Variational Analysis and Dynamic Optimization

摘要：本文的目的是通过实际提出不同的最优性准则和提供稳定性分析结果，为这三类问题的数学进步做出重要贡献。这三个类分为三个主要题目：广义Bolza问题、连续时间最优控制和离散时间最优控制。第一类，广义bolza问题，落在非光滑分析的中心，其中被积函数的正则性是缺乏的，但由其密尔顿量的相对正则性来补偿。尽管在某些假设下，第二类可以看作是第一类的一个子类，但是直接单独研究这一类对于获得考虑到问题的特殊而丰富的结构的结果是非常有益的。这一研究方向产生的最优控制问题，结果是，一般的，不同于那些由应用Bolza的广义问题的结果。此外，这两组结果对不同假设集合下的最优控制设置都成立。第三类问题对于离散化连续时间最优控制问题的数值求解很重要。对于这三类，将利用二次泛函、共轭点理论和里卡蒂方程导出二阶充分必要条件。此外，还将研究解对扰动的稳定性和灵敏度的结果。Hamilton-Jacobi理论，以及变分分析和非光滑分析的最新技术将被采用。动态结构优化问题作为时间演化系统的数学模型，在许多学科中都占有重要地位。特别是，最优控制理论是在20世纪50年代发展起来的一门学科，主要是为了处理来自几个学科的应用。这包括工程、运营管理和经济。大多数问题的表述最初是从工程考虑出发的，这导致了控制和状态约束的存在。“软着陆问题”和“火箭车问题”就是这种情况。最近，最优控制问题的表述已被调整为包括管理和经济学等其他领域的考虑。这解释了对包含状态和控制变量约束的最优控制问题的兴趣增加。因此，提供新的工具和更好的技术来处理这些问题，并帮助计算它们的解决方案，将对来自这些学科的应用产生重要影响。这正是本提案的目标。实际上，其目的是为具有不同类型约束的控制问题导出最优性准则。此外，还分析了这些判据在小扰动下的稳定性。由于测量中的误差，后者对于应用至关重要。

VARIATIONAL ANALYSIS AND DYNAMIC OPTIMIZATIONDMS 0072598ABSTRACT:The aim of this proposal is to make an essential contribution to the mathematical progress of three classes of such problems by actually producing different optimality criteria and furnishing stability analysis results. The three classes are under three main headings: the generalized problem of Bolza, the continuous-time optimal control, and the discrete-time optimal control. The first class, the generalized problem ofBolza, falls in the heart of nonsmooth analysis where the regularity of the integrand is lacking but compensated for by the relative regularity of itsHamiltonian. Despite the fact that under certain hypotheses the second class can be viewed as a subclass of the first, it is quite beneficial tostudy this class directly on its own to obtain results that take into account the special and rich structure of the problem. This direction of research produces for the optimal control problems, results that are, ingeneral, distinct from those obtained by applying the results of the generalized problem of Bolza. Furthermore, these two sets of results hold for the optimal control setting under different sets of hypotheses. The third class of problems is important for numerically solving thecontinuous-time optimal control problem via discretization.For the three classes, second-order necessary and sufficient conditions would be derived in terms of a quadratical functional, conjugate point theory, and a Riccati equation. Furthermore, results on the stability and sensitivity of solutions to perturbations would be investigated. The Hamilton-Jacobi theory would be employed, as well as the latest techniques from variational and nonsmooth analysis.Optimization problems with dynamical structure have become prominent in a variety of disciplines as mathematical models of systems with time evolution. In particular, optimal control theory is a subject that was developed in the 1950's mainly to deal with applications arising from several disciplines. This includes engineering, operationsmanagement and economics. Most problem formulation were initially derived from engineering considerations which lead to the presence of control and state constraints. This is the case with the "soft moon landing problem" and the "rocket car problem". Most recently, the formulation of the optimal control problem has been adapted to include considerations from other areas like management and economics. This accounts for the increased interest in optimal control problems with constraints containing both the state and the control variable. Therefore, providing new tools and better techniques to tackle these problems and to help in computing their solutions would have an important impact on the applications emanating from those disciplines. This is exactly the goal of the present proposal. In fact, the intent is to derive optimality criteria for control problems with different types of constraints. Furthermore, the stability of these criteria under small perturbations is analysed. This latter is crucial for applications, due to the error in measurements.