Conic Distributions and Accessible Sets

Motivated by nonlinear control theory, we introduce the notion of conic distributions on a smooth manifold. We study topological and smoothness aspects of the set of accessible points associated with a conic distribution. We introduce the notion of abnormal paths and study their relation to boundary points of the accessible set. In particular, we provide sufficient conditions for the accessible set to be a maximal integral of the smallest integrable vector distribution containing the conic distribution. Under rather strong conditions, we are able to prove that the accessible set has the structure of a ‘manifold with corners’.      

Conjugate Points in the Euler Elastic Problem

For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. This work was supported by the Russian Foundation for Basic Research, project No. 06-01-00330.     

On the Motion Planning Problem, Complexity, Entropy, and Nonholonomic Interpolation

We consider the sub-Riemannian motion planning problem defined by a sub-Riemannian metric (the robot and the cost to minimize) and a non-admissible curve to be ε-approximated in the sub-Riemannian sense by a trajectory of the robot. Several notions characterize the ε-optimality of the approximation: the “metric complexity” MC and the “entropy” E (Kolmogorov-Jean). In this paper, we extend our previous results. 1. For generic one-step bracketgenerating problems, when the corank is at most 3, the entropy is related to the complexity by E = 2πMC. 2. We compute the entropy in the special 2-step bracket-generating case, modelling the car plus a single trailer. The ε-minimizing trajectories (solutions of the “ε-nonholonomic interpolation problem”), in certain normal coordinates, are given by Euler's periodic inflexional elastica. 3. Finally, we show that the formula for entropy which is valid up to corank 3 changes in a wild case of corank 6: it has to be multiplied by a factor which is at most 3/2. 2000 Mathematics Subject Classification. 53C17, 49J15, 34H05. The second author is supported by grants RFBR 050100458 and UR 0401128.     

Singular Curves Determine Generic Distributions of Corank at Least 3

We prove that generic distribution germ of even rank not divisible by 4 and of corank at least 3 is strongly determined by singular curves. 2000 Mathematics Subject Classification. Primary 58A30, 58K05; Secondary 53B99. The author was partially supported by Committee for Scientific Research (KBN), Poland (project No. 2P03A 00124).     

On the Existence of Codimension-One Nonorientable Expanding Attractors

We prove that a structurally stable diffeomorphism of any closed (2m + 1)-manifold, m ≥ 1, has no codimension one nonorientable expanding attractors. However, given any d ≥ 3, there exists an 9-stable A-diffeomorphism of a closed d-manifold with a codimension one nonorientable expanding attractor. 2000 Mathematics Subject Classification. Primary 37D20; Secondary 37C70, 37C15. This research was partially supported by the Russian Foundation for Basic Research (project Nos. 02-01-00098, 05-01-00501.     

Meromorphic Connections on ℙ1 and the Multiplicity of Abelian Integrals

Dedicated to Professor César Camacho on his 60th birthday In this paper, we introduce the notion of Abelian integrals in differential equations for an arbitrary vector bundle on ℙ¹ with a meromorphic connection. In this general context, we give an upper bound for the maximum multiplicity of Abelian integrals. 2000 Mathematics Subject Classification. 14F05, 32S40. Supported by MPIM-Germany.     

On The Stabilization of Homogeneous Perturbed Systems

In this paper, using some results on manifolds, we establish some conditions for stabilization of single-input homogeneous by dilation systems.      

On the Codimension One Motion Planning Problem

In this paper, we improve the results of [5] related to motion planning problems for corank one sub-Riemannian (SR) metrics. First, we give the exact estimate of the metric complexity, in the generic 3-dimensional case. (Only bounds from above and from below were given in [5].) Second, we show that the general expression for the metric complexity (that was proven to hold generically in the C case, or under certain nonvanishing condition (C) in the analytic case) is, in fact, always true under condition (C), on the complement of a subset of codimension infinity, in the set of C motion planning problems. Both results are constructive, i.e., an asymptotic optimal synthesis is exhibited in both cases.2000 Mathematics Subject Classification. 53C17, 49J15, 34H05.     

Navier-Stokes Equation on the Rectangle: Controllability by Means of Low Mode Forcing

We study controllability issues for the Navier-Stokes equation on a two-dimensional rectangle under so-called Lions boundary conditions. The Navier-Stokes equation is controlled by forcing applied to a small number of harmonic modes. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low mode forcing of finite-dimensional Galerkin approximations of this system. Proving the continuity of the “control ↦ solution” mapping in the so-called relaxation metric we use it to prove both solid controllability on the observed component and L ²-approximate controllability of the Navier-Stokes equation (full system) by low mode forcing. 2000 Mathematics Subject Classification. 35Q30, 93C20, 93B05, 93B29. Supported by FCT (Portuguese Foundation for Science and Technology).     

