非线性最优控制问题的保辛多层次求解方法

将非线性系统的最优控制问题导向Hamilton系统,提出了求解非线性最优控制问题的保辛多层次方法．首先,以时间区段两端状态为独立变量并在区段内采用Lagrange插值近似状态和协态变量,通过对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解．然后,在保辛算法的具体实施过程中提出了多层次求解思想,以2N类算法为基础由低层次到高层次加密离散时间区段,利用Lagrange插值得到网格加密后的初始状态与协态变量作为求解非线性方程组的初值,可提高计算效率．数值算例验证了算法在求解效率与求解精度上的有效性．     

基于拟线性化方法的非线性系统闭环反馈控制保辛算法

提出了一种求解非线性系统闭环反馈控制问题的保辛算法．首先,通过拟线性化方法将非线性系统最优控制问题转化为线性非齐次Hamilton系统两端边值问题的迭代格式求解．然后,通过作用量变分原理与生成函数构造了保辛的数值算法,且该算法保持了原Hamilton系统的辛几何性质．最后,通过时间步的递进完成状态与控制变量的更新,进而达到闭环控制的目的．数值算例表明:保辛算法具有较高的计算精度和较快的收敛速度．此外,将闭环反馈控制与开环控制分别应用于驱动小车上的倒立摆控制系统中,结果表明:在存在初始偏差的情况下,开环控制会导致稳定控制任务的失败,而闭环反馈控制能够在一段时间后消除初始偏差的影响,并使系统达到稳定状态．     

刚-柔体动力学方程的保辛摄动迭代法

针对刚-柔体动力学方程,提出保辛摄动迭代算法.该方法把刚-柔体动力学方程的低频运动和高频振动分开处理,用保辛摄动的思想来处理低、高频耦合作用,从而可以采用较大时间步长进行数值积分,即可给出满意的数值结果,很好地解决了刚性积分问题.数值算例表明该方法是可行的.     

求解无限维线性二次最优控制问题的一种新的途径

本文借助Ｃ_０半群的Ｙｏｓｉｄａ近似构造无限维线性二次最优控制问题的相应近似，证明了后者的最优控制、Ｒｉｃｃａｔｉ方程之解（从而反馈算子）和最优状态函数均一致强收敛，极限即为原问题的解．     

线性Hamilton系统边值问题的保辛数值方法

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.     

含分布及非线性离散时滞Volterra积分系统的近最优控制

研究由关于状态为(仿射)线性的兼含分布及非线性离散时滞Volterra积分方程系统、紧控制域约束和控制与状态分离型目标泛函构成的最优控制问题.证明了近最优控制的必要条件和充分条件,并将之用于求近最优控制的算法设计.     

半线性椭圆方程支配系统的最优性条件

本文讨论了可能具有多值解的椭圆型偏微分方程支配系统的最优控制问题,我们通过构造一个抛物方程控制问题的逼近序列,并利用抛物方程控制问题的结果,得到了椭圆系统最优控制的必要条件．     

求解Navier-Stokes方程最优控制问题的移动网格方法

提出了一种受偏微分方程约束最优控制问题的移动网格方法,并以NavierStokes方程为状态方程进行了研究.所采用的网格移动策略中节点距离的移动是通过求解一个扩散方程得到.设计出了有效的求解流体力学最优控制问题的算法,给出了算法的实施过程.提供的数值算例说明所提算法可以在保证高精度数值解的前提下稳定、高效的求解最优控制问题.     

机械多体系统动力学非线性最优控制问题的Noether理论

基于群不变性原理求解了机械多体动力学系统非线性最优控制问题的Noether型守恒定律.该文主要研究一类理想完整约束下的受控机械多刚体系统,通过增广向量法将动力学Euler-Lagrange方程以状态空间形式表示,利用变分法得到最优控制问题最优解的状态方程、伴随方程和控制方程,对系统性能指标泛函进行包含时间、状态变量、协态变量和控制变量的Noether对称无限小变换,进而得到最优解方程组的守恒量,使最优解关系以一组代数方程形式表达,为最优解的积分方法以及各种数值算法都奠定了坚实基础.最后,以基础振动下机械臂非线性动力学的能量最优控制实例分析,说明了该文对称性方法的正确性.     

