Properties of the Bellman Function in Time-Optimal Control Problems

In this paper, time-optimal control problems with closed terminal sets are considered. We give conditions which guarantee the Bellman function to be Hölder and Lipschitz continuous. We then prove that the condition for Lipschitz continuity is also necessary.     

具有生产性与福利性公共支出的随机内生增长模型

文章建立一个随机内生增长模型来阐明主要政策参数对经济增长与社会福利的影响.若对生产函数、效用函数、偏好及随机干扰作一些特殊的假设,我们证明了主要政策参数的均衡值能被模型参数唯一决定.进一步我们还得到了期望增长率与储蓄的清晰解.文章的最后,我们证明了政府支出直接影响个体决策者的决策:即提高经济增长率将减少福利;反之,增加福利将减少增长率.     

Optimal Consumption in a Growth Model with the CES Production Function

Abstract

This article studies the optimization problem of maximizing the expected discounted present value of lifetime utility of consumption in the framework of one-sector neoclassical growth model with the Constant Elasticity of Substitution (CES) production function. We establish the existence of a classical solution of the Hamilton–Jacobi–Bellman (HJB) equation associated with this problem by the technique of viscosity solutions under the strict concavity of the utility function, and hence derive an optimal consumption from the optimality conditions in the HJB equation.     

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

The optimal value function of the quadratic program , where is a given symmetric matrix, a given matrix, and are the linear perturbations, is considered. It is proved that is directionally differentiable at any point in its effective domain . Formulae for computing the directional derivative of at in a direction are obtained. We also present an example showing that, in general, is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.     

Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

一类高阶超双曲型方程的中量定理及其逆定理

Asgeirsson中量定理表明超双曲型方程的Cauchy问题一般是不适定的,对Asgeirsson中量定理的推广就有重要意义。目前关于高阶方程解的中量只有初步探讨,尚未得到具体结果,本文直接利用Asgeirsson中量定理结果和积分、微分的性质与关系,得到了高阶方程解的中量满足广义双轴对称位势方程,同时还证明了其逆定理。利用关于广义双轴对称位势方程正则解的表达式及雅可比多项式的特殊性质,得到了高阶方程解的中量公式,从而使得关于解的拓展性和适定性的讨论将有可能。     

非线性经济增长模型的最优控制问题

研究了带有中性技术进步生产函数边界条件的非线性经济增长模型的最优控制问题.利用Banach空间不动点原理,得到了系统解的存在唯一性,利用Gron-wall不等式得到了系统解关于控制序列的连续依赖性,借助于法锥和共轭系统,得到了控制问题最优解存在的必要条件.     

Optimal Dynamics of Innovation in Models of Economic Growth

A nonlinear model of economic growth which involves production, technology stock, and their rates as the main variables is considered. Two trends (growth and decline) in the interaction between the production and R&D investment are examined in the balanced dynamics. The optimal control problem of R&D investment is studied for the balanced dynamics and the utility function with the discounted consumption. The Pontryagin optimality principle is applied for designing the optimal nonlinear dynamics. An existence and uniqueness result is proved for an equilibrium of the saddle type and the convergence property of the optimal trajectories is shown. Quasioptimal feedbacks of the rational type for balancing the dynamical system are proposed. The growth properties of the production rate, R&D, and technology intensities are examined on the generated trajectories.     

On the existence of an optimal element for a delay control problem

The existence theorems of the optimal element are proved for a nonlinear control problem with constant delay in phase coordinates and with general functional. Here element implies the collection of delay parameter and initial function, initial moment and vector, control and finally moment.     

Optimal recovery of solutions of the generalized heat equation in the unit ball from inaccurate data

We consider the problem of optimal recovery of solutions of the generalized heat equation in the unit ball. Information is given at two time instances, but inaccurate. The solution is to be constructed at some intermediate time. We provide the optimal error and present an algorithm which achieves this error level.     

On the optimality of the logistic growth

The rate at which a population should grow is determined by finding the best trade-off between the loss due to the deviation from a target population size and the loss associated to the growing effort. It is also shown that, in the case of infinite-time horizon and quadratic loss functions, the optimal growth is logistic. This research was supported by the Centro Teoria dei Sistemi, CNR, and by the Italian Ministry of Public Education.     

