Variational Problems: Nonsmooth Penalties and Time Scales

变分问题出现在各种学科中，包括工程、自然资源和经济学。在这类优化模型的学派中，最优控制类占据着重要的地位，因为它源于各种领域的应用。这个类的特点是存在“硬”约束，比如控制约束和状态约束。由R.T. Rockafellar引入的广义Bolza问题包含了一类最优控制。在那里，约束通过“非光滑惩罚”项被构建到目标函数（或拉格朗日函数）中。观察到该模型的哈密顿量表现出较好的平滑行为。第一部分的研究目的是从广义Bolza问题的哈密顿量的角度来完成对广义Bolza问题的研究。计划是建立一个全面的二阶最优理论，并在存在状态约束时进行稳定性分析。本研究的第二个目标是启动“时间尺度”上的最优控制问题的研究，也就是说，当底层时间属于紧集，而不一定是连通区间时。具体而言，其目的是制定必要和充分的最佳标准。在连续时间和离散时间最优控制以及非光滑分析方面的最新技术的实质性改进将有助于实现本研究的目标。近年来，连续或离散运行时间系统因其广泛的应用而成为研究的热点。允许这样的组合和更多的时间模型被称为“时间尺度”。这种类型的模型与“混合系统”直接相关，这种系统出现在人口动态、汽车电子、自动化系统、空中交通管理系统、集成系统设计和多媒体中。在这些学科中，人们会遇到一些应用，这些应用被数学地表述为具有输入和/或输出约束的连续时间最优控制问题，或作为时间尺度上的最优控制问题。对于前者，关于找到准确的标准来确定最佳候选人的重要问题仍然没有答案。另一方面，由于后者是一个全新的研究方向，因此已知的结果很少，需要回答几个开放的问题。这项研究的目的是制定标准，以确定这类问题的最佳候选者。

Variational problems arise in a variety of disciplines, including engineering, natural resources, and economics. In the school of such optimization models, the class of Optimal Controls occupies an essential position since it stems from applications in a variety of fields. This class is distinguished by the presence of "hard" constraints such as the control and the state constraints. The generalized problem of Bolza, introduced by R.T. Rockafellar, includes under its wing the class of optimal controls. There, constraints are built into the objective function (or Lagrangian) through "nonsmooth penalty" terms. It is observed that better smoothness behavior is displayed by this model's Hamiltonian. The aim of the first part of the research is to complete the study of the generalized Bolza problem from the point of view of its Hamiltonian. The plan is to build a comprehensive second-order optimality theory and a stability analysis when the state constraints are present. The second goal of this research is to launch the study of optimal control problems over "time scales," that is, when the underlying time belongs to a compact set, not necessarily a connected interval. In particular, the aim is to develop necessary and sufficient criteria for optimality. Substantial modification of the latest techniques in continuous- and discrete-time optimal controls and in nonsmooth analysis will be instrumental in achieving the goals of this research.In recent years, systems in which the running time could be continuous or discrete have taken the front row due to their appearance in a wide range of applications. A time model that allows for such a combination and many more is known as a "time scale." This type of model is directly related to "hybrid systems," which occur in population dynamics, automotive electronics, automated systems, air traffic management systems, integrated system design, and multi-media. In these disciplines, one encounters applications that are formulated mathematically as continuous-time optimal control problems with constraints on the input and/or output, or as optimal control problems over time scales. For the former, important questions remain unanswered regarding finding accurate criteria to identify optimal candidates. On the other hand, since the latter is a completely new direction of research, very few results are known and several open questions need to be answered. The aim of this research is to develop criteria that serve in identifying optimal candidates for each of these classes of problems.