理想塑性的最优控制第一部分：应力跟踪
Optimal Control of Perfect Plasticity Part I: Stress Tracking
摘要：
本文研究了率无关拟静态理想弹塑性系统的最优控制问题。其目的是通过控制边界规定部分的位移来优化应力场。因此，控制进入系统的狄利克雷边界条件。因此，安全载荷条件自动满足，使得系统允许应力场唯一的解。这就产生了一个定义良好的控制到状态的运营商，这是连续的，但不是盖特可微的。因此，首先通过Yosida正则化，然后通过第二次平滑来正则化控制到状态映射，以获得平滑问题。证明了原非光滑最优控制问题的全局极小解的逼近性，并建立了正则化问题的最优性条件。数值算例说明了该方法的可行性。
英文摘要：
The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gateaux-differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.
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