Volume 6 (2000), No. 3 (July), pp. 311--459

P. Mormul. Goursat Flags: Classification of Codimension-One Singularities. J. Dynam. Control Systems 6 (2000), no. 3, 311--330.

A distribution D of corank r \ge 2 on a manifold M is Goursat when its Lie square [D, D] is a distribution of constant corank r-1, the Lie square of [D, D] is of constant corank r-2 and so on. Any such D, according to von Weber [21] and E. Cartan [3], behaves in a well-known way at generic points of M: in certain local coordinates it is the chained model (C) given below, a classical object in the control theory. Singularities concealed in Goursat distributions have emerged for the first time in [8]; by now the complete local classification of these objects of coranks not exceeding 7 is known, plus some isolated facts for coranks 8, 9, and 10.

In the present paper we deal with the Goursat distributions of any corank r and obtain a complete classification of the first occurring singularities of them, located at points outside a stratified codimension-2 submanifold of M. Off this set there are (on top of (C)) only r-2 non-equivalent singular behaviours possible.

S.A. Schapovalov. A New Solution of One Birkhoff Problem. J. Dynam. Control Systems 6 (2000), no. 3, 331--339.

In 1926, G. D. Birkhoff in [1] has formulated the following problem: describe the set of all ordinal numbers for which there exist dynamical systems with center depths (the definition will be given later) equal to them. In this case, Birkhoff used the term ``ordinal numbers of central trajectories" instead of ``center depths"; however, the both terms mean the same. In 1948, A.G. Maier gave an answer to this question. In [2], he presented a construction of a flow having an arbitrary (given) center depth the phase space of which is a subset of R^3.

The present paper gives a new solution of Birkhoff's problem based on some methods of symbolic dynamics. The mentioned methods were earlier used by the author ([3], [4]) in his study of one topological invariant of symbolic dynamical systems, which obtained the name of depth. (To avoid any misunderstanding, we emphasize that terms ``depth" and ``center depth" stand for different concepts though sound likely; the depth of the center of some dynamical system and its center depth need not to be the same ordinal numbers.)

A new construction of dynamical systems with arbitrary center depth, which is presented below, seems to be more simple and clear than Maier's construction.

V.S. Afraimovich, T. Young. Mather Invariants and Smooth Conjugacy on S^2. J. Dynam. Control Systems 6 (2000), no. 3, 341--352.

We construct a ``Mather invariant'' for certain classes of diffeomorphisms of the sphere. We show that two such maps f and g are smoothly conjugate if and only if the eigenvalues of Df and Dg at the fixed points agree and the Mather invariants are equivalent. We also show that the Mather invariant is onto as a functional and give conditions on the invariant under which a diffeomorphism is embeddable in a flow.

P.O. Schorygin. On the Controllability Problem Arising in Financial Mathematics. J. Dynam. Control Systems 6 (2000), no. 3, 353--363.

This paper deals with the approximate controllability problem for a parabolic equation defined for x \in R, t \in[0,T]. It is possible to find a sequence of initial conditions with given compact support such that traces at x=0 of corresponding solutions converge to the trace of a given solution. We obtain estimates of the rate of this convergence. These estimates are constructed in terms of eigenvalues of certain compact operator.

A.A. Agrachev, G. Charlot, J.P.A. Gauthier, V.M. Zakalyukin. On Sub-Riemannian Caustics and Wave Fronts for Contact Distributions in the Three-Space. J. Dynam. Control Systems 6 (2000), no. 3, 365--395.

In a number of previous papers of the first and third authors, caustics, cut-loci, spheres, and wave fronts of a system of sub-Riemannian geodesics emanating from a point q_0 were studied. It turns out that only certain special arrangements of classical Lagrangian and Legendrian singularities occur outside q_0. As a consequence of this, for instance, the generic caustic is a globally stable object outside the origin q_0. Here we solve two remaining stability problems.

The first part of the paper shows that in fact generic caustics have moduli at the origin, and the first module that occurs has a simple geometric interpretation.

On the contrary, the second part of the paper shows a stability result at q_0. We define the ``big wave front'': it is the graph of the multivalued function arclength -> wave-front reparametrized in a certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this ``big wave front'' on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.

A.Yu. Zhirov. Complete Combinatorial Invariants for Conjugacy of Hyperbolic Attractors of Diffeomorphisms of Surfaces. J. Dynam. Control Systems 6 (2000), no. 3, 397--430. Combinatorial description of one-dimensional hyperbolic attractors of diffeomorphisms of surfaces is given in the general case (including the case of diffeomorphisms of nonorientable surfaces). On the base of it, complete invariants for conjugacy of diffeomorphisms on some neighborhoods of attractors are defined. It is shown that solution of the problem of conjugacy of two attractors by using these invariants can be reduced to a finite algorithm.

V.V. Filippov. Remarks on Periodic Solutions of Ordinary Differential Equations. J. Dynam. Control Systems 6 (2000), no. 3, 431--451.

The paper is written in comparison of the power of the Leray--Schauder method and the method of translation along trajectories in the boundary values problems theory for ordinary differential equations. The author suggests a new version of the continuation principle for the method of translation along trajectories. Then he shows how to use the new approach to obtain in a simpler way a reinforced analog of a theorem of J. Mawhin.

D. Mittenhuber. Controllability of Solvable Lie Algebras. J. Dynam. Control Systems 6 (2000), no. 3, 453--459.

We call a Lie-algebra g SID-controllable if there exist A, B \in g such that the invariant control system on the simply connected group G given by \dot g=g(A+uB), u \in R, is controllable. This is tantamount to saying that the semigroup S(A,\pm B)= \subseteq G is all of G.

In his papers [4]--[7] Yu. Sachkov analyzed and characterized solvable SID-controllable Lie algebras. In this paper we improve his results by proving the following simple reduction criterion: a solvable Lie algebra g is SID-controllable iff the factor-algebra g/D^2 g is SID-controllable.

For solvable g this reduces the whole problem to metabelian algebras (D^2 g=0), a case which is completely solved in Sachkov's work.

Our result also generalizes immediately to systems with multiple inputs.