CLOSED EXTREMALS

Nonregular Abnormal Extremals

We study existence and rigidity (W1-isolatedness) of nonregular abnormal extremals of completely nonholonomic 2-distribution (nonregularity means that such extremals do not satisfy the strong generalized Legendre-Clebsch condition). Introducing the notion of diagonal form of the second variation, we generalize some results of A. Agrachev and A. Sarychev about rigidity of regular abnormal extrem...

Injectivity and adjoint extremals

The following investigation deals with injectivity results for general diffeomorphisms as well as for Haar’s transformation. We apply these results to adjoint variational problems introduced by Haar [6]. In particular, we prove a generalization of Krust’s theorem for two-dimensional adjoint extremals. AMS Classification Code: 26B10, 49B, 49L, 53A05, 53A10, 70G, 70H

Optimality of Broken Extremals

In this paper we analyse the optimality of broken Pontryagin extremal for an n -dimensional affine control system with a control parameter, taking values in a k dimensional closed ball. We prove the optimality of broken normal extremals when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the controllable fields is locally orthogonal to the sing...

Singular extremals on Lie groups

Extremals of Determinants of Laplacians

The modern quantum geometry of strings is concerned to a large extent with the set of all surfaces, with varying metrics on those surfaces, and with determinants of associated Laplacians. Our aim in this paper is to study this determinant as a function of the metric on a given surface and in particular its extreme values when the metric is suitably restricted. This kind of problem has actually ...