A Novel Regularization-Based Computational Framework for State-Constrained Optimal Control

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The research plan for this project is motivated by a broad range of practical applications. A relevant example is in the localized heat treatment of cancer in which the intent is to heat the tumor cells, but at the same time assure that nearby healthy cells are not heated, and hence not damaged. A goal of this kind is called an optimal control problem with pointwise state constraints. The principal investigators will develop novel computational models for the solution of these control problems. Their research is based on the cross-fertilization of ideas from diverse disciplines of mathematics and application domains, and has strong potential for impact in engineering domains such as the optimization of the process of producing hot steel profiles without the generation of cracks. The research team will also integrate their research in the university educational program in the mathematical sciences and will produce basic software that can be made available for solution of other significant applications that fall into the same modeling framework. The principal investigators aim to develop a novel regularization approach for the solution of pointwise constrained optimal control problems. Such problems are a focus of considerable recent research and pose serious challenges for finding reliable solutions. One of the main issues is that the associated Lagrange multipliers are Radon measures so that the control has low regularity. This causes adverse effects at the analytical level when obtaining optimality conditions for the control problem, and at the numerical level when performing discretization. The lack of regularity can be attributed to the fact that the underlying ordering cone has an empty interior. Consequently, no general Karush-Kuhn-Tucker theory is available. In fact, the failure of a Slater-type constraint qualification is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, non-smooth optimization, and variational inequalities. Conical regularization provides a unified framework to study optimization problems for which a Slater-type constraint qualification fails to hold due to the empty interior of the ordering cone associated with the inequality constraints. The investigators plan to develop new error estimates for the conical regularization for optimal control of partial differential equations and variational inequalities with pointwise state constraints. The project will test the new theoretical results for Nash equilibrium problems, linear elasticity, and supply chains on networks. The project also has an educational impact in the training of graduate students. The investigators will integrate education with research and design courses to teach state-of-the-art techniques on optimal control.