Optimal control of vibrations of a dynamic Gao beam in contact with a reactive foundation

In this paper, we investigate the optimal control of vibrations of a nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a reactive foundation underneath it. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle of the system governed by the Gao beam equation. And the first-order necessary optimality condition is presented for the optimal control problem in fixed final horizon case. Finally, we also sketch the numerical solution based on the obtained theoretical results.     

Necessary conditions of fractional optimal control problems with state constraints in the sense of Riemann‐Liouville

This paper is devoted to the optimal control problem of a fractional dynamic system with state constraints in the sense of Riemann‐Liouville. By means of the needle variation, we establish the Pontryagin's maximum principle for the optimal control problem. Moreover, when such a necessary condition is singular in some sense, we investigate the “second‐order” necessary conditions accordingly. As an application, two examples are presented to demonstrate the accuracy and efficiency of the result.     

A multiple-criteria decision problem for optimal management of farm resources under uncertainty - a case study

An attempt is made to minimize the total operational cost and specific energy for rice cultivation, when the constraints relating to the availability of power sources and crop yield are supposed to be stochastic in nature. The Sequential Linear Goal Programming algorithm has been used to solve the resulting multiple objective optimization problem.     

Necessary optimality conditions for quasisingular controls in a step control problem

Necessary second-order optimality conditions are obtained for one class of optimal control problems for step systems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 101–115, January–February 2008.     

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.     

一类混杂系统的最优控制

研究了一类脉冲依赖于状态的混杂系统的最优控制问题.与传统的变分方法不同,通过将跳跃瞬间转化为一个新的待优化参数,得到了该混杂系统的必要最优性条件,从而将最优控制问题转化为一边界值问题,该边界值问题可由数值方法或解析方法解决.此外,利用广义微分的理论,将该必要最优性条件推广到Frechet微分形式.结论表明,在混杂动态系统运行的连续部分,最优解所满足的必要性条件和传统的连续系统相同.在混杂动态系统的脉冲点处,哈密尔顿函数满足连续性条件,协态变量则满足一定的跳跃条件.最后,通过两个实例分析,表明该方法是有效的.     

Analogue of the Kelley condition for optimal systems with retarded control

In this paper, we consider an optimal control problem with retarded control and study a larger class of singular (in the classical sense) controls. The Kelley and equality type optimality conditions are obtained. To prove our main results, we use the Legendre polynomials as variations of control.     

Optimal control of mobile sources for heat conductivity processes

We consider a problem of optimal control of the mobile sources for heat conductivity processes, described by parabolic equation and systems of ordinary differential equations. Sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined. The necessary optimality condition was established in the form of the integral principles of maximum. The theoretical conclusions are illustrated by the solution of a numerical example.     

最优控制的计算原理:评论最小值原理

提出连续最优控制计算原理和离散最优控制计算原理,两者都可用于最优控制的数值优化. Pontryagin最小值原理和离散最小值原理,由于其含有的信息不完整,形式特殊,所以都不能用于最优控制的数值优化.此外Pontryagin最小值原理和离散最小值原理分别为连续最优控制计算原理和离散最优控制计算原理的特殊情况.     

Near-maximum principle for general recursive utility optimal control problem

In this study, we introduce a general recursive utility optimal control problem driven by a fully nonlinear stochastic differential system. In order to investigate Pontryagin's stochastic maximum principle for this new optimal control problem, we establish the near-optimal control system for the original problem and obtain the related near-maximum principle. Two examples are also provided to illustrate the near-maximum principle.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Global maximum principle for the forward–backward stochastic optimal control problem with poisson jumps

This paper is concerned with the forward–backward stochastic optimal control problem with Poisson jumps. A necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented under the assumption that the diffusion and jump coefficients do not contain the control variable, and the control domain need not be convex. The case where there are some state constraints is also discussed. A financial example is discussed to illustrate the application of our result. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

年龄相关的种群空间扩散系统的广义解与收获控制

研究了由积分偏微分方程描述的年龄相关的种群空间扩散系统的收获控制问题.首先利用不动点方法证明了对于有界死亡率μ的系统广义解的存在性,但这是预备的结果.进一步,运用上述结果、先验估计和紧性定理,证明了对于在r=A附近无界的μ的系统解的存在惟一性.其次,利用类似方法得到系统最优收获控制的存在性.最后,利用G^ateax微分和Lions的变分不等式理论,推得了控制为最优的必要条件;从而得到了由积分偏微分方程和变分不等式构成的最优性组.最优性组能够确定最优控制.还建立了表征最优控制的Euler_Lagrange组.这些结果可为种群系统控制问题的实际研究作为理论参考.     

Legendre wavelet method for solving variable-order nonlinear fractional optimal control problems with variable-order fractional Bolza cost

This paper deals with a numerical method for solving variable-order fractional optimal control problem with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a variable-order Riemann–Liouville fractional integral. Using the integration by part formula and the calculus of variations, the necessary optimality conditions are derived in terms of two-point variable-order boundary value problem. Operational matrices of variable-order right and left Riemann–Liouville integration are derived, and by using them, the two-point boundary value problem is reduced into the system of algebraic equations. Additionally, the convergence analysis of the proposed method has been considered. Moreover, illustrative examples are given to demonstrate the applicability of the proposed 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.     

Necessary optimality conditions in a nonsmooth control problem with variable structure

An optimal control problem with nonsmooth performance criterion described by a system of ordinary differential equations is considered. Necessary first-order optimality conditions are obtained.     

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.