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Optimality Conditions for Variational Problems in Incomplete Functional Spaces

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Abstract

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class \({{\mathcal {C}}}^k\) as \(k\ge 1\).

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Funding

Research of this author was partly supported by the National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research under grant #15RT04, and by the Australian Research Council under Discovery Project DP-190100555.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20007, USA

    Ashkan Mohammadi

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

Authors
  1. Ashkan Mohammadi
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  2. Boris S. Mordukhovich
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Corresponding author

Correspondence to Boris S. Mordukhovich.

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Communicated by Massimo Pappalardo.

Dedicated to Professor Franco Giannessi in the occasion of his 85th birthday.

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Mohammadi, A., Mordukhovich, B.S. Optimality Conditions for Variational Problems in Incomplete Functional Spaces. J Optim Theory Appl 193, 139–157 (2022). https://doi.org/10.1007/s10957-021-01964-2

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