基于不确定控制理论的最优策略研究

本文在不确定理论的框架下,研究一类带背景状态变量的最优控制模型.在乐观值准则下,利用不确定动态规划的方法,证明了不确定最优性原则,得到最优性方程.作为应用,求解一个固定缴费(DC)型养老金的最优投资策略问题,在乐观值准则下,以工资变量为背景状态变量,建立养老金模型.通过求解不确定最优性方程得到最优投资策略和最优支付率.     

Optimal Harvesting in an Age-Structured Predator–Prey Model

We investigate optimal harvesting control in a predator–prey model in which the prey population is represented by a first-order partial differential equation with age-structure and the predator population is represented by an ordinary differential equation in time. The controls are the proportions of the populations to be harvested, and the objective functional represents the profit from harvesting. The existence and uniqueness of the optimal control pair are established.     

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

Use of a range scaling method to determine alanine/water stopping power ratios

A phantom composed of alanine dosimeter material has been constructed and depth–dose measurements made in a 10 MeV electron beam. The results have demonstrated the feasibility of using relative depth–dose measurements to determine stopping power ratios in materials of dosimetric interest. Experimental stopping power ratios for alanine dosimeter material and water agreed with the data of ICRU Report 37 within the uncertainty of the experiment (±1.2% at a 95% confidence level).     

Differentiability of the Value Function of Nonclassical Optimal Growth Models

We consider an optimal growth (multi-sector) model with nonconvex technology. Using the Clarke results on generalized gradients, we prove that the value function has left and right derivatives with respect to the initial capital stock, without requiring supermodularity assumptions.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

一类经典”秘书问题”的推广

”秘书问题”在最优停时理论的发展中曾起过重要作用 ,实际中的一类问题与”秘书问题”有类似之处 ,但比”秘书问题”更复杂 .本文将经典”秘书问题”进行推广 ,建立了一类比经典”秘书问题”更有实际意义的模型 ,并给出了该类模型的解 .     

Optimal purchase and advertising for a product with immediate sale start

Summary We consider the problem of maximizing the discounted net profit of a firm which purchases a quantity of some product at a given time and afterwards advertises and sells the product progressively. We distinguish among the three possibilities of assuming the final time to be either fixed, or bounded, or free. In all cases, after stating the problem in the optimal control theory framework, we prove the existence of an optimal solution and characterize it using the Maximum Principle necessary conditions. Furthermore, we prove that the convexity of the purchase cost function is a sufficient condition for the uniqueness of the optimal solution. Partially supported by MURST.     

Optimal vaccine scheduling in cancer immunotherapy

Cancer immunotherapy aims at stimulating the immune system to react against cancer stealth capabilities. It consists of repeatedly injecting small doses of a tumor-associated molecule one wants the immune system to recognize, until a consistent immune response directed against the tumor cells is observed.

We have applied the theory of optimal control to the problem of finding the optimal schedule of injections of an immunotherapeutic agent against cancer. The method employed works for a general ODE system and can be applied to find the optimal protocol in a variety of clinical problems where the kinetics of the drug or treatment and its influence on the normal physiologic functions have been described by a mathematical model.

We show that the choice of the cost function has dramatic effects on the kind of solution the optimization algorithm is able to find. This provides evidence that a careful ODE model and optimization schema must be designed by mathematicians and clinicians using their proper different perspectives.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.