Characterizations of hamiltonian geodesics in sub-riemannian geometry

Let (M, ε, g) be a sub-Riemannian manifold. The geodesics of (M, ε, g) are either Hamiltonian or srictly abnormal. In this paper we give necessary and sufficient conditions for a geodesic curve to be Hamiltonian, and we apply this to the study of totally geodesic submanifolds of Lie groups. We also give an example of a foliation by totally geodesic submanifolds, each leaf containing a one parameter family of geodesics which are Hamiltonian in the leaf but stricly abnormal in the ambient space.     

On regularity properties of extremal controls

We prove some regularity properties of the optimal controls for the smooth bracket generating systems with scalar control parameters, and show that the Cantor sets cannot be the sets of switching points.Partially supported by the Russian Fund for Fundamental Research and grant No. 95-01-00310a.     

The dynamic interpolation problem: On Riemannian manifolds,Lie groups,and symmetric spaces

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM 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 trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.Work supported in part by NSF grant No. 89-14643 and NATO project CRG 910926.     

A note on Carnot geodesics in nilpotent Lie groups

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct a group, with a left invariant bracket-generating distribution, for which some Carnot geodecics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, which we derive (for any Lie group) explicitly in terms of the structure constants.The first author was partially supported by an NSF Postdoctoral Fellowship, and the second author was partially supported by a Rothschild Postdoctoral Fellowship.     

Rigid and abnormal line subdistributions of 2-distributions

A line subdistributionL of the distributionE on then-manifoldM is said to be rigid (abnormal) if anyL-curve is rigid (abnormal), i.e., is an isolated (singular) point of the set of allE-curves joining two fixed points ofM. This paper contains the necessary and sufficient conditions for a line subdistribution to be rigid (abnormal), formulated in geometrical and algebraic terms, the existence theorems, and the analysis of the structure of rigid (abnormal), line subdistributions of generic 2-disrtibutions.Research partly supported by the Technion VRP Fund.     

Generalized Rolle theorem in ℝ n and ℂ

The Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if [0,1] ⁿ ,tx(t), is a closed smooth spatial curve and L() is the length of its spherical projection on a unit sphere, then for thederived curve [0,1], ⁿ , the following inequality holds: L() L(). For the analytic functionF(z) defined in a neighborhood of a closed plane curve ² this inequality implies that (F) (F) + (), where (F) is the total variation of the argument ofF along , and () is the integral absolute curvature of .As an application of this inequality, we find an upper bound for the number of complex isolated zeros of quasipolynomials. We also establish a two-sided inequality between the variation index (F) and another quantity, called theBernstein index, which is expressed in terms of the modulus growth of an analytic function.     

Approximate Invariance and Differential Inclusions in Hilbert Spaces

Consider a mappingF from a Hilbert spaceH to the subsets ofH, which is upper semicontinuous/Lipschitz, has nonconvex, noncompact values, and satisfies a linear growth condition. We give the first necessary and sufficient conditions in this general setting for a subsetS ofH to be approximately weakly/strongly invariant with respect to approximate solutions of the differential inclusion . The conditions are given in terms of the lower/upper Hamiltonians corresponding toF and involve nonsmooth analysis elements and techniques. The concept of approximate invariance generalizes the well-known concept of invariance and in turn relies on the notion of an ∈-trajectory corresponding to a differential inclusion. Dedicated to Professor R. V. Gamkrelidze The first and the third authors are supported in part by the Natural Sciences and Engineering Research Council of Canada and Le Fonds FCAR du Québec. The second author is visiting Rutgers University. Supported in part by the Russian Foundation for Fundamental Research, Grant 96-01-00219, by the Natural Sciences and Engineering Research Council of Canada, Le Fonds FCAR du Québec, and by the Rutgers Center for Systems and Control (SYCON).     

First- and Second-Order Integral Functionals of the Calculus of Variations which Exhibit the Lavrentiev Phenomenon

