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| Master Thesis | FZJ-2025-03614 |
2025
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Please use a persistent id in citations: doi:10.34734/FZJ-2025-03614
Abstract: Witsenhausen’s counterexample is a well known problem from control theory illustrating, linear controllers are not always the best choice. Studies on theoretical and numerical results have been conducted for now more than 50 years and mathematicians are still searching for new attempts gaining better controllers for the problem. The performance of these controllers is compared on a benchmark based on the problem’s underlying cost functional. In this thesis first a new method to evaluate the named cost functional was developed. Hereby the method was built as it works adaptively, requiring only as much computing capacity as is necessary. Moreover, the method includes a discontinuity detection to handle step functions which are often used for Witsenhausen’s counterexample. Next, it was shown that Witsenhausen’s counterexample is a problem from variational analysis and a necessary criterion for optimality, based on the Euler-Lagrange, equation was derived. Based on this result, a basis function fulfilling the gained criterion was computed. In the first performed optimization step, the described basis functions were combined to gain an approximation for an optimal controller. The next optimization step was created based on the insights from previous papers indicating that adding a curve to each step improves the results. The result on the one hand was an evaluation method computing the cost for an analytically known result in less than a second for a precision of $10^{-8}$. Moreover, this method was able to determine the value up to a precision of $10^{-14}$. On the other hand, the optimization yielded the fourth best value known up to now, with an absolute difference of $3.159\cdotp 10^{-5}$ to the best known.
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