Showing papers in "Journal of Dynamical and Control Systems in 1997"
TL;DR: In this article, the authors review some known results and open problems related to the growth of groups and discuss the notions of finite generating set and finite generative set, given whenever necessary.
Abstract: 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
157 citations
TL;DR: In this paper, a feedback-invariant approach to smooth optimal control problems is considered, analogous to the differential geometric method of investigating Riemannian geodesics in terms of the Levi-Civita connection and the curvature tensor.
Abstract: A feedback-invariant approach to smooth optimal control problems is considered. A Hamiltonian method of investigating regular extremals is developed, analogous to the differential-geometric method of investigating Riemannian geodesics in terms of the Levi-Civita connection and the curvature tensor.
117 citations
TL;DR: In this article, the authors present the properties of mappings of bounded variation defined on a subset of the real line with values in metric and normed spaces and show that major aspects of the theory of realvalued functions of bounded variations remains valid in this case.
Abstract: We present the properties of mappings of bounded variation defined on a subset of the real line with values in metric and normed spaces and show that major aspects of the theory of realvalued functions of bounded variation remains valid in this case. In particular, we prove the structure theorem and obtain the continuity properties of these mappings as well as jump formulas for the variation. We establish the existence of Lipschitz continuous geodesic paths and prove an analog of the well-known Helly selection principle. For normed space-valued smooth mappings we obtain the usual integral formula for the variation without the completeness assumption on the space of values. As an application of our theory we show that compact set-valued mappings (=multifunctions) of bounded variation admit regular selections of bounded variation.
63 citations
TL;DR: Isomonodromic deformations of Fuchsian systems are considered in this article, where a description of all possible forms of such deformations is presented. But the authors do not consider the Fuchsians in this paper.
Abstract: Isomonodromic deformations of Fuchsian systems are considered. A description of all possible forms of such isomonodromic deformations is presented.
63 citations
TL;DR: In this paper, the continuity properties of the bundle of solutions to a differential inclusion subject to a singular perturbation were studied, i.e., with respect to a scalar parameter multiplying a part of the derivatives.
Abstract: We study the continuity properties of the bundle of solutions to a differential inclusion subject to a singular perturbation, i.e., with respect to a scalar parameter ɛ multiplying a part of the derivatives. We give conditions under which every solution of the singularly perturbed inclusion is close, in a certain sense and for a sufficiently small ɛ, to a solution of the degenerate inclusion obtained for ɛ=0. These conditions include both stability and structural requirements (the later having no counterpart in the case of a differential equation). The main result obtained generalizes the well-known Tikhonov theorem for singularly perturbed differential equations.
58 citations
TL;DR: The concept of approximate invariance generalizes the well-known concept of invariance and in turn relies on the notion of an e-trajectory corresponding to a differential inclusion as discussed by the authors.
Abstract: Consider a mapping F from a Hilbert space H to the subsets of H, 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 subset S of H to be approximately weakly/strongly invariant with respect to approximate solutions of the differential inclusion \dot{x}(t) \in F(x). The conditions are given in terms of the lower/upper Hamiltonians corresponding to F 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 e-trajectory corresponding to a differential inclusion.
42 citations
TL;DR: Lower level sets of continuous selections of C2-functions defined on a smooth manifold in the vicinity of a non-degenerate critical point in the sense of l11r are studied in this article.
Abstract: Lower level sets of continuous selections of C2-functions defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of l11r are studied. It is shown that the lower level set is homotopy equivalent to the join of the lower level sets of the smooth and the nonsmooth part, respectively, of the corresponding normal form. Some generalized Morse inequalities are deduced from this result.
34 citations
TL;DR: In this article, a general concept of multifractality and a complete description of the multifractal spectra for Gibbs measures on two-dimensional horseshoes are discussed, as well as a characterization of surface diffeomorphisms.
Abstract: We discuss a general concept of multifractality, and give a complete description of the multifractal spectra for Gibbs measures on two-dimensional horseshoes. We discuss a multifractal characterization of surface diffeomorphisms.
31 citations
TL;DR: In this article, the authors studied the Lavrentiev gap for the calculus of variations and provided a negative answer to the question of whether functionals with regular second-order integrands should only have minimizers with essentially bounded second derivative.
Abstract: We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations {\cal J}(x)= \int\limits^1_0 L(t,x(t),\dot{x}(t))dt \rightarrow \inf, \quad x(0)=x_0, \quad x(1)=x_1, when the infimum of the problem in the class of absolutely continuous functions W1,1l0, 1r is strictly less than the infimum of the same problem in the class of Lipshitzian functions W1,∞l0,1r. 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 autonomous C1-differentiable integrands L(x,\dot{x}, \ddot{x}) of the calculus of variations which are regular, i.e., convex, coercive w.r.t. \ddot{x}, and exhibit the W2,1 – W2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem {\cal J}(x(\cdot))=\int\limits^1_0L(x(t),\dot{x}(t),\ddot {x}(t))dt \rightarrow \inf, x(0)=x_0, \quad \dot {x}(0)=v_0,\quad x(1)=x_1, \quad \dot {x}(1)=v_1, the infimum of this problem over the space W2,1l0, 1r is strictly less than its infimum over the space W2,∞l0, 1r. 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.
25 citations
TL;DR: In this paper, it was shown that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices.
Abstract: Graded nilpotent Lie groups, or Carnot groups, are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices. As a corollary, we prove that the centralizer for the corresponding quadratic “quantum” Hamiltonian in the universal enveloping algebra of this Lie algebra is “as small as possible.”
