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@ARTICLE{HannemannTams:826314,
      author       = {Hannemann-Tamás, Ralf and Marquardt, Wolfgang},
      title        = {{H}ow to verify optimal controls computed by direct
                      shooting methods? – {A} tutorial},
      journal      = {Journal of process control},
      volume       = {22},
      number       = {2},
      issn         = {0959-1524},
      address      = {Amsterdam [u.a.]},
      publisher    = {Elsevier Science},
      reportid     = {FZJ-2017-00547},
      pages        = {494 - 507},
      year         = {2012},
      abstract     = {For the solution of optimal control problems, direct
                      methods have been established in the process engineering
                      community. If set up correctly they robustly provide more or
                      less accurate approximations of the exact solution. In the
                      usual engineering practice, neither the distance to the
                      exact solution is reflected, nor the compliance with the
                      continuous necessary conditions in form of Pontryagin's
                      Minimum Principle is checked. At the end, some approximate
                      solution is available but its quality is at question.This
                      tutorial addresses the problem of the verification of
                      optimal controls computed by direct shooting methods. We
                      focus on this popular transcription method though the
                      results are also relevant for other solution strategies. We
                      review known results spread in the mathematical literature
                      on optimal control to show how the output of the nonlinear
                      programs (NLPs) resulting from single shooting
                      transcriptions of optimal control problems can be
                      interpreted in the context of Pontryagin's Minimum
                      Principle. In particular, we show how to approximate
                      continuous adjoint variables by means of the dual
                      information provided by the NLP solver. Based on this
                      adjoint approximation we use a multi-level setting to
                      construct an estimate of the distance to a true extremal
                      solution satisfying the continuous necessary conditions of
                      optimality. A comprehensive case study illustrates the
                      theoretical results.},
      cin          = {VS-V / GRS Jülich ; German Research School for Simulation
                      Sciences},
      ddc          = {004},
      cid          = {I:(DE-Juel1)VS-V-20090406 / I:(DE-Juel1)GRS-20100316},
      pnm          = {899 - ohne Topic (POF3-899)},
      pid          = {G:(DE-HGF)POF3-899},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000301688000014},
      doi          = {10.1016/j.jprocont.2011.11.002},
      url          = {https://juser.fz-juelich.de/record/826314},
}