Relaxation Gaps in Optimal Control Processes with State Constraints

  • Sandra Butzek
  • Werner H. Schmidt
Part of the International Series of Numerical Mathematics book series (ISNM, volume 124)


There are a lot of very simple examples of control processes which have no optimal solution. A well-known fundamental existence theorem is that of Roxin-Fillipov; unfortunately, the assumptions in this theorem are rather strong. Sometimes one can prove existence in absence of convexity by studying “bigger” problems obtained by relaxation, that means by convexification of the sets of speed vectors. Then we try to choose a special optimal solution of the relaxed problem and apply PONTRYAGIN’S maximum principle in order to discuss whether certain derived controls are optimal ones of the original problem (or not). BALDER [1] used a similar idea to prove new existence results for optimal control problems without convexity. He applies BAUER’S extremal principle instead of the maximum principle.


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Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Sandra Butzek
    • 1
  • Werner H. Schmidt
    • 1
  1. 1.Institut für Mathematik und InformatikErnst-Moritz-Arndt-Universität GreifswaldGreifswaldGermany

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