随机椭圆最优控制问题的随机配置方法

本文讨论求解随机系数泊松方程约束最优控制问题的有效数值方法.通过应用有限元方法和随机配置法,将原最优控制问题离散转化为最优化问题,再利用交替方向乘子法求解最优化问题.之后,对所提出的算法进行了收敛性分析,并通过数值实验验证了算法的有效性.     

多尺度辛格式求解复杂介质波传问题

在哈密顿体系中引入小波分析,利用辛格式和紧支正交小波对波动方程的时、空间变量进行联合离散近似,构造了多尺度辛格式——MSS（Multiresolution Symplectic Scheme）．将地震波传播问题放在小波域哈密顿体系下的多尺度辛几何空间中进行分析,利用小波基与辛格式的特性,有效改善了计算效率,可解决波动力学长时模拟追踪的稳定性与逼真性．     

基于径向基逼近的KdV方程的无网格辛算法

基于径向基逼近理论,本文为KdV方程构造了一个无网格辛算法.首先借助径向基空间离散Hamilton函数以及Poisson括号,把KdV方程转化成一个有限维的Hamilton系统.然后用辛积分子离散有限维系统,得到辛算法.文章进一步讨论了所构造辛算法的收敛性和误差界.数值例子验证了理论分析.     

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic,chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.     

An energy-momentum conserving scheme for Hamiltonian wave equation based on multiquadric trigonometric quasi-interpolation

Based on multiquadric trigonometric quasi-interpolation, the paper proposes a meshless symplectic scheme for Hamiltonian wave equation with periodic boundary conditions. The scheme first discretizes the equation in space using an iterated derivative approximation method based on multiquadric trigonometric quasi-interpolation and then in time with an appropriate symplectic scheme. This in turn yields a finite-dimensional semi-discrete Hamiltonian system whose energy and momentum (approximations of the continuous ones) are invariant with respect to time. The key feature of the scheme is that it conserves both the energy and momentum of the Hamiltonian system for both uniform and scattered centers, while classical energy-momentum conserving schemes are only for uniform centers. Numerical examples provided at the end of the paper show that the scheme is efficient and easy to implement.     

Method of separation of variables and Hamiltonian system

In the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables ususally leads to the Sturm–Liouville-type eigenproblems of self-adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite-dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration. © 1993 John Wiley & Sons, Inc.     

Geometric integration of the paraxial equation

Evolution of solitary waves in photovoltaic-photorefractive crystal satisfy the paraxial equation. The paraxial equation is transformed into the symplectic structure of the infinite dimensional Hamiltonian system. The symplectic structure of the paraxial equation is discretizated by the symplectic method. The corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation. The symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation. Evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated.     

哈密尔顿系统的有限元法

利用常微分方程的连续有限元法,结合函数的M-型展开,对非线性哈密尔顿系统证明了连续一、二次有限元分在3阶量、5阶量意义下近似保辛,且保持能量守恒.在数值实验中结合庞加莱截面,哈密尔顿混沌数值试验结果与理论相吻合.     

Feedback control optimization of nonlinear systems under random excitations

This paper uses stochastic averaging method to design an optimal feedback control for nonlinear stochastic systems. The method of stochastic averaging is used to reduce the dimension of the state space and to derive the Itô stochastic differential equation for the response amplitude process. Two approaches to optimization, namely, with the exact steady state probability density function of the amplitude process and the Rayleigh approximation are compared. The cost function is a steady state response measure. Numerical examples are studied to demonstrate the performance of the control both in transient and steady-state. The effect of the control on the system response and control performance is studied. The regions where the controls are conservative and unconservative are pointed out.     

Poisson系统的辛结构

当Poisson系统中的Poisson矩阵是非常数时,经典的辛方法如辛Runge-Kutta方法,生成函数法一般不能保持Poisson系统的Poisson结构,利用非线性变换可把非常数Poisson结构转化成辛结构,然后任意阶的辛方法可以长时间计算Poisson系统的辛结构．自由刚体问题中Euler方程被转换成辛结构并用辛中点格式进行数值求解,数值结果给出了这种非线性变换的有效性．