Abstract

[For the first part of this paper see ESAIM, Control, Optimisation and Calculus of Variations.] Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, e.g. nonconvex correspondence problems from image processing, we raise the question how to understand to notion of quasiconvexity for a continuous function ff with a convex body K⊂Rnm\mathrm{K} \subset \mathbb{R}^{nm} instead of the whole space Rnm\mathbb{R}^{nm} as the range of definition. Extending ff by (+∞)(+\infty) to the complement Rnm∖ K\mathbb{R}^{nm}\setminus\,\mathrm{K}, the appropriate quasiconvex envelope turns out to be f(qc)(w)=sup⁡ {g(w) ∣ g ⁣:Rnm→R∪{(+∞)}f^{(qc)}(w) = \sup\, \bigl\{g(w)\, \big\vert\, g \colon \mathbb{R}^{nm} \to \mathbb{R} \cup \{(+\infty)\} quasiconvex and lower semicontinuous, g(v)≤f(v) ∀v∈Rnm}.g(v) \le f(v)\ \forall v \in \mathbb{R}^{nm}\bigr\}. In the present paper, we prove that f(qc)f^{(qc)} admits a representation as f(qc)(w)=f^{(qc)}(w) = Min {∫Kf(v) dν(v) ∣ ν∈S(qc)(w)}∀w∈K\bigl\{\int_{\mathrm{K}} f(v)\,d\nu(v)\, \big\vert\, \nu \in \mathrm{S}^{(qc)}(w)\bigr\} \quad \forall w \in \mathrm{K} where the sets S(qc)(w)\mathrm{S}^{(qc)} (w) are nonempty, convex, weak∗^*-sequentially compact subsets of probability measures. This theorem, forming a natural counterpart to the author's previous results about the representation of f(qc)f^{(qc)} in terms of Jacobi matrices, has been proven indispensable for the derivation of Jensens' integral inequality as well as of differentiability theorems for the envelope f(qc)f^{(qc)}. The paper is mainly concerned with a detailed analysis of the set-valued map S(qc)\mathrm{S}^{(qc)}, which will be explicitely described in terms of averages of generalized controls.

Author information

Contact details are reproduced from the original publication and may be historical.

Marcus Wagner

Brandenburg University of Technology, Dept. of Mathematics, P. O. Box 101344, 03013 Cottbus, Germany

wagner@math.tu-cottbus.de

Keywords

  • Unbounded function
  • quasiconvex envelope
  • probability measure
  • generalized control
  • mean value theorem
  • set-valued map
  • representation theorem

Mathematics Subject Classification

26B25, 26B40, 26E25, 49J45, 49J53

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Suggested citation

M. Wagner. “On the Lower Semicontinuous Quasiconvex Envelope for Unbounded Integrands (II): Representation by Generalized Controls.” Journal of Convex Analysis 16 (2009), No. 2, 441–472. https://doi.org/10.68381/jca16024

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press.