The maximum principle without the adjoint equation

The Classical Pontryagin maximum principle for boundary trajectories of control systems consists of two parts: the maximum condition and the adjoint equation. In this paper, we study the maximum condition and minimal assumptions under which it holds for boundary trajectories of differential inclusions. Not surprisingly, the maximum condition alone is satisfied under much weaker assumptions than the full maximum principle. We prove it under mild hypotheses of measurability and lower continuity. In particular, we allow inclusions with nonconvex, unbounded values, not continuous in the state variable.     

Optimization of clamped plates with discontinuous thickness

We consider a new variational method for a clamped plate model and related shape optimization problems. Our approach allows the study of plates with discontinuous thickness.     

Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

A frequency‐constrained geometric Pontryagin maximum principle on matrix Lie groups

We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).     

Application of Pontryagin’s principle to bimodal optimization problems

By using a version of the Pontryagin’s principle allowing bimodal optimization, we formulate necessary optimality conditions and determine numerically the shape of optimal columns for two different problems. In all cases, we have to deal with bimodal optimization. The nonlinear system of differential equations is solved numerically. A first integral (Hamiltonian) is used to monitor the accuracy of integration. New numerical results are obtained, and comparison with some previously known cases is given, showing efficiency and accuracy of the suggested method.     

A maximum principle based combined method for scheduling in a flexible manufacturing system

A continuous time dynamic model of discrete scheduling problems for a large class of manufacturing systems is considered in the present paper. The realistic manufacturing based on multi-level bills of materials, flexible machines, controllable buffers and deterministic demand profiles is modeled in the canonical form of optimal control. Carrying buffer costs are minimized by controlling production rates of all machines that can be set up instantly. The maximum principle for the model is studied and properties of the optimal production regimes are revealed. The solution method developed rests on the iterative approach generalizing the method of projected gradient, but takes advantage of the analytical properties of the optimal solution to reduce significantly computational efforts. Computational experiments presented demonstrate effectiveness of the approach in comparison with pure iterative method.     

Nearly time-optimal paths for a ground vehicle

It is well known that the sufiqdent family of time-optimal paths for both Dubins‘ as well as Reeds-Shepp‘ s car models consist of the concatenation of circular arcs with nrtximum curvature and straight line segments, all tangentially connected.These time-optimal solutions suffer from some drawbacks. Their discontinuous curvature profile, together with the wear and impairment on the control equipment that the bang-bang solutions induce, calls for “smoother” and more supple reference paths to follow. Avoiding the bang-bang solutions also raises the robttstness with respect to any possible uncertainties. In this paper, ourmain tool for generating these “nearly firne-optunal”, but nevertheless continuous-curvature paths, is to use the Pontryagin Maximum Principle (PMP) and make an appropriate and cmming choice of the Lagrangmn function. Despite some rewarding sinmlation results, this concept turns out to be numerically divergent at some instances. Upon a more careful investigation, it can be concluded that the problem at hand is nearly singular. This is seen by applying the PMP to Dubins‘ car and studymg the corresponding two point boundary value problem, which turn out to be singular. Realizing this, one is able to contradict the widespread behef that all the information about the motion of a mobile platform lies in the initial values of the attxiliary variables associated with the PMP.     

Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.     

带Markov跳的离散时间随机控制系统的最大值原理

本文研究一类同时含有Markov跳过程和乘性噪声的离散时间非线性随机系统的最优控制问题, 给出并证明了相应的最大值原理. 首先, 利用条件期望的平滑性, 通过引入具有适应解的倒向随机差分方程, 给出了带有线性差分方程约束的线性泛函的表示形式, 并利用Riesz定理证明其唯一性. 其次, 对带Markov跳的非线性随机控制系统, 利用针状变分法, 对状态方程进行一阶变分, 获得其变分所满足的线性差分方程. 然后, 在引入Hamilton函数的基础上, 通过一对由倒向随机差分方程刻画的伴随方程, 给出并证明了带有Markov跳的离散时间非线性随机最优控制问题的最大值原理, 并给出该最优控制问题的一个充分条件和相应的Hamilton-Jacobi-Bellman方程. 最后, 通过 一个实际例子说明了所提理论的实用性和可行性.     

