A Cornucopia Of Four-Dimensional Abnormal Subriemannian Minimizers

. We study in detail the local optimality of abnormal sub-Riemannian extremals for a completely arbitrary sub-Riemannian structure on a four-dimensional manifold, associated to a two-dimensional bracket-generating regular distribution. Using a technique introduced in earlier work with W. Liu, we show that large collections of simple (i.e. without double points) nondegenerate extremals exist, and are always uniquely locally optimal. In particular, we prove that the simple abnormal extremals parametrized by arc-length foliate the space (i.e. through every point there passes exactly one of them) and they are all local minimizers. Under an extra nondegeneracy assumption, these abnormal extremals are strictly abnormal (i.e. are not normal). (In the forthcoming paper [6] with W. Liu we show that in higher dimensions there are large families of "nondegenerate abnormal extremals" that are local minimizers as well. In dimension 3, for a regular distribution there are no nontrivial abnormal extr..

Growth or environmental concern: which comes first? Optimal control with pure stock pollutants

\ua9 1999, Springer Japan. This paper models an economy with a stock pollution problem that must choose between productive and environmental investments. Both increase consumption, but only the former leads to economic growth. An optimal control model is solved giving four different paths depending on the initial parameters. For small values of pollution maximizing economic growth is optimal; and for massive pollution all investments are dedicated to environmental abatement. Similarly, the role of discount and savings rates, the relative profitability of abatement and productive investments, and the length of the time horizon are analyzed. Optimal control models simple enough to solve analytically often give intuitively unsatisfactory, boundary-type solutions. Our model, however, does have a large domain of parameter values for which “interior” solutions are optimal. These may start with a period of exclusive productive or environmental investments and then switch over to a mix of investments that corresponds to real-life expectations