Mathematical Sciences: Lipschitz Stability and Its Application to Numerical Analysis in Optimal Control

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9404431 Hager/Dontchev This research will pursue the applications of recent implicit function-type results for set-valued maps in an infinite dimensional setting to numerical algorithms for optimal control problems and the Lipschitz properties of their solutions. This work considers control problems with state and control constraints for which there are many open questions. Log-barrier approaches to control constraints will be investigated, and error estimates for discretizations will be derived. This research will develop and analyze numerical algorithms for solving optimal control problems with control and state constraints. In particular, the possible Lipschitz properties of optimal solutions will be investigated. Knowledge of the Lipschitz continuity of the solution is important, for instance, in the study of the convergence behavior of various numerical approximations for the solution. Anticipated applications of the research include problems in computer aided geometric design, and optimal motion planning of a flexible structure (such as a robot or a crane) in the presence of obstacles.