带有小参数变分问题的极小化序列

针对一类含有小参数的变分问题构造了零次渐近解,并证明了当小参数趋向于0时,该零次渐近解就是原问题的极小化序列.     

非定常经济系统控制模型的最优控制问题

利用Banach空间理论和Banach-Saks-Mazur定理,讨论了一类非定常经济系统中,资产相对折旧率的最优控制问题,给出了其最优解的存在唯一性.     

固定资产投资系统的稳定性分析及最优控制问题

给出带有时滞的一类固定资产模型，此模型为含有非局部和时滞边界条件的分布参数系统。通过Lyapunov函数，对系统的稳定性进行了分析，给出系统稳定的充分条件，然后，讨论了积累率的最优控制问题。根据Banach空间的一些理论，证明了其最优解的存在唯一性。     

具有劳动力增长的资产发展方程的最优控制问题

经济系统是一个复杂巨系统,具有复杂的层次结构.近年来,系统科学理论的发展为研究经济系统提供了新的思路和方法,已有了很大进展.劳动力是资产发展系统中的一个重要参数,对具有劳动力增长的非线性资产发展方程中劳动力的最优控制问题进行了研究.利用Banach空间理论,对极小化序列中的弱收敛序列,构造一强收敛极小化序列,得到了其最优解的存在性和唯一性,结果推广和改进了最近文献的一些主要结果.这个问题的研究对于促进我国经济高速、稳定持续增长具有重要的理论意义和现实指导价值.     

一类带指数权最优控制问题的Turnpike性质

该文主要研究一类带指数权的有限维最优控制问题的长时间渐近行为.主要方法是依据Pontryagin最大值原理,代数Riccati理论,对最优控制问题的Hamilton系统进行解耦分析,分别在两种不同的状态约束条件下建立最优轨道和最优控制的指数turnpike性质.     

具有备用部件的并联可修复系统稳态解的最优控制问题

针对一种具有两个运行部件和一个储备部件,考虑系统通常故障的发生,且系统故障服从一般分布的人—机系统模型.在Banach空间中,用泛数指标函数作为衡量系统可控的标准,给出了可修复系统最优控制的判别条件.     

刮板沉降箱式除尘可修复系统的最优控制

研究了刮板沉降箱式除尘可修复系统稳态解的最优控制问题.通过利用范数指标泛函作为控制变量的标准,并采用极小化序列的方法,得到了系统最优控制元.     

一个稳态PDE系统的状态约束极小能最优控制问题

本文研究一个由偏微分方程描述的调和型稳态系统带不等式或光顺约束条件的极小能最优控制问题。基于由笔者新近发展的样条函数最优解技术,我们对所论问题可以给出完整的显式解析型最优解。问题的最优解可以通过一种优雅的再生核结构来表达。这种新技巧还可以用来对其他不同领域如图象和信号的最优再现等问题作统一的处理,体现出样条函数新理论的优越性。 这种样条函数技巧的前身只能解决由常微分方程描述的集中参数系统问题。有关这方面的历史发展和评述可参阅笔者下面的综合报告。本文的结果是这一研究方向的新发展。     

一个备用部件的可修复人机系统的最优控制

讨论了两个运行部件并联和一个储备部件组成的可修复系统的非负解以及系统在某一时刻T的最优控制问题.     

可修复人机储备系统稳态解的最优控制

针对可修复人机储备系统的模型,以范数指标泛函作为衡量系统可控性的标准,利用Banach空间理论讨论系统稳态解达到预期概率分布的最优控制问题,给出了其最优解存在唯一性.     

Minimizing sequences in optimal control with approximately given input data and the regularizing properties of the Pontryagin maximum principle

Results related to the optimal control theory for systems with approximately given input data are presented. The basic (desired) element in the theory is the minimizing sequence of feasible controls rather than the classical optimal control. Necessary and sufficient conditions for minimizing sequences are established. The regularizing properties of the Pontryagin maximum principle and of minimizing sequences are discussed. Three basic regularization levels are singled out that are characteristic of any optimal control problem. The stability of the optimal value in a problem depending on the constraint parameter is discussed. Illustrative examples are considered in detail.     

Necessary extremum conditions in the optimal control problem with state constraints

The optimal control problem with state constraints is examined. An alternative to the available approaches to the study of this problem is proposed. The maximum principle and second-order necessary conditions are proved.     

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear–quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388–2399, 1978; Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129–148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour. Research partially supported by MEC grant MTM2004-07090-C03-03. SIMUMAT-CM, UC3M-MTM-05-028 and CCG06-UC3M/ESP-0850.     

An inverse optimal problem in discrete-time stochastic control

In this paper, we study an inverse optimal problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain optimal control problem. Our results extend to the stochastic case the work of Dechert. In particular, we present a stochastic version of an important principle in welfare economics.     

An optimal control problem for the multidimensional diffusion equation with a generalized control variable

The present paper is concerned with an optimal control problem for then-dimensional diffusion equation with a sequence of Radon measures as generalized control variables. Suppose that a desired final state is not reachable. We enlarge the set of admissible controls and provide a solution to the corresponding moment problem for the diffusion equation, so that the previously chosen desired final state is actually reachable by the action of a generalized control. Then, we minimize an objective function in this extended space, which can be characterized as consisting of infinite sequences of Radon measures which satisfy some constraints. Then, we approximate the action of the optimal sequence by that of a control, and finally develop numerical methods to estimate these nearly optimal controls. Several numerical examples are presented to illustrate these ideas.     

An optimal control problem with a random stopping time

This paper deals with a stochastic optimal control problem where the randomness is essentially concentrated in the stopping time terminating the process. If the stopping time is characterized by an intensity depending on the state and control variables, one can reformulate the problem equivalently as an infinite-horizon optimal control problem. Applying dynamic programming and minimum principle techniques to this associated deterministic control problem yields specific optimality conditions for the original stochastic control problem. It is also possible to characterize extremal steady states. The model is illustrated by an example related to the economics of technological innovation.This research has been supported by NSERC-Canada, Grants 36444 and A4952; by FCAR-Québec, Grant 88EQ3528, Actions Structurantes; and by MESS-Québec, Grant 6.1/7.4(28).     

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.     

Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.     

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.     

一类状态受限的随机延迟最优控制问题的最大值原理

本文考虑一类状态受限的随机延迟最优控制问题,其中控制域为凸集且扩散项系数中含有控制变量.控制域可以是无界集合.用最大值原理方法建立了最优控制满足的必要条件.也给出了充分最优性条件,从而有助于找到最优控制.