奇异线性二次型微分鞍点对策的小波逼近解法

研究广义状态系统中线性二次型微分对策鞍点策略的数值求解问题。基于小波多尺度多分辨逼近特性 ,提出了一种数值求解新方法。该法基于Daubechies小波的优良性质 ,特别是将Daubechies小波基的积分运算矩阵、乘积矩阵和快速离散小波变换系数矩阵应用于原问题的主要方程 ,将原问题转化为矩阵代数优化问题 ,避免直接计算耦合Riccati微分方程。算法简洁明了 ,适合于计算机求解。实例计算结果显示 ,该算法是可行的     

Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

For a nonlinear finite time optimal control problem, a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper. This algorithm allows one to obtain the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively. Furthermore, the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition. Once a generating function is found, it can be used to generate a family of optimal control for different boundary conditions. Since the generating function is computed off-line,the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method. It is useful to online optimal trajectory generation problems. Numerical examples illustrate the effectiveness of the proposed algorithm.     

导弹追逃博弈微分对策建模与求解

针对导弹攻防对抗过程中拦截器追击具备较强机动能力弹头的追逃问题,建立了双方追逃微分对策模型并给出求解方法.一是给出导弹追逃质点动力学模型;二是基于微分对策理论,建立了导弹攻防对抗微分对策模型,模型以推力角为控制变量,高度,速度和经度角为状态变量,并考虑了地球重力和自转的影响;三是针对模型获得解析解的困难,给出高精度四阶Gauss-Lobatto多项式配点法来逼近非线性方程,通过离散化节点和配点上的状态量和控制量将微分方程组转换为代数约束;四是为采用配点法求解模型,给出了将双边最优对策问题转化为单边最优对策问题的具体方法.最后实例分析对本文研究进行了仿真验证.     

基于Lanchester方程的作战混合动态系统最优控制

针对作战过程的混合动态特性, 利用Lanchester方程建立了一类作战混合动态系统模型, 在合理战术假设的基础上, 讨论了一类变招顺序固定的作战决策方最优控制问题. 利用动态规划原理给出了解决问题的基本框架与途径. 最后, 通过应用算例验证了所建立模型和所设计最优控制方案的可行性. 研究结果对作战指挥决策和对策的研究具有理论指导意义.     

车辆路径问题(VRP)的蚂蚁搜索算法

车辆路径问题(vehicle routing problem，VRP)是组合优化中一个典型的NP难题，理论上，目前仅能保证一些相对小规模的问题可求得最优解．基于近些年出现的新型智能优化思想：人工蚂蚁系统，给出了一种可快速求解VRP的蚂蚁搜索算法．通过定义基本的人工蚂蚁状态转移概率，并结合局部搜索策略，用迭代次数控制算法的运行时间，从而使该方法具有实用意义和可操作性．经一系列数据测试和验证，并与若干已有的经典算法相比较．获得了较好的结果．     

FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS, LINEAR QUADRATIC STOCHASTIC OPTIMAL CONTROL AND NONZERO SUM DIFFERENTIAL GAMES

In this paper, we use the solutions of forward-backward stochastic differential equations to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and the open-loop Nash equilibrium point for nonzero sum differential games problem. We also discuss the solvability of the generalized Riccati equation system and give the linear feedback regulator for the optimal control problem using the solution of this kind of Riccati equation system.     

Inertial projection algorithms for convex feasibility problem

The purpose of this paper is to apply inertial technique to string averaging projection method and block-iterative projection method in order to get two accelerated projection algorithms for solving convex feasibility problem.Compared with the existing accelerated methods for solving the problem,the inertial technique employs a parameter sequence and two previous iterations to get the next iteration and hence improves the flexibility of the algorithm.Theoretical asymptotic convergence results are presented under some suitable conditions.Numerical simulations illustrate that the new methods have better convergence than the general projection methods.The presented algorithms are inspired by the inertial proximal point algorithm for finding zeros of a maximal monotone operator.     

