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Sustainable Optimal Controls for Nonlinear Partial Differential Equations with Applications

非线性偏微分方程的可持续最优控制及其应用

基本信息

批准号：
405372818
负责人：
Professor Dr. Günter Leugering
金额：
--
依托单位：
The State Fund for Fundamental Researches
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2019
资助国家：
德国
起止时间：
2018-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/405372818?language=en
关键词：
Sustainable Optimal Controls Nonlinear Partial

项目摘要

Optimization and Control under constraints typically leads to full exploitation of the resources. Therefore, as a rule, the resulting optimal controls exert significant inputs to the system under consideration. In particular, in the context of continuum mechanics and, say, time-minimum final profile control, large control forces may, in the long run, introduce damage to the system until complete failure occurs. We are, thus, faced with a dilemma: On the one hand we want to achieve our objective, say, in minimum time, while on other hand we would like to save the infrastructure. In other words, we would like to apply controls that maintain or enhance sustainability. Obviously, the interaction of controls and the evolution of damage in the corresponding material or structure heavily depends on the particular type of process and the material characteristics. The project, therefore focuses on a detailed analysis of different aspects (such as relaxation, approximation, regularization) of optimal controls problems for parabolic and hyperbolic initial-boundary value problems for elastic bodies arising, e.g. in contact mechanics, coupled systems, composite materials, where 'life-cycle-optimization' appears as a challenge. Instead of the standard statements of optimal control problems for PDEs, the effect of sustainability in the setting of the optimal control problems for elastic materials typically leads to the consideration of degeneration in hyperbolic (or parabolic) equations, where the evolution of the damage field is described by a parabolic inclusion or equation with a damage source function depending on the mechanical compression or tension. Material damage typically is reflected by degenerating coefficients in the leading coefficients of the underlying differential operator. In case of strong degeneration, the corresponding mixed initial-boundary value problem may lack uniqueness (or even the existence) of weak solutions due to e.g. the Lavrentieff-gap phenomenon. However, damage may also take place in coupling conditions, such that the degenerating coefficients are not factors of the highest order differential expressions, but rather appear in connection with lower coupling terms at the coupling interface. This is true e.g. for plate and beam structures where the coupling between two structural elements, realized via welding or glueing, may suffer deterioration to due excessive stresses at the coupling interface.The project is concentrated on the existence of sustainable optimal controls, deriving of the corresponding optimality conditions, development of the scheme for their approximation, study of the possible ways for the relaxation of the original statements of optimal control problems. In cases, where the strict fulfilment of the constraints appears to be infeasible, we proposed approximate solutions to the sustainable optimal control problems under rather general assumptions on the evolution of damage field.

约束条件下的优化和控制通常会导致资源的充分利用。因此，作为一项规则，由此产生的最优控制施加显着的输入到所考虑的系统。特别是，在连续介质力学的背景下，也就是说，时间最小的最终轮廓控制，大的控制力，从长远来看，可能会引入损坏的系统，直到完全故障发生。因此，我们面对一个两难的局面：一方面，我们希望在最短的时间内达到我们的目标，另一方面，我们希望节省基础设施。换句话说，我们希望实施维持或增强可持续性的控制措施。显然，控制的相互作用和相应材料或结构中损伤的演变在很大程度上取决于特定类型的过程和材料特性。因此，该项目侧重于详细分析弹性体抛物和双曲初边值问题的最优控制问题的不同方面（如松弛，近似，正则化），例如在接触力学，耦合系统，复合材料中，“生命周期优化”似乎是一个挑战。而不是偏微分方程的最优控制问题的标准陈述，在弹性材料的最优控制问题的设置的可持续性的影响通常会导致考虑退化的双曲（或抛物线）方程，其中的损伤场的演变是由一个抛物线包含或方程的损伤源函数取决于机械压缩或拉伸。材料损伤通常通过基础微分算子的前导系数中的退化系数来反映。在强退化的情况下，相应的混合初边值问题可能缺乏弱解的唯一性（甚至存在性），例如由于Lavrentieff间隙现象。然而，损伤也可能发生在耦合条件下，使得退化系数不是最高阶微分表达式的因子，而是与耦合界面处的较低耦合项结合出现。例如，对于板和梁结构，通过焊接或胶合实现的两个结构元件之间的耦合可能会由于耦合界面处的过度应力而恶化。该项目集中在可持续最优控制的存在上，导出相应的最优性条件，开发其近似方案，研究放松最优控制问题原始陈述的可能途径。在严格满足约束条件似乎不可行的情况下，我们提出了近似解的可持续最优控制问题的损伤场的演变相当一般的假设下。