Point and state constrained optimal control parabolic problems

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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.