Optimal pollution control,irreversibilities, and the value of future information

This paper focuses on the issue of optimal pollution control when either pollution itself is irreversible or when some characteristic of the environmental resource is irreversibly destroyed in the course of growing pollution. It is shown that exhausting the assimilative capacity through too heavy pollution is never optimal unless the rate of social time preference is sufficiently high. The paper also investigates the case that decisions about irreversible developments have to be made under uncertainty today when the decision maker faces the prospect of better information about the irreversible damage at some future point in time. A non-negative quasi-option value is shown to exist as in the Arrow-Fisher-Henry model that relates to natural resource deletion by projects of industrial development.     

The Value Function of Singularly Perturbed Control Systems

The limit as ɛ→ 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton—Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton—Jacobi equation are applicable in cases where the reduction of order, namely setting ɛ = 0 , does not yield an optimal behavior. Accepted 18 November 1999     

Lipschitz Continuity of the Value Function in Optimal Control

For optimal control problems in with given target and free final time, we obtain a necessary and sufficient condition for local Lipschitz continuity of the optimal value as a function of the initial position. The target can be an arbitrary closed set, and the dynamics can depend in a measurable way on the time. As a limit case of this condition, we obtain a characterization of the viability property of the target, in terms of perpendiculars to the target instead of tangent cones. As an application, we analyze the convergence of certain discretization schemes for time-optimal problems.     

Existence of the Boundary Value of a Polyharmonic Function

We establish a criterion for existence of an L₂-limit and a weak L₂-limit of a polyharmonic function on a regular analytic boundary of a bounded plane domain.     

Optimal financing and dividend control of the insurance company with proportional reinsurance policy

We consider the optimal control problem of the insurance company with proportional reinsurance policy. The management of the company controls the reinsurance rate, dividends payout as well as the equity issuance processes to maximize the expected present value of the dividends minus the equity issuance until the time of bankruptcy. This is the first time that the financing process in an insurance model has been considered, which is more realistic. To find the solution of the mixed singular-regular control problem, we firstly construct two categories of suboptimal models, one is the classical model without equity issuance, the other never goes bankrupt by equity issuance. Then we identify the value functions and the optimal strategies corresponding to the suboptimal models depending on the relationships between the coefficients.     

Optimal distribution of the internal null control for the one-dimensional heat equation

This paper addresses the problem of optimizing the distribution of the support of the internal null control of minimal L²-norm for the 1-D heat equation. A measure constraint is imposed on the support but no topological assumption such as the number of connected components. Therefore, the problem typically lacks of solution in the class of characteristic functions and needs of relaxation. We show that the relaxed formulation is obtained by replacing the set of characteristic functions by its convex envelope. The proof requires that the observability constant related to the control problem be uniform with respect to the support, property which is obtained by the control transmutation method. The optimality conditions of the relaxed problem as well as the case where the number of connected components is fixed a priori are also discussed. Several numerical experiments complete the study and suggest the ill-posedness of the problem in contrast to the wave situation.     

投资项目的期权评价与最优投资规则

本文介绍了不确定环境下的投资项目的期权评价方法和最优投资规则，研究了单期项目和连续投资项目的投资决策问题，探讨了实物期权评价方法与传统的净现值评价方法中最优投资规则的差异，并对影响最优投资规则的差异因素进行了敏感性分析，得出了直观而有实用价值的结论。     

Optimal investment and reinsurance policies in insurance markets under the effect of inside information

In this paper, we study the problem of optimal investment and proportional reinsurance coverage in the presence of inside information. To be more precise, we consider two firms: an insurer and a reinsurer who are both allowed to invest their surplus in a Black–Scholes‐type financial market. The insurer faces a claims process that is modeled by a Brownian motion with drift and has the possibility to reduce the risk involved with this process by purchasing proportional reinsurance coverage. Moreover, the insurer has some extra information at her disposal concerning the future realizations of her claims process, available from the beginning of the trading interval and hidden from the reinsurer, thus introducing in this way inside information aspects to our model. The optimal investment and proportional reinsurance decision for both firms is determined by the solution of suitable expected utility maximization problems, taking into account explicitly their different information sets. The solution of these problems also determines the reinsurance premia via a partial equilibrium approach. Copyright © 2011 John Wiley & Sons, Ltd.