Infinite Horizon Problems in the Calculus of Variations

The Role of Transformations with an Application to the Brachistochrone Problem
  • Sabine Pickenhain


In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume that the objective is convex. The problem setting implies a weighted Sobolev space as state space. For this class of problems we establish necessary optimality conditions in form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to establish sufficient optimality conditions. We apply the theoretical results proven to the problem of the Brachistochrone.


Weighted functional spaces Calculus of variations Infinite horizon 

Mathematics Subject Classification (2010)

49J40 49K15 46N10 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.BTU Cottbus-SenftenbergCottbusGermany

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