Cosmically Lipschitz set-valued mappings (Q1863702)

In view of applications to necessary optimality conditions in optimal control, to the theory of Hamilton-Jacobi equations and to invariant sets for differential inclusions, the author proves a large number of results (some 9 theorems, 6 propositions and 8 lemmas) concerning, mainly, the so-called ``cosmic Lipschitzianity'' of set-valued mappings introduced in [\textit{R. T. Rockafellar} and \textit{R. J.-B. Wets}, ``Variational analysis'', (1998; Zbl 0888.49001)]. Defined in the usual way in terms of the so-called ``cosmic distance'' between two sets (with a too complicated definition to be presented here), the cosmic Lipschitzianity is proved in Prop. 2.9 to be an intermediate property between the usual Lipschitz property (expressed in terms of the well-known Pompeiu-Hausdorff distance) and the weaker ``sub-Lipschitz'' property expressed in terms of the \(\rho\)-pseudometrics. Among the multitude of interesting results in the paper, one may note the one in Theorem 3.1 according to which cosmic Lipschitzianity of the epigraph \(x\mapsto \text{epi}(L(x,.))\) is preserved ``under the Legendre-Fenchel transform'', \(H(x,p):= \sup_{v\in \mathbb{R}^m}[\langle p,v\rangle-L(x,v)]\) (in the sense that \(x\mapsto \text{epi}(H(x,.))\) has the same property), provided the Lagrangean \(L(x,.)\) is proper, lsc and convex for any \(x\in \mathbb{R}^n\).