JDCS, vol. 1, #2 (1995), pp. 139-294

1. A.A. Agrachev, A.V. Sarychev
STRONG MINIMALITY OF ABNORMAL GEODESICS FOR 2 - DISTRIBUTIONS pp. 139 - 176

We investigate the local length minimality (by the W-1,1 or H-1 topology) of abnormal sub-Riemannian geodesics for rank 2 distributions. In particular, we demonstrate that this kind of local minimality is equivalent to the rigidity for generic abnormal geodesics, and introduce an appropriate Jacobi equation in order to compute conjugate points. As a corollary, we obtain a recent result of Sussmann and Liu about the global length minimality of short pieces of the abnormal geodesics.

2. P. Crouch, F. Silva Leite
THE DYNAMIC INTERPOLATION PROBLEM: ON RIEMANNIAN MANIFOLDS, LIE GROUPS, AND SYMMETRIC SPACES, pp. 177 - 202

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifold M. In this problem we are given an ordered set of points in M and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the rajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases where M is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

3. W. Balser
FIRST - LEVEL FORMAL SOLUTIONS AND MULTISUMMABILITY, pp. 203 - 227

The concept of first-level formal solutions has been introduced and studied in earlier papers of the author. Here, a different but equivalent definition is given, and their existence proof is outlined. Furthermore, it is shown how this concept leads to a proof for the multisummability of formal solutions (in the usual sence) for linear systems of ODE.

4. A.A. Bolibruch
VECTOR BUNDLES ASSOCIATED WITH MONODROMIES AND ASYMPTOTICS OF FUCHSIAN SYSTEMS, pp. 229 - 252

Connections between the splitting type of a holomorphic vector bundle and the asymptotics of solutions to the corresponding Fuchsian system P-1(C) are presented. As an application, the exact number of apparent singularitites arising as a result of the attempt to construct a scalar Fuchsian equation with a given monodromy is obtained.

5. M. Zhitomirskii
RIGID AND ABNORMAL LINE SUBDISTRIBUTIONS OF 2-DISTRIBUTIONS, pp. 253 - 294

A line subdistribution L of the distribution E on the n-manifold M is said to be rigid (abnormal) if any L-curve is rigid (abnormal), i.e., is an isolated (singular) point of the set of all E-curves joining two fixed points of M. This paper contains the necessary and sufficient conditions for a line distribution to be rigid (abnormal), formulated in geometrical and algebraic terms, the existence theorems, and the analysis of the structure of rigid (abnormal), line distributions of generic 2-distributions.