Numerical analysis and discretization strategies for optimal control problems with singularities

工艺过程优化在科学和工程中发挥着越来越重要的作用。该项目处理由椭圆型或抛物型偏微分方程控制的不同类型的最优控制问题，并以控制和状态的附加点向不等式约束为特征。特别感兴趣的是各种奇点的问题，包括可重入角和边、非光滑系数、小参数和不等式约束。该项目有两个目标：首先，从先验误差估计开始，生成网格族，确保最佳近似率。其次，开发可靠的后验误差估计器，并将其用于自适应网格细化。一个挑战是将控制和状态的逐点不等式约束结合起来。两种方法均能保证计算结果的高效、可靠。有了一个成功的策略，就有可能以低成本计算出给定精度的最优控制问题的数值解。在第一期中，我们主要关注线性状态方程的控制问题，而在第二期中，我们计划考虑半线性状态方程。

Optimization of technological processes plays an increasing role in science and engineering. This project deals with different types of optimal control problems governed by elliptic or parabolic partial differential equations and characterized by additional pointwise inequality constraints for control and state. Of particular interest are problems with all kinds of singularities including those due to reentrant corners and edges, nonsmooth coefficients, small parameters, and inequality constraints. The project targets two goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, reliable posteriori error estimators are developed and used for adaptive mesh refinement. A challenge is the incorporation of pointwise inequality constraints for control and state. Both techniques can ensure efficient and reliable numerical results. With a successful strategy it is possible to calculate numerical solutions of the optimal control problems with given accuracy at low cost. While we concentrate on control problems with a linear state equation in this proposal for the first period, we plan to consider semilinear state equations in the second period.