动力学系统数值仿真并行算法的发展

本文主要综述动力学系统仿真的非刚性、刚性系统和微分代数系统并行数值方法的一些最近发展。     

导弹攻防对抗中追逃对策模型与配点求解法

未来的来袭导弹可能具备较强的机动性,其弹道不可预测,针对拦截弹追击此类目标的追逃问题,基于微分对策(differential-game, DG)理论建立追逃博弈模型并给出求解方法。模型在分析两者相对运动的基础上,考虑地球重力和自转的影响,以推力角为控制变量,离地高度、速度和经度角为状态变量,建立微分方程组。然后将追逃DG模型转化为单边最优对策问题;并给出改进的高精度五阶Gauss-Lobatto多项式配点法来近似状态变量对时间的导数,将微分方程组转换为代数约束,降低非线性规划问题复杂程度。最后给出了本文研究的仿真实例。     

切向加速度与弦误差约束的CNC系统时间最优轨迹规划

针对计算机数控(CNC)系统给定参数化路径, 给出了一种求解时间最优轨迹规划问题的凸优化方法. 轨迹规划问题考虑切向加速度约束与弦误差约束. 通过建立两种约束下的状态容许空间, 分析约束对时间最优轨迹的影响. 通过非线性变量代换, 时间最优轨迹规划问题被表述为一个与时间无关的凸最优控制问题. 基于控制向量参数化(CVP)方法, 问题被进一步转化为易于求解的凸优化问题. 以路径参数对时间的二阶导数(参数加速度)为优化变量, 序列二次规划(SQP)方法获得问题数值解. 文末通过求解两个测试路径的时间最优轨迹规划问题, 验证方法的有效性.     

非线性控制系统单支方法的IS稳定性

控制系统在实际问题中有广泛应用,众多文献对系统本身及其数值方法的稳定性进行了深入研究。将单支方法用于求解非线性控制系统,获得了方法IS稳定的条件,可视为单支方法关于非线性常微分方程的稳定性分析在非线性控制系统的进一步推广。最后给出了一些常用的单支方法IS稳定的条件。     

Variational discretization for parabolic optimal control problems with control constraints

This paper studies variational discretization for the optimal control problem governed by parabolic equations with control constraints.First of all,the authors derive a priori error estimates where |||u-U₍h|||_L∞（J;L²（Ω））=O（h²+k）.It is much better than a priori error estimates of standard finite element and backward Euler method where |||μ-U_h|||₍L∞（J;L²（Ω））=O（h+ k）.Secondly,the authors obtain a posteriori error estimates of residual type.Finally,the authors present some numerical algorithms for the optimal control problem and do some numerical experiments to illustrate their theoretical results.     

Interest rate risk premium and equity valuation

<正> The authors employ the recent stochastic-control-based approach to financial mathematicsto solve a problem of determination of the risk premium for a stochastic interest rate model,andthe corresponding problem of equity valuation.The risk premium is determined explicitly,by meansof solving a corresponding partial differential equation (PDE),in two forms:one,time-dependent,corresponding to a finite time contract expiration,and the simpler version corresponding to perpetualcontracts.As stocks are perpetual contracts,when solving the problem of equity valuation,the latterform of the risk premium is used.By means of solving the general pricing PDE,an efficient equityvaluation method was developed that is a combination of some sophisticated explicit formulas,and anumerical procedure.     

基于动力系统的无约束优化问题的方法分析

通过解由常微分方程构成的动力系统的稳定点得到等价的无约束优化问题的局部极小点 ,而动力系统的稳定点可以沿动力系统轨线上的任一点通过路径跟踪得到。我们发现 ,在用Euler方法求解二次优化问题的等价动力系统的方程时 ,由方法的步长确定的稳定区域对应于这些方法所得到的迭代公式的步长满足单调下降算法的条件确定的单调下降区域 ,因此我们可以利用这个性质构造解无约束优化问题的数值方法而不采用标准的常微分方程的数值求解公式。分析了一些基于微分方程的无约束优化方法并举例说明这些方法有些是数值不可行的。     

Approximation methods of mixed l_1/H_2 optimization problems for MIMO discrete-time systems

