可迹图的谱充分条件

设G是一个简单图,A(G),Q(G)以及Q(G)分别为G的邻接矩阵,无符号拉普拉斯矩阵以及距离无符号拉普拉斯矩阵,其最大特征值分别称为G的谱半径,无符号拉普拉斯谱半径以及距离无符号拉普拉斯谱半径.如果图G中有一条包含G中所有顶点的路,则称这条路为哈密顿路;如果图G含有哈密顿路,则称G为可迹图;如果图G含有从任意一点出发的哈密顿路,则称G从任意一点出发都是可迹的.主要研究利用图G的谱半径,无符号拉普拉斯谱半径,以及距离无符号拉普拉斯谱半径,分别给出图G从任意一点出发都是可迹的充分条件.     

哈密顿矩阵的逆特征值问题

该文探讨了哈密顿矩阵的逆特征值问题, 得到了有解的充要条件、通解的表达式以及最小范数解.并给出了最佳逼近解的求法. 给出了相应的算法, 数值实例说明算法是可行的.     

多分量AKNS方程的新可积分解

从一个任意阶矩阵谱问题出发,多分量AKNS方程的新可积分解被导出.通过利用迹恒等式建立了其双哈密顿结构.同时,证明了空间与时间的约束流在刘维尔意义下是两个完全可积的哈密顿系统.     

关于正规矩阵对广义特征值新的扰动界限

本文考虑了正规矩阵对的任意扰动时广义特征值的变化情况,给出了正规矩阵对任意扰动的Hoffman-Wielandt型扰动界,推广了正规矩阵对的相应的扰动结果.     

输出反馈二次型最优的可解性问题

本文研究了用输出反馈实现二次型最优控制的问题,指出任何最优输出反馈都是对应最优状态反馈的衍生解和在一般情况下最优输出反馈所满足的线性矩阵方程是不可解的.并讨论了输出矩阵含有待定参数的情形,给出了最优输出反馈存在的必要条件,对于单输入系统证明了该条件几乎是充分的.     

0-1对策的完全混合Nash均衡的代数求解法

将求解一般0-1策略对策的完全混合Nash均衡的问题转化为求解根为正的纯小数的高次代数方程组的问题.作为一种特殊而重要的情形,利用Pascal矩阵,Newton矩阵(对角元素为Newton二项式系数的对角矩阵)和Pascal-Newton矩阵(Pascal矩阵和Newton矩阵的逆阵的乘积)将求解对称0-1对策的完全混合Nash均衡的问题转化为求解根为正的纯小数的高次代数方程的问题,并给出第二问题的反问题(由完全混合Nash均衡求解对称0-1对策族)的求解方法.同时,给出了一些算例来说明对应问题的算法.     

奇异线性哈密顿算子自伴扩张的新描述

研究了具有一个奇异端点的线性哈密顿算子的白伴扩张的解析描述.设最小哈密顿算子h的亏指数为(d,d),将Im(h~*y,y)表示为秩为2d的二次型,该文利用二次型的表示矩阵得到了最小哈密顿算子h的自伴扩张域的一种新的完全描述.     

一些结构线性系统的SBE与P-SBE的比较

很多重要的结构矩阵都属于Rn或Cn上纯量积定义的Jordan代数J或者是Lie代数L.本文比较线性系统AX=B关于近似解的范数型结构向后误差(SBE)与偏结构向后误差(P-SBE).这里系数矩阵A∈J或A∈L.在给出若干预备性结果后,先对单右端项情形比较SBE与P-SBE,然后对多右端项情形比较.部分结果是Sun近期的一些结果的推广.     

极大加代数的对称代数S上互补基本矩阵

主要研究了极大加代数的对称代数S上互补基本矩阵,给出本征积的概念,证明了S上的Laplace定理,由此推出所有互补基本矩阵的行列式相等,且任意两个互补基本矩阵的行列式中的非零项均一一对应相等.在一个互补基本矩阵的行列式中,对于确定非零项的任一置换,给出了在另一个互补基本矩阵的行列式中找到置换使其确定相同非零项的方法.     

