变质量非线性非完整系统的Gibbs-Appell方程

本文首先将Gibbs-Appell方程推广到最一般的变质量非完整系统.得到变质量非线性非完整系统在广义坐标、准坐标下的Gibbs-Appell方程和积分变分原理,最后给出一个例子.     

广义LQ最优控制问题的极小化序列

将经典LQ问题的评价泛函中关于控制变量的二次型推广为一类偶次多项式,证明了这类广义LQ无约束最优控制问题的一个等价扩张逼近可由一列半径递增的球约束最优控制问题加以实现.进而利用P0ntryagin极值原理建立相应的球约束最优控制问题的二次规划,并通过Canonical倒向微分流及不动点定理,求解常微分方程边值问题,得到球约束最优控制问题的最优值.随着约束球半径趋于无穷大,形成原广义LQ最优控制问题的一个极小化序列,从而得到原问题的最优值.     

Extremal Paths in the Nilpotent sub-Riemannian Problem on the Engel Group (Subcritical Case of Pendulum Oscillations)

We consider a left-invariant sub-Riemannian problem on an Engel group. This problem arises as a nilpotent approximation of nonholonomic systems in the four-dimensional space with twodimensional control (e.g., the system describing the motion of a mobile robot with a trailer). For the sub-Riemannian problem on the Engel group, abnormal extremal paths are calculated. The subsystem for conjugate variables of normal Hamiltonian system of Pontryagin’s maximum principle is reduced to the pendulum equation. Normal extremal paths corresponding to subcritical pendulum oscillations were calculated.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

Dual regularization and Pontryagin’s maximum principle in a problem of optimal boundary control for a parabolic equation with nondifferentiable functionals

A problem of optimal boundary control is considered for a divergent linear parabolic equation. Equality constraints of the problem are given by nondifferentiable functionals. A dual regularization algorithm stable to errors in initial data is constructed for solving the problem. Pontryagin’s maximum principle plays the key role in this algorithm.     

Noether’s theorem of nonholonomic systems in optimal control

In this paper, we extend Noether’s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control problems with nonholonomic constraints, which are invariant under the parameter groups of infinitesimal transformations that change all (time, state, control) variables. Meanwhile, the Noether equalities corresponding to the conservation laws are given. Then, we obtain a new version of Noether’s theorem to optimal control systems. An example is given to illustrate the application of these results.     

基于Gauss伪谱法的欠驱动航天器姿态优化控制

现代航天器一般可以通过三正交反作用动量飞轮对其进行姿态机动控制并任意定位.研究了当其中某一个动量飞轮失效而不能输出完整三轴控制力矩时的欠驱动航天器姿态优化控制问题.在系统动量矩等于零时,其姿态控制问题可以转化为无漂移系统的非完整运动规划问题.采用Gauss伪谱法(GPM)将带有两个反作用动量飞轮的航天器姿态非完整运动规划问题转换为非线性规划问题(NLP),再通过SQP(sequential quadratic programming)算法求解.通过数值仿真、优化控制能达到设计的零边界控制要求,方便伺服电机对动量飞轮的控制;规划得到的姿态曲线与数值积分得到的曲线几乎完全重叠;权衡最终的优化目标值、运行时间和精度三因素找到合适的插值配点个数;结果表明了该方法对欠驱动航天器的姿态优化控制是有效的.     

New proof and some generalizations of the minimum principle in optimal control

An optimal control problem is reduced to the finite-dimensional problem of minimizing the terminal payoff over the intersection of the target set with the reachable set. The pointwise Pontryagin minimum principle is derived from two simple preliminary results: the first states that the intersection of two inseparable derived cones at a common point of two given sets is contained in the quasitangent cone (hence, in the contingent cone) to their intersection; the second identifies a derived cone to the reachable set. The standard variants of the minimum principle are easily generalized to problems defined by non-differentiable terminal payoffs on arbitrary target sets.     