Asymptotics of Solutions of Periodic Problem for Generalized Nonlinear Hamiltonian System Singularly Perturbed by a Matrix

The paper deals with the periodic problem for a generalized nonlinear Hamiltonian system, which appears from the control optimality condition for the periodic problem of the minimization of the functional on trajectories of the equation with the operator A + ɛB standing before the derivative, where the operator A is singular and A + ɛB is invertible for sufficiently small ɛ > 0. Under some conditions, the asymptotic solution of such problem has been constructed in the form of the series with respect to nonnegative integer powers of ɛ. The solvability of the considered problem has also been established for sufficiently small ɛ > 0. This work was supported by the Russian Fundamental Research Foundation (project No. 06–01–00296).     

Invariants of binary differential equations

In this paper, we study binary differential equations a(x, y)dy ² + 2b(x, y) dx dy + c(x, y)dx ² = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf’s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F _p  = 0, and F _pp  ≠ 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form ω = dy − pdx defined on the singular surface F = 0. This work was partially supported by Fapesp grant No. 02/09157-5.     

Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

An efficient and accurate computational approach is proposed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete time necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two-point boundary-value problem. This formulation wherein the optimal control problem is solved based on discretization of the attitude dynamics and derivation of discrete time necessary conditions, rather than development and discretization of continuous time necessary conditions, is shown to have significant advantages. In particular, the use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. The first and second authors have been partially supported by NSF (project Nos. DMS-0504747 and DMS-0726263). The first and third authors have been partially supported by NSF (project Nos. ECS-0244977 and CMS-0555797.     

On the One-Step-Bracket-Generating Motion Planning Problem

We consider the general motion planning problem for a sub-Riemannian metric with one-step bracket-generating distribution. Our results generalize earlier results in the corank-one case. Mostly, we completely solve the problem in generic situation for corank smaller or equal to 3. Our results are constructive: we explicitly construct the asymptotically optimal solutions.2000 Mathematics Subject Classification. 53C17, 53C99, 93B29.The second author was partially supported by the Russian Foundation for Basic Research (project No. 0201099) and UR 03099.     

Reachable Sets for the Heisenberg Sub-Lorentzian Structure On ℝ3. An Estimate for the Distance Function

In this paper we compute reachable sets from a point for the Heisenberg sub-Lorentzian metric on ℝ³ and give an estimate (from below) for the distance function. 2000 Mathematics Subject Classification. 53C50.     

Iterated Integrals, Gelfand—Leray Residue, and First Return Mapping

Recently, one of the authors gave an algorithm for calculating the first nonzero Poincaré-Pontryagin function of a small polynomial perturbation of a polynomial Hamiltonian, under a generic hypothesis. We generalize this algorithm and show that any Poincaré-Pontryagin function of order l, denoted by M _l, can be written as a sum of an iterated integral of length at most l and of a combination of all previous Poincaré-Pontryagin functions, M ₁, M ₂, …, M _l-1, and their derivatives. This extends some results obtained recently and allows to identify the Bautin ideal with the ideal generated by iterated integrals. 2000 Mathematics Subject Classification. 34C35, 34D30.     

Stabilization of Generic Trees of Strings

We study the energy decay of a tree-shaped network of vibrating elastic strings when the pointwise feedback acts in the root of the tree. We show that the strings are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.2000 Mathematics Subject Classification. 35B37, 93B07, 93D15.     

Rigid Carnot Algebras: A Classification

A Carnot algebra is a graded nilpotent Lie algebra L = L₁ ⊕ … ⊕ L_r generated by L₁. The bidimension of the Carnot algebra L is the pair (dim L₁, dim L). A Carnot algebra is said to be rigid if it is isomorphic to any of its small perturbations in the space of Carnot algebras of the prescribed bidimension. In this paper, we give a complete classification of rigid Carnot algebras. In addition to free nilpotent Lie algebras, there are two infinite series and 29 exceptional rigid algebras of 16 exceptional bidimensions. 2000 Mathematics Subject Classification. 58A30,58K50.     

Oscillation of Linear Ordinary Differential Equations: On a Theorem of A. Grigoriev

We give a simplified proof and an improvement of a recent theorem of A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions of systems of linear equations with polynomial or rational coefficients. To Yulij Sergeevich Ilyashenko on his 60th birthday. 2000 Mathematics Subject Classification. 34C08, 34C10, 34M10. Gershon Kekst professorial chair in Mathematics.     

On the coincidence of two manifolds associated with the Calogero model

Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the “universal” Dunkl operators. These manifolds were defined as subsets of the complex space ℂ ^N of dimension equal to the number of roots of the root system under consideration. The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the “universal” Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the “universal” Dunkl operators commute. In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in ℂ ^N/2. A theoremon the coincidence of Bethe and Dunkl manifolds is proved.      

Multiple positive solutions of singular dirichlet second-order boundary-value problems with derivative dependence

The existence of multiple positive solutions for the singular Dirichlet boundary-value problem