Sustainable Optimal Controls for Nonlinear Partial Differential Equations with Applications

在约束条件下的优化和控制通常会导致资源的充分开发。因此，作为一种规则，所得到的最优控制对所考虑的系统施加重要的输入。特别是，在连续介质力学的背景下，比如，时间最小的最终剖面控制，从长远来看，大的控制力可能会对系统造成损害，直到完全失效。因此，我们面临着一个两难境地：一方面，我们希望在最短的时间内实现我们的目标，而另一方面，我们希望节省基础设施。换句话说，我们希望应用维持或增强可持续性的控制。显然，控制的相互作用和相应材料或结构的损伤演变在很大程度上取决于特定的过程类型和材料特性。因此，该项目侧重于对弹性体出现的抛物型和双曲型初始边值问题的最优控制问题的不同方面（如松弛、近似、正则化）的详细分析，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。与pde最优控制问题的标准表述不同，弹性材料最优控制问题设置中可持续性的影响通常导致考虑双曲（或抛物线）方程中的退化，其中损伤场的演变由抛物线包含或方程描述，损伤源函数取决于机械压缩或张力。材料损伤通常由微分算子前导系数中的退化系数反映出来。在强退化情况下，相应的混合初边值问题可能由于Lavrentieff-gap现象等原因而缺乏弱解的唯一性（甚至存在性）。然而，在耦合条件下也可能发生损伤，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有关。例如，对于板和梁结构，通过焊接或粘合实现的两个结构元件之间的耦合可能由于耦合界面处的过大应力而恶化。该项目集中于可持续最优控制的存在性，推导相应的最优性条件，开发其逼近方案，研究最优控制问题原始陈述的松弛可能方法。在严格满足约束条件似乎不可行的情况下，我们在相当一般的损伤场演化假设下提出了可持续最优控制问题的近似解。

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.