Derivation of the Lagrange equations for nonholonomic chetaev systems from a modified Pontryagin maximum principle

Summary The generalized Lagrange equations for nonholonomic systems of Chetaev's type are derived from a maximum principle by introducing a modification of Pontryagin's classical method in optimal control theory. The method is illustrated by Appell's example dealing with a nonlinear nonholonomic constraint.

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Résumé Les équations généralisées de Lagrange pour des systèmes non-holonomes du type de Chetaev sont obtenues par un principe du maximum en introduisant une modification de la méthode classique de Pontryagin dans la théorie du contrôle optimal. La méthode est illustrée par l'exemple d'Appell traitant d'une contrainte non-holonome et non-linéaire.

     

Optimal strategies to diminish a pest population via bilinear controls

In this work we model and analyze two control strategies to diminish a pest population using traps. The action of any trap depends on its age. Both problems contain bilinear control rates. The large-time behavior of the model with time-periodic inflow is investigated. The first control strategy deals with a finite horizon problem while the second one is related to a time-periodic problem. We obtain Pontryagin’s principle for both control problems. A special attention is given to the periodic problem. Pontryagin’s principle is used to derive a conceptual gradient-type algorithm to approximate the optimal solution. Numerical tests are given.     

带一类第一积分的非完整系统的Szebehely问题

研究非完整系统动力学的一类逆问题·给出非完整系统的运动方程及其显式，考虑一类仅受齐次非完整约束的力学系统的Ｓｚｅｂｅｈｅｌｙ问题，研究已知一类第一积分的一般非完整系统的情形·最后举例说明其应用·     

损伤粘弹性力学的广义变分原理及应用

从粘弹性材料的Boltzmann迭加原理和带空洞材料的线弹性本构关系出发,提出了一种损伤粘弹性材料具有广义力场的本构模型．应用变积方法得到了以卷积形式表示的泛函,并建立了损伤粘弹性固体的广义变分原理和广义势能原理．把它们应用于带损伤的粘弹性Timoshenko梁,得到了Timoshenko梁的统一的运动微分方程、初始条件和边界条件． 这些广义变分原理为近似求解带损伤的粘弹性问题提供了一条途径．     

A note on the classical brachistochrone

It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.     

Isomorphism and Hamilton representation of some nonholonomic systems

We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.     

The decomposition method for a control problem for an underactuated Lagrangian system

The control problem for an underactuated Lagrangian system is considered. A system of smooth non-linear functions of the generalized coordinates is introduced into the treatment and the number of functions is equal to the number of generalized control forces. The aim of the control is to bring the system in a finite time to a terminal set specified by the level lines of the selected functions, and it is required that the motion at the terminal instant occurs along the level lines. As a result, a development and extension of Chernous’ko's decomposition method is given. This method was proposed for designing feedback control for Lagrangian systems when the number of controls in a system is equal to the number of its degrees of freedom.     

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.     

Cosserat生长弹性杆动力学的Gauss最小拘束原理

以自然界中具有生长、变形和运动特征的细长体为背景,用经典力学中的Gauss最小拘束原理研究生长弹性杆的动力学建模问题.在为生长弹性杆动力学建模提供新方法的同时,扩大了Gauss原理的应用范围.以Cosserat弹性杆为对象,分析弹性杆生长和变形的几何规则,表明生长应变和弹性应变是非线性耦合的;本构方程给出了截面的内力与弹性变形的线性关系;利用逆并矢,将经典力学中的Gauss原理和Gauss最小拘束原理用于生长弹性杆动力学,得到等价的两种表现形式,反映了时间和弧坐标在表述上的对称性,由此导出了封闭的动力学微分方程.给出了两种形式的最小拘束函数,表明生长弹性杆的实际运动使拘束函数取驻值,且为最小值.最后讨论了生长弹性杆的约束与条件极值等问题.     

Properties of extremals in optimal control problems with state constraints

An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time.     

On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems

A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of nonsmooth and nonconvex partial differential equation (PDE) optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, nonconvex, and discontinuous costs of the controls, L¹ tracking terms, and the case of state constraints. The SQH method is based on the characterization of optimality of PDE optimal control problems by the Pontryagin’s maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.