On the Lower Semicontinuous Quasiconvex Envelope for Unbounded Integrands (II): Representation by Generalized Controls

\def\K{\mathord{{\rm K}}} \def\R{{\mathbb R}^{nm}} \def\S{\mathord{{\rm S}}} [For the first part of this paper see ESAIM, Control, Optimisation and Calculus of Variations.]\par Motivated by the study of multidimensional control problems of Dieudonn\'e-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 $f$ with a convex body $\K \subset \R$ instead of the whole space $\R$ as the range of definition. Extending $f$ by $(+\infty)$ to the complement $\R\setminus\,\K$, the appropriate quasiconvex envelope turns out to be\par \medskip\hskip10mm $f^{(qc)}(w) = \sup\, \bigl\{g(w)\, \big\vert\, g \colon \R \to \mathbb{R} \cup \{(+\infty)\}$ quasiconvex\par \vskip-2mm\hskip10mm and lower semicontinuous, $g(v) \le f(v)\ \forall v \in \R\bigr\}.$\par In the present paper, we prove that $f^{(qc)}$ admits a representation as\par \medskip\hskip10mm $f^{(qc)}(w) =$ Min $\bigl\{\int_{\K} f(v)\,d\nu(v)\, \big\vert\, \nu \in \S^{(qc)}(w)\bigr\} \quad \forall w \in \K$\par where the sets $\S^{(qc)} (w)$ 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 $f^{(qc)}$ 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 $f^{(qc)}$. The paper is mainly concerned with a detailed analysis of the set-valued map $\S^{(qc)}$, which will be explicitely described in terms of averages of generalized controls.