betweentwo fixed points and the derivatives of the second fundamental form. By use of a new maximumprinciple, some geometric properties at infinity are obtained.

On Stabilization of It Stochastic Time-Varying Systems

The stabilization with receding horizon control （RHC） of It5 stochastic time-varying systems is studied in this paper. Based on monotonically non-increasing of optimal cost and stochastic Lyapunov stability theory, a necessary and sufficient stabilization condition on the terminal weighting matrix is proposed, which guarantees the mean-square stability of the closed-loop system. The explicit receding horizon controller is obtained by employing stochastic maximum principle. Simulations demonstrate the effectiveness of the proposed method.

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland's variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.

A Geometric Problem and the Hopf Lemma. II

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.

A Maximum Principle for General Backward Stochastic Differential Equation

In this paper,the authors consider a stochastic control problem where the system is governed by a general backward stochastic differential equation.The control domain need not be convex,and the diffusion coefficient can contain a control variable.The authors obtain a stochastic maximum principle for the optimal control of this problem by virtue of the second-order duality method.

NONCONSTANT PREY HARVESTING IN RATIO-DEPENDENT PREDATOR-PREY SYSTEM INCORPORATING A CONSTANT PREY REFUGE

This paper deals with the problem of nonconstant harvesting of prey in a ratio-dependent predator-prey system incorporating a constant prey refuge. Here we use the reasonable catch-rate function instead of usual catch-per-unit-effort hypothesis. The existence, as well as the stability of possible equilibria, is carried out. Bionomic equilibrium of the system is determined and optimal harvest policy is studied with the help of Pontryagin＇s maximum principle. The key results developed in this paper are illustrated using numer- ical simulations. Our results indicate that dynamic behavior of the system very much depends on the prey refuge parameter and increasing amount of refuge could increase prey density and may lead to the extinction of predator population density.

Optimal Reinsurance Under Distortion Risk Measures and Expected Value Premium Principle for Reinsurer

This paper discusses optimal reinsurance strategy by minimizing insurer's risk under one general risk measure:Distortion risk measure.The authors assume that the reinsurance premium is determined by the expected value premium principle and the retained loss of the insurer is an increasing function of the initial loss.An explicit solution of the insurer's optimal reinsurance problem is obtained.The optimal strategies for some special distortion risk measures,such as value-at-risk(VaR) and tail value-at-risk(TVaR),are also investigated.

On Optimal Mean-Field Control Problem of Mean-Field Forward-Backward Stochastic System with Jumps Under Partial Information

This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When the coefficients of the system and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed. The authors also investigate the mean-field type optimal control problem for the system driven by mean-field type forward-backward stochastic differential equations(FBSDEs in short) with jumps, where the coefficients contain not only the state process but also its expectation under partially observed information. The maximum principle is established using convex variational technique. An example is given to illustrate the obtained results.

Latest advances in discontinuous deformation analysis method

Discontinuous deformation analysis （DDA） method, proposed firstly by Shi [1] in 1988, is a novel numerical approach to simulate the discontinuous deformation behaviors of blocky rock structures. In DDA, the domain of interest is represented as an assemblage of discrete blocks and the joints are treated as interfaces between blocks. The governing equations of DDA are derived from Newton’s Second Law of Motion and the Principle of Minimum Potential Energy.