Numerical analysis and discretization strategies for optimal control problems with singularities

工艺过程的优化在科学和工程中的作用越来越大。这个项目处理不同类型的最优控制问题，由椭圆型或抛物型偏微分方程组控制，并以控制和状态的附加逐点不等式约束为特征。特别令人感兴趣的是各种奇点的问题，包括由于折返角和边、非光滑系数、小参数和不等式约束引起的问题。该项目的目标有两个：首先，从先验误差估计出发，生成确保最佳逼近速度的网格族。其次，开发了可靠的后验误差估计器，并将其用于自适应网格加密。一个挑战是纳入针对控制和国家的逐点不平等约束。这两种技术都能保证高效可靠的数值结果。有了一个成功的策略，就有可能以低成本计算出具有给定精度的最优控制问题的数值解。当我们在第一阶段集中讨论具有线性状态方程的控制问题时，我们计划在第二阶段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.