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@MASTERSTHESIS{Noffke:1045815,
      author       = {Noffke, René},
      title        = {{W}itsenhausen’s {C}ounterexample – {A} {R}efined
                      {A}pproach using {V}ariational {A}nalysis},
      school       = {FH Aachen},
      type         = {Masterarbeit},
      reportid     = {FZJ-2025-03614},
      pages        = {97 p.},
      year         = {2025},
      note         = {Masterarbeit, FH Aachen, 2025},
      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.},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {5112 - Cross-Domain Algorithms, Tools, Methods Labs (ATMLs)
                      and Research Groups (POF4-511)},
      pid          = {G:(DE-HGF)POF4-5112},
      typ          = {PUB:(DE-HGF)19},
      doi          = {10.34734/FZJ-2025-03614},
      url          = {https://juser.fz-juelich.de/record/1045815},
}