BOULIGAND DIFFERENTIABILITY OF SOLUTIONS TO PARAMETRIC OPTIMAL CONTROL PROBLEMS*

A family of parameter dependent optimal control problems for nonlinear ODEs is considered. The problems are subject to pointwise control constraints. It is shown that the standard conditions, used in stability analysis of optimal control problems, ensure not only Lipschitz continuity, but also Bouligand differentiability of the solutions with respect to the parameter. The Bouligand differentials are characterized as the solutions to the accessory linear-quadratic optimal control problems.

     

一类连续最优控制问题的有限维近似

1　引　　言本文考虑具有状态终端约束、控制受限的非线性连续最优控制问题min　h0(x(0))+∫T0f0(x(t),u(t))dt+g0(x(T))(1.1)s.t.　x(t)=f(x(t),u(t)),　　t∈[0,T](1.2)D(x(0))=0,(1.3)E(x(T))=0,(1.4)S(u(t))≤0,　　t∈[0,T](1.5)其中,h0:Rn→R,f0:Rn×Rm→R,f:Rn×Rm→Rn,g0:Rn→R,D:Rn→Rp,E:Rn→Rq,S:Rm→Rr均为二次连续可微函数.T为终端时间(固定),p,q≤n,x(t)∈W1,∞[0,T]n,u(t)∈L∞[0,T]m分别为状态函数和控制函数.U(t)={u:S(u(t))≤0}为紧凸集.问题(1.1)—(1.5)要求寻找最佳控制u(t)使得目标函数(1.1)达到极小.…     

具有跳跃的扩散过程的最优控制及相关的积分-微分型分布参数系统的最优控制问题

§1.引言在[1]、[2]及[3]中考虑了可以用偏微分方程方法处理的一类具有连续轨道的扩散过程的最优控制问题.近来[4]研究了具有跳跃的扩散过程的最优控制问题,证明了 reward函数满足 Bellman 方程.     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.     

关于人口系统妇女总和生育率的范数最优控制问题

本文讨论人口系统妇女总和生育率的范数最优控制问题。本文将妇女总和生充臃当作控制变量，在一定条件下证得最优控制的存在和唯一性，并给出其相应的优化条件。     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

一类半鞅状态的平稳型脉冲随机控制

本文提出了一类新的随机控制模型,这类模型不但在费用结构上推广了此前的平稳型脉冲随机控制,而且首次将一类半鞅引入脉冲控制模型的状态结构从而推广了相应的状态过程.通过对一类相当复杂的变分方程问题的研究并利用其有关结论,我们证明了新模型最佳控制的存在性并刻划出其结构.     

具有点态控制约束热方程的时间与范数最优控制问题的等价性

本文研究了具有点态控制热方程的等价性问题.利用变分法分析时间最优控制的唯一性,能控性以及范数最优控制的特征,获得了具有点态控制约束热方程的时间与范数最优控制问题之间的等价性,推广了现有文献的结果.     

New optimality conditions for nonsmooth control problems

This work considers nonsmooth optimal control problems and provides two new sufficient conditions of optimality. The first condition involves the Lagrange multipliers while the second does not. We show that under the first new condition all processes satisfying the Pontryagin Maximum Principle (called MP-processes) are optimal. Conversely, we prove that optimal control problems in which every MP-process is optimal necessarily obey our first optimality condition. The second condition is more natural, but it is only applicable to normal problems and the converse holds just for smooth problems. Nevertheless, it is proved that for the class of normal smooth optimal control problems the two conditions are equivalent. Some examples illustrating the features of these sufficient concepts are presented.     

关于一类连续最优控制问题的一种可实现的离散方法

本文对具有状态终端约束、控制受限的非线性连续最优控制问题给出一种新的可实现的离散方法，此方法通过求解非线最小二乘问题避免这类问题离散后出现的不可行现象，文中给出这种做法的理论证明和实现方案。     

A general theorem on error estimates with application to a quasilinear elliptic optimal control problem

A theorem on error estimates for smooth nonlinear programming problems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while different discretization methods are used for the control functions. The distance of locally optimal controls to their discrete approximations is estimated.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

THE APPLICATION OF OPTIMAL CONTROL METHODOLOGY TO A WELL-STIRRED BIOREACTOR

We examine a model for a well-stirred bioreactor with dynamics governed by two ordinary differential equations. The equations model bacteria and contaminant growth and decay. We solve two optimal control problems, taking the input nutrient as the control.     

Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.     

Differential sensitivity of solutions of convex constrained optimal control problems for discrete systems

A family of optimal control problems for discrete systems that depend on a real parameter is considered. The problems are strongly convex and subject to state and control constraints. Some regularity conditions are imposed on the constraints.The control problems are reformulated as mathematical programming problems. It is shown that both the primal and dual optimal variables for these problems are right-differentiable functions of a parameter. The right-derivatives are characterized as solutions to auxiliary quadratic control problems. Conditions of continuous differentiability are discussed, and some estimates of the rate of convergence of the difference quotients to the respective derivatives are given.     

OPTIMAL CONTROL THEORY APPLIED TO JOINT PRODUCTION OF TIMBER AND FORAGE

Renewable natural resource systems often represent examples of joint production. Optimal control theory is employed using the linear variational method to derive the general solution to the timber-forage joint production problem, with the objective of maximizing present value of revenue. The results indicate that optimal control theory can successfully solve such problems. The functional forms of the solution provide insight into how changes in parameters will influence the optimal joint production system.     

A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM

In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.