The mixed l1/H2 optimization problem for MIMO (multiple input-multiple output) discrete-time systems is considered. This problem is formulated as minimizing the l1-norm of a closed-loop transfer matrix while maintaining the H2-norm of another closed-loop transfer matrix at prescribed level. The continuity property of the optimal value in respect to changes in the H2-norm constraint is studied. The existence of the optimal solutions of mixed l1/H2 problem is proved. Because the solution of the mixed l1/H2 problem is based on the scaled-Q method, it avoids the zero interpolation difficulties. The convergent upper and lower bounds can be obtained by solving a sequence of finite dimensional nonlinear programming for which many efficient numerical optimization algorithms exist.     

基于变阶长迭代动态规划的聚合物驱方案动态优化

聚合物驱是一种重要的提高原油采收率技术. 针对聚合物驱的注入方案优化问题, 建立了聚合物驱最优控制模型. 该模型以利润最大为性能指标, 以聚合物驱的渗流力学方程为支配方程, 并以聚合物最大用量约束和聚合物注入浓度的上下限约束为不等式约束条件. 各段塞间的注入浓度、段塞尺寸及驱油结束时间为该最优控制问题的优化变量. 对于该分布参数最优控制问题, 提出了一种变阶长迭代动态规划算法. 首先引入标准化的时间变量, 将原自由终端时间的聚合物驱最优控制问题转化为固定终端时间问题, 然后采用经典的迭代动态规划算法求解, 实现了注入浓度、段塞尺寸及驱油结束时间的同时优化, 最后通过实例验证了所提出算法的有效性.     

裂隙-岩溶介质空间水流数值仿真与流场优化

裂隙-溶隙水流系统的数值仿真具有明显的优点，数值仿真试验可以通过多种数值方法，识别流场特征，刻画流场时空演化规律，本文基于研究区地下水系统概念模型。以非均质各向异性渗流模型仿真裂隙-岩溶水流，用有限元法求数值解，先解逆问题反演求参，再解正问题计算地下水开采资源量，耦合地下水数值仿真模型与线性规划，建立地下水优化管理模型。通过设计地下水降深分布求最优开采量的方法，实现对地下水渗流场的优化调控。     

利用高斯伪谱法求解具有最大横程的再入轨迹

为了使升力式飞行器再入大气层后取得最大横程,采用高斯伪谱方法求解最优再入轨迹。利用微分形式高斯伪谱方法将飞行器三自由度再入轨迹优化问题转化为非线性规划问题,选取高斯节点上的状态量和控制量作为待优化参数,并将最优性能指标选为横程最大,然后对再入轨迹进行了求解。通过与按最大升阻比飞行方案所得结果进行对比,表明按所提方法求取的再入轨迹优于后者。此外,仿真过程还说明高斯伪谱法对状态猜测值并不敏感,算法容易收敛,适用于轨迹优化问题的求解。     

Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations

1 .INTRODUCTIONStudies have shownthat some systems are highlysen-sitive :a small perturbationin the data can result in alarge changeinthe solutions .Such systems are calledill-conditioned systems .Ill-conditionedlinear systemsof equations have a wide application in many fieldssuch asi magine processing,deconvolution, model pa-rameters esti mation.Because the condition number ofill-conditionedlinear systems of equationsis very big,the data error andthe rounding error inthe computa-tional p…     

Optimal control using microscopic models for a pollutant elimination problem

Optimal control problem with partial derivative equation (PDE) constraint is a numerical-wise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a new approach to solve this type of problem by space discretization and transform it into a standard optimal control for a multi-agent system. This resulting problem is formulated from a microscopic perspective while the solution only needs limited the macroscopic measurement due to the approach of Hamilton-Jacobi-Bellman (HJB) equation approximation. For solving the problem, only an HJB equation (a PDE with only terminal boundary condition) needs to be solved, although the dimension of that PDE is increased as a drawback. A pollutant elimination problem is considered as an example and solved by this approach. A numerical method for solving the HJB equation is proposed and a simulation is carried out.