巴拿赫空间中演化算子的非一致多项式膨胀性

给出了Banach空间中演化算子多项式膨胀性的三个概念,讨论了演化算子的非一致多项式膨胀性的积分特征.且作为应用,利用Lyapunov函数给出了相应概念的刻画.从而将演化型算子指数膨胀性理论的相关经典结论推广到了多项式膨胀的情形.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

Optimal feedback control policies for stochastic,distributed-parameter systems

This paper considers optimal feedback control policies for a class of discrete stochastic distributed-parameter systems. The class under consideration has the property that the random variable in the dynamic systems depends only on the time and possesses the Markovian property with stationary transition probabilities. A necessary condition for optimality of a feedback control policy, which has form similar to the Hamiltonian form in the deterministic case, is derived via a dynamic programming approach.     

The Serret-Andoyer formalism in rigid-body dynamics: II. Geometry,stabilization, and control

This paper continues the review of the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, commenced by [1], and presents some new result. We discuss the applications of the SA formalism to control theory. Considerable attention is devoted to the geometry of the Andoyer variables and to the modeling of control torques. We develop a new approach to Stabilization of rigid-body dynamics, an approach wherein the state-space model is formulated through sets of canonical elements that partially or completely reduce the unperturbed Euler-Poinsot problem. The controllability of the system model is examined using the notion of accessibility, and is shown to be accessible. Based on the accessibility proof, a Hamiltonian controller is derived by using the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian controller is both passive and inverse optimal with respect to a meaningful performance-index. Finally, we point out the possibility to apply methods of structure-preserving control using the canonical Andoyer variables, and we illustrate this approach on rigid bodies containing internal rotors.      

辛映射与Hamilton控制系统

给定等价辛流形 ,即辛同态或形变等价的辛流形 ,研究了建立在这些辛流形上的Hamilton控制系统之间的一些性质的联系 ,诸如 (局部 )能观测性 ,强可接近性 ,(拟 )极小性等 .而且 ,利用Cort啨ｓ介绍的 (弱 )外等价系统的概念 ,给出使得两个Hamilton控制系统是辛同态的一个充分条件     

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

     

Deterministic near-optimal control,part 1: Necessary and sufficient conditions for near-optimality

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal control, for systems governed by deterministic ordinary differential equations. Necessary and sufficient conditions for near-optima control are studied. It is shown that any near-optimal control nearly maximizes the Hamiltonian in some integral sense, and vice versa, if some additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximality of the Hamiltonian are obtained. A number of examples are presented to illustrate these results.This work was supported by the RGC Earmarked Grant CUHK 249/94E. Helpful comments from L. D. Berkovitz are gratefully acknowledged.     

Homogenization in L

Homogenization of deterministic control problems with L^∞ running cost is studied by viscosity solutions techniques. It is proved that the value function of an L^∞ problem in a medium with a periodic micro-structure converges uniformly on the compact sets to the value function of the homogenized problem as the period shrinks to 0. Our main convergence result extends that of Ishii (Stochastic Analysis, control, optimization and applications, pp. 305-324, Birkhäuser Boston, Boston, MA, 1999.) to the case of a discontinuous Hamiltonian. The cell problem is solved, but, as non-uniqueness occurs, the effective Hamiltonian must be selected in a careful way. The paper also provides a representation formula for the effective Hamiltonian and gives illustrations to calculus of variations, averaging and one-dimensional problems.     

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     

Optimization of functionality development

The problem of regulation of logistic growth trends is examined in the framework of control theory for problems with infinite horizon. The problem statement is related to microeconomic models of dynamic optimization of a company’s indicator of functionality development. Various control regimes of functionality development are studied for identification of plausible production trends. Optimal control problem is posed to optimize the utility function of logarithmic consumption index of the system entropy type. Solution of the problem is constructed in analysis of the Hamiltonian system and its algebraic properties. Based on this analysis nonlinear stabilizers are elaborated which lead the system to the steady state with the same growth rates as the optimal control regime. The model is tested on the real time series of dynamic trends of functionality development for two generations of mobile phones in Japan.