Point and state constrained optimal control parabolic problems

PI将研究一类具有逐点控制的最优控制问题。这些问题是经典的并且具有广泛的应用，例如在废水处理、河流污染、心脏细胞中的钙波和噪声控制中。有限元方法是解决这类问题的最流行的方法数值，但在这方面的先验误差估计的结果很少。困难在于基本方程的解的粗糙性。 我们的目的是了解如何以及这种粗糙的解决方案可以近似数值，并获得尖锐的先验误差估计。这个研究领域需要从古典和现代分析，偏微分方程，有限元方法和优化的许多工具，并提供了各种各样的令人兴奋的问题，非常适合研究和教育目的。PI将研究逐点控制和/或状态约束的问题。这些问题通常是由狄拉克δ函数在源项，和控制和状态变量是在一些非平凡的容许集。由于状态方程的解的正则性低，对这些问题的分析是具有挑战性的。在存在状态约束的情况下，拉格朗日乘子仅仅是测度，伴随方程的解也具有很低的正则性。为了显示最佳误差估计，必须建立非标准范数下有限元解的最佳逼近性质，例如空间上的逐点逼近和时间上的全局逼近。这样的误差估计是不可用的有限元文献，需要开发。在非标准范数下得到这些尖锐误差估计的关键思想是首先给出一类全离散间断Galerkin方法的离散最大正则性结果。这些新的结果将提供一个更深入的了解常用的数值方法来解决这些问题，也可能是有用的其他问题，使用各向异性空间。目前，各向异性空间上的有限元误差估计的结果很少，这些尖锐的，最佳逼近型的结果将推进当前的有限元知识。

The PI will investigate a class of optimal control problems with pointwise controls. These problems are classical and have a wide range of applications, for instance in water waste treatment, river pollution, calcium waves in a heart cell, and noise control. The finite element method is the most popular method to solve such problems numerically, but there are very few results in this area on a priori error estimates. The difficulty lies in the roughness of the solutions of the underlying equations. The aim is to understand how well such rough solutions can be approximated numerically and to obtain sharp a priori error estimates. This area of research requires many tools from classical and modern analysis, partial differential equations, finite element methods, and optimization, and offers a wide variety of exciting problems well suited for research and educational purposes.The PI will study problems with pointwise controls and/or state constraints. These problems are usually modeled by Dirac delta functions in the source term, and control and state variables are in some nontrivial admissible sets. Analysis of such problems is challenging due to low regularity of solutions of the state equations. In the presence of state constraints, the Lagrange multipliers are merely measures and solutions of the adjoint equation have very low regularity as well. To show optimal error estimates one has to establish sharp best approximation properties of the finite element solution in non-standard norms, such as pointwise in space and global in time. Such error estimates are not available in the finite element literature and need to be developed. The key idea in obtaining these sharp error estimates in such non-standard norms is first to show discrete maximum regularity results for a class of fully discrete discontinuous Galerkin methods. These new results will provide a deeper insight into numerical methods commonly used to solve such problems and may also be useful for other problems where anisotropic spaces are used. Presently, there are very few results on finite element error estimates on anisotropic spaces and those sharp, best approximation type results will advance the current finite element knowledge.