Pontryagin Maximum Principle and Stokes Theorem

We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived in a new insightful way. It also suggests generalizations in diverse directions of such famous principle.Comment: 21 pages, 7 figures; we corrected a few minor misprints, added a couple of references and inserted a new section (Sect. 7); to appear in Journal of Geometry and Physic

On the localization principle for the automorphisms of pseudoellipsoids

We show that Alexander's extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $f$ of a pseudoellipsoid \E^n_{(p_1, ..., p_{k})} \= \{z \in \C^n : \sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + ... + |z_n|^{2 p_{k}} < 1 \}, provided that $f$ is defined on a region \U \subset \E^n_{(p)} such that: i) \partial \U \cap \partial \E^n_{(p)} contains an open set of strongly pseudoconvex points; ii) \U \cap \{z_i = 0 \} \neq \emptyset for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.Comment: 7 pages; to appear on Proceedings of AM

On the singular loci and the images of proper holomorphic maps from pseudoellipsoids

We prove a generalisation of Rudin's theorem on proper holomorphic maps from the unit ball to the case of proper holomorphic maps from pseudoellipsoids.Comment: 15 pages; to appear on Mathematica Scandinavic