基于最大流原理的全球工业系统演化模型

随着全球化势力的扩张,全球工业系统发展模式作为全球核心问题之一有着尤为重要的研究价值。本文从信息视角,将全球工业系统看作一个信息实在及信息过程,由最大信息熵和最大流原理构建起描述和控制全球工业系统演化的新模型,进而以全球工业系统下发达、发展中国家这两个局部工业模式为研究案例。结果表明该模型可为经济全球化大背景下各国家工业模式的演化评价、预测、调控以及实现全球可持续发展等问题提供了全新的借鉴。     

Design of a complex system based on the maximum entropy principle

A complex system is a system composed of many dynamic elements with mutual interactions. This paper proposes a unified approach for the design of an information processing system using a complex system. The method of design is based on the maximum entropy principle. After a detailed explanation, the proposed method is applied to the design of a spatial filter using a complex system. This work was presented, in part, at the International Symposium on Artificial Life and Robotics, Oita, Japan, February 18–20, 1996.     

Pontryagin principle of maximum for the quantum problem of speed

The Pontryagin principle of maximum as originally formulated for the classical dynamic systems, was extended to the quantum systems with a finite number of states of the discrete spectrum. The principle was derived on the basis of equivalence of the matrix representation of the Schrödinger equation to the system of ordinary differential equations. An example of the problem of speed for generation of the given-structure wave package was considered.     

Stochastic maximum principle for delayed doubly stochastic control systems and their applications

ABSTRACT

In this paper, we investigate the optimal control problems for delayed doubly stochastic control systems. We first discuss the existence and uniqueness of the delayed doubly stochastic differential equation by martingale representation theorem and contraction mapping principle. As a necessary condition of the optimal control, we deduce a stochastic maximum principle under some assumption. At the same time, a sufficient condition of optimality is obtained by using the duality method. At the end of the paper, we apply our stochastic maximum principle to a class of linear quadratic optimal control problem and obtain the explicit expression of the optimal control.     

带反馈多传感器模糊最大熵单目标跟踪算法

针对矩阵加权融合算法计算量大、传感器数量不易扩充的特点,提出了一种带反馈的模糊最大熵融合算法。该算法采用模糊C-均值算法和最大熵原理计算状态向量中每一分量的权值,不但从整体考虑各分量对融合估计的影响,而且减少了复杂的矩阵运算过程,实时性较好。与矩阵加权算法相比,该融合算法还具有容易扩充的特点,能够直接应用于传感器数量大于2时的融合计算。实验仿真结果表明,融合估计的准确性与矩阵加权融合算法基本一致,算法的有效性得到了验证。     

A stochastic maximum principle for systems with jumps, with applications to finance

We consider a stochastic control problem with linear dynamics with jumps, convex cost criterion, and convex state constraint, in which the control enters the drift, the diffusion, and the jump coefficients. We allow these coefficients to be random, and do not impose any L^p-bounds on the control.

We obtain a stochastic maximum principle for this model that provides both necessary and sufficient conditions of optimality. This is the first version of the stochastic maximum principle that covers the consumption–investment problem in which there are jumps in the price system.     

Joint maximum principle in the problem of synthesizing an optimal control of nonlinear systems

The problem of optimization of dynamic systems is considered. It is shown that, unlike the well-known methods of optimal control, through use of the maximum principle it becomes possible to synthesize an efficient control law that substantially reduces computational complexity, as is demonstrated by the results of numerical simulation.     

基于最大隶属度原则的基因表达式编程分类

提出了一种基于最大隶属度原则的基因表达式编程(Gene Expression Programming,GEP)分类方法MDM-GEP。引入模糊集合中的隶属度描述分类的模糊性,在训练集上得到逼近各类别隶属函数的GEP分类器。对于待分类实例,计算其在各模糊集中的隶属度,基于最大隶属度的模糊模式识别原则确定最终归属类,并在三个UCI数据集上对该算法进行了实验。实验结果表明,MDM-GEP不仅具有较好的分类性能,而且有效解决了传统的简单GEP分类方法中存在的拒分区域问题。     

基于极大熵原理的理性公平交换协议

鉴于理性交换协议是一个动态博弈模型, 在完全不完美动态博弈中, 力图用极大熵原理来解决理性参与者的策略行为推断问题。扩展了一个基于信息熵的理性交换协议模型, 通过引入期望收益函数和期望均衡的方法, 给出理性交换协议的公平性描述; 基于最大熵原理构造了一种新的理性交换协议; 证明该协议的安全性, 利用博弈树的方法对整个交换过程进行分析并给出了理性公平性证明, 结果表明该协议能达到期望均衡。协议交换过程中无须可信第三方的参与, 该协议实现了理性公平性且具有更好的适应性。