Optimal Control of Dissipative Solids: Viscosity Limits and Non-Smooth Algorithms

提出的项目涉及耗散固体的最优控制。我们的出发点是一个热力学一致的材料模型，它考虑了损伤效应和热弹塑性。这种具有速率无关分量的系统的现代解理论是建立在所谓的平衡粘度解的基础上的，这种平衡粘度解的存在性是通过粘性正则化和随后通过粘度参数的极限来证明的。在这个项目中，我们打算在这个解决方案的概念下分析损伤和热塑性变形过程的优化。除了最优控制的存在性外，我们主要对用粘性正则化方法求解局部最优控制的逼近性感兴趣。速率相关的粘性问题本身具有物理意义，并且它们仍然是非光滑的，因为相关的控制-状态算子通常不是Gateaux可微的。此外，粘性问题为开发一种高效的优化算法——函数空间中的束法提供了基础。为了应用它，需要在粘性模型问题的方向导数的基础上确定Clarke意义上的次微分的元素。通过使用路径跟踪方法来消除粘度，我们期望能够计算出甚至与速率无关的相关问题的最优解。

The proposed project concerns the optimal control of dissipative solids. Our point of departure is a thermodynamically consistent material model which takes into account damage effects as well as thermo-elastoplasticity. A modern solution theory for such systems with rate-independent components is based on so-called balanced viscosity solutions whose existence is proved by means of viscous regularization and a subsequent passage to the limit in the viscosity parameters.Within the proposed project, we intend to analyze the optimization of damage and thermo-plastic deformation processes under this solution concept. Besides the existence of optimal controls, we are mainly interested in the approximability of locally optimal controls by viscous regularization.The rate-dependent, viscous problems have a physical meaning in their own right, and they are still non-smooth in the sense that the associated control-to-state operator is, in general, not Gateaux differentiable. Moreover, the viscous problems serve as a basis for the development of an efficient optimization algorithm, a bundle method in function space. To apply it, elements of the subdifferential in the sense of Clarke are to be determined on the basis of directional derivatives for the viscous model problems.By using a path-following approach for vanishing viscosity, we expect to be able to compute optimal solutions even of the associated rate-independent problems.