We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations $$\mathcal{J}(x) = \int\limits_0^1 {L(t,x(t),\dot x(t))dt \to \inf , x(0) = x_0 } , x(1) = x_1$$ ,when the infimum of the problem in the class of absolutely continuous functionsW _1,1[0, 1] is strictly less than the infimum of the same problem in the class of Lipshitzian functionsW _1,∞[0, 1]. We suggest an approach to constructing new classes of integrands which exhibit the Lavrentiev phenomenon (Theorem 2.1). A similar method is used to construct (Theorem 3.1) a class of autonomousC ¹-differentiable integrandsL(x, .x, ..x) of the calculus of variations which are regular, i.e., convex, coercive w.r.t. ..x, and exhibit theW _2,1–W _2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem $$\begin{array}{*{20}c} {\mathcal{J}(x( \cdot )) = \int\limits_0^1 {L(x(t),\dot x(t),\ddot x(t)) dt \to \inf ,} } \\ {x(0) = x_0 , \dot x(0) = \upsilon _0 , x(1) = x_1 , \dot x(1) = \upsilon _1 } \\ \end{array}$$ ,the infimum of this problem over the spaceW _{2, 1}[0, 1] is strictly less than its infimum over the spaceW _2,∞[0, 1]. This provides a negative answer to the question of whether functionals with regular autonomous second-order integrands should only have minimizers with essentially bounded second derivative.     

An analytic classification of saddle resonant singular points of holomorphic vector fields in the complex plane

An analytic classification of general saddle resonant points of holomorphic vector fields in the complex plane is obtained. This classification has two functional moduli more than an analytic orbital classification.This paper is partly supported by the Soros Fund, grant M98000.     

Homogeneous tangent vectors and high-order necessary conditions for optimal controls

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems , is proved. In connection with a suitable dilation on the state space ℝ ⁿ , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information. This research was partially supported by Istituto Nazionale di Alta Matematica “F. Severi.”     

On the central connection problem for equations with an irregular singular point of a single level

We study the central connection problem for linear systems of equations with two singularities: one at the origin which is assumed to be regular-singular, and another one at infinity having a formal fundamental solution of only one level (in the Newton polygon).     

On problems related to growth, entropy, and spectrum in group theory

We review some known results and open problems related to the growth of groups. For a finitely generated group Γ, given whenever necessarytogether with a finite generating set, we discuss the notions of

Nonabelian statistics for calogero-sutherland systems of particles

The mathematical aspects of the theory of nonabelian statistics and anyons are discussed. The dynamics and kinematics of these objects are defined by Hamiltonians which are the sums of terms associated with different roots of a given root system. The equations of Fuchsian type which are the so-called generalized Knizhnik-Zamolodchikov equations associated with the root systemsA _n ,B _n , andC _n , are presented. Some results the generalized pure braid groups and their application for defining the generalized anyon are discussed. The factorization of the spin Calogero-Sutherland operators and their connection with the theory of nonabelian statistics and anyons are given. To Dmitrii Anosov with admiration and deep respect on his sixtieth birthday The research was supported by Grants RFFI-INTAS 00418 and RFFI-Germany 96-01-00008G.     

Topological obstructions to the representability of functions by quadratures

A topological variant of Galois theory, in which the monodromy group plays the role of the Galois group, is described. It turns out that there are topological restrictions on the way the Riemann surface of a function represented by quadratures covers the complex plane.This work was partially supported by Grant MBF000 from the International Science Foundation.     

On the confluence phenomenon for fuchsian equations

In the paper we present a method of investigation of asymptotic behavior of solutions to parameter-dependent degenerate differential equations both with regular and irregular points of singularity. We examine the confluence phenomenon for such points, which is the process of the coincidence of different points at a certain value of the parameter. This examination is based on a resurgent representation of the corresponding solutions which also depends on the parameter. In particular, the confluence of the representations in question is considered.     

Output Controllability for Semi-Linear Distributed Parabolic Systems

The purpose of this paper is to explore regional controllability developed for linear system [5, 15] to the semi-linear case. Under the hypothesis that the approximate regional controllability holds, we find a control which steers a system to a desired state in a subregion of the system domain and the approach is based on an extension of the Hilbert uniqueness method and the Schauder fixed-point theorem. The analytical case is then discussed using generalized inverse techniques and successfully illustrated by simulations.      

Geometric Orbital Transfer Using Averaging Techniques

In this article, we focus on the coplanar orbital transfer of a satellite between Keplerian orbits, with low propulsion. We compute and analyze averaged systems associated with the energy minimization problem with the full control or control oriented in a single direction, tangential-normal or radial-orthoradial. This allows to understand the properties of each control direction in the optimal control problems.      

Parametric Stability of Impulsive Functional Differential Equations

The purpose of this paper is to obtain sufficient conditions for parametric stability of impulsive functional differential equations with impulse perturbations at fixed moments of time. The concept of parametric stability provides a mathematical framework for solving the joint problem of feasibility and stability of equilibrium states. This problem arises whenever uncertain parameters can cause the equilibrium states to shift or disappear. The main results are obtained by means of piecewise continuous functions, which are analogs of the classical Lyapunov functions, and via the Razumikhin technique. Two examples are given to illustrate our results.