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.     

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 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.     

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.     

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.     

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.     

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。     

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.     

Optimal control solutions to the maximum volume isoperimetric pillars problem

This paper extends the isoperimetric problem. The problem is to find an enclosed cross-sectional/base region of a pillar defined by a simple closed curve of fixed perimeter such that the volume of the constructed pillar, bounded above by a relatively smooth ceiling, is maximized. Green’s Theorem is applied in the formulation of the problem such that the problem can be transformed into canonical form handled by MISER3. For the case of multiple pillars, a novel elliptic separation technique is developed for multiple pillars constructions. This technique is used to ensure that the cross-sectional regions of any pillars are separated. Illustrative examples are provided to demonstrate the effectiveness of the technique developed.     

Stochastic maximum principle for SPDEs with noise and control on the boundary

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problem can be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.     

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract.     

Chaos, estimation and optimal control of habitat destruction model with uncertain parameters

The present paper discusses chaos, estimation and optimal control of the habitat destruction model with unknown parameters. The linear stability analysis of the steady states of the model will be discussed. Further, the chaotic behavior of this system will be investigated. The dynamic estimators of the unknown parameters and their updating rules are derived. Using Pontryagin principle, the optimal control inputs are derived with respect to a selected measure. Finally, a numerical simulation study for various parameters and different initial densities is presented.     

Solution of Fuller’s problem on the basis of the joint Pontryagin-Hamilton-Ostrogradskii principle

The present study is devoted to the synthesis of the optimal control of a dynamic system. It is proved that application of the method of needle variation to the invariant features of effective motion is a promising approach to increasing the efficiency with which problems involved in the synthesis of optimal control are solved. The Hamilton-Ostrogradskii action integral, to which the Pontryagin needle variation is applied, is adopted as such a feature in the study. As a result, a minium condition for the objective functional is obtained; this condition is termed the joint maximum principle.     

New results on the relationship between dynamic programming and the maximum principle

The dynamic programming approach to optimal control theory attempts to characterize the value functionV as a solution to the Hamilton-Jacobian-Bellman equation. Heuristic arguments have long been advanced relating the Pontryagin maximum principle and dynamic programming according to the equation (H(t, x ^* (t), u ^* (t), p(t)),−p(t))=√V(t,x ^* (t)), where (x^*, u^*) is the optimal control process under consideration,p(t), is the coextremal, andH is the Hamiltonian. The relationship has previously been verified under only very restrictive hypotheses. We prove new results, establishing the relationship, now expressed in terms of the generalized gradient ofV, for a large class of nonsmooth problems.