Applied Mathematics & Optimization

, Volume 77, Issue 3, pp 463–497 | Cite as

Pointwise Constraints in Vector-Valued Sobolev Spaces

with Applications in Optimal Control


We consider a set \(\mathcal {C}\) with pointwise constraints in a vector-valued Sobolev space. We characterize its tangent and normal cone. Under the additional assumption that the pointwise constraints are affine and satisfy the linear independence constraint qualification, we show that the set \(\mathcal {C}\) is polyhedric. The results are applied to the optimal control of a string in a polyhedral tube.


Tangent cone Normal cone Polyhedricity Vector-valued function Vector-valued measure 

Mathematics Subject Classification

46N10 49K21 



The author would like to thank Daniel Wachsmuth for the idea of the proof of Theorem 7.2. Moreover, the author appreciates the fruitful discussions with Frank Göring and Thomas Jahn, which led to the proof of Lemma 9.1. Finally, the associate editor drew the attention to the references [4, 13] and this is gratefully acknowledged.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculty of MathematicsTechnische Universität ChemnitzChemnitzGermany

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