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起着越来越重要的作用。该项目处理不同类型的最优控制问题所管辖的椭圆或抛物偏微分方程，其特点是控制和状态的附加逐点不等式约束。特别感兴趣的是各种奇异性的问题，包括那些由于凹入的角落和边缘，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，生成确保最佳逼近率的网格族。其次，可靠的后验误差估计器的开发和自适应网格细化。一个挑战是将逐点不等式约束控制和状态。这两种方法都能保证计算结果的有效性和可靠性。有了一个成功的策略，它是可能的计算数值解的最优控制问题与给定的精度在低成本。虽然我们集中在控制问题的线性状态方程在这个建议的第一阶段，我们计划考虑半线性状态方程在第二阶段。

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.