19 citations
TL;DR: In this paper, it was shown that 2-fold mixing automorphisms with minimal self-joinings of order 2 have minimal self joinings of all orders, and that any 2-simple action commuting with an uncountable family of weakly mixing automomorphisms is simple of all the orders.
Abstract: We show that 2-fold mixing automorphisms with minimal self-joinings of order 2 have minimal self-joinings of all orders; an order 3 quasi-simple action is quasi-simple of all orders; any 2-simple action commuting with an uncountable family of weakly mixing automorphisms is simple of all orders.
TL;DR: In this article, the integrable Hamiltonian dynamical systems of differential geometric origin of homogeneous Carnot-Caratheodory metrics have been described and the relation to previous descriptions of such flows via non-Hamiltonian methods and to problems of analytic mechanics is discussed.
Abstract: It is shown how to produce new examples of integrable Hamiltonian dynamical systems of differential geometric origin. These are normal geodesic flows of homogeneous Carnot-Caratheodory metrics. The relation to previous descriptions of such flows via non-Hamiltonian methods and to problems of analytic mechanics is discussed.
TL;DR: The authors apply a method, going back to E Nelson, for simplifying the expression of vector fields to a normal form, for example near a partially hyperbolic fixed point, in the case where some additional properties have to be respected by the change of variables.
Abstract: We apply a method, going back to E Nelson, for simplifying the expression of a vector field to a normal form, for example near a partially hyperbolic fixed point, in the case where some additional properties have to be respected by the change of variables We illustrate the method by solving recent questions encountered in desingularization problems
TL;DR: In this paper, the controllability of right-invariant control systems on Lie groups is studied in terms of the root decomposition corresponding to the adjoint operator of a Lie group.
Abstract: We study controllability of right-invariant control systems \Gamma = A+{\Bbb R}B on Lie groups. Necessary and sufficient controllability conditions for Lie groups not coinciding with their derived subgroup are obtained in terms of the root decomposition corresponding to the adjoint operator ad B. As an application, right-invariant systems on metabelian groups and matrix groups, and bilinear systems are considered.
TL;DR: In this article, the maximal number of basic sets of orientable and non-orientable surfaces is given, which depends on both the topological type of the surface and geometric properties of the basic sets.
Abstract: ForA-diffeomorphisms of orientable and nonorientable surfaces estimates of the maximal number of their one-dimensional basic sets are given. This number depends on both the topological type of the surface and geometric properties of basic sets. It is shown that these estimates cannot be improved.
TL;DR: In this article, a method of investigation of asymptotic behavior of solutions to parameter-dependent degenerate differential equations both with regular and irregular points of singularity is presented, which is the process of the coincidence of different points at a certain value of the parameter.
Abstract: 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.
TL;DR: In this article, it was shown that the Cartesian square T×T can have a uniform rank if the columns at each stage can be chosen of equal height, and that for a given positive integer m ≥ n, the identity automorphism can be conjugated to its inverse.
Abstract: An automorphismT: (X,F,μ) → (X,F,μ) on a standard Borel probability space is said to have uniform rankn, if it has rankn and the columns at each stage can be chosen of equal height. We show that for such an automorphismT, ifS is an automorphism conjugatingT to its inverse (i.e.,ST=T
−1
S), thenS
2m
=I for some positive integerm≤n (I denotes the identity automorphism). Related results for transformations having local rank one (also called positive β-rank maps) with no partial rigidity are given. Particular attention is given to the question of whether the Cartesian squareT×T can have finite rank, and this question is answered in certain cases. The transformationsR(x,y)=(y,Tx) and the symmetric Cartesian squareT
⊙2 are shown to have some analogous properties.
TL;DR: In this paper, the structural stability of a diffeomorphism of a two-dimensional smooth closed orientable manifoldM of genusg ≥ 0 and asymptotic behavior of stable and unstable manifolds of points of one-dimensional basic sets is studied.
Abstract: In this paper, the interrelation between structural stability of a diffeomorphismf of a two-dimensional smooth closed orientable manifoldM of genusg≥0 and asymptotic behavior of stable and unstable manifolds of points of one-dimensional basic sets is studied. For a manifoldM of genusg≥1 with the universal covering\(\overline M \) we study also the problem of deviation from geodesics on\(\overline M \) of preimages of stable and unstable manifolds of the points of exteriorly situated one-dimensional basic sets.
TL;DR: In this article, necessary and sufficient conditions for a geodesic curve to be Hamiltonian were given for the study of totally geodesically sub-Riemannian submanifolds of Lie groups.
Abstract: Let (M, e, g) be a sub-Riemannian manifold. The geodesics of (M, e, 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.
TL;DR: In this paper, a classification of generic synthesis of analytic systems in dimensionn≥3 is given, where the trajectories are not tangent to a terminal submanifold.
Abstract: In previous papers, we gave in dimension 2 and 3 a classification of generic synthesis of analytic systems\(\dot v(t) = X(v(t)) + u(t)Y(v(t))\) with a terminal submanifoldN of codimension one when the trajectories are not tangent toN. We complete here this classification in all generic cases in dimension 3, giving a topological classification and a model in each case. We prove also that in dimensionn≥3, out of a subvariety ofN of codimension there, we have described all the local synthesis.
TL;DR: In this article, the classical Hopf-Lax formula for an explicit solution of a Hamilton-Jacobi equation is extended to equations of the formu t +H(u,Du)=0 with terminal datau(T, x)=g(x) assumed to be merely quasiconvex, i.e., having convex level sets.
Abstract: The classical Hopf-Lax formula for an explicit solution of a Hamilton-Jacobi equation is extended to equations of the formu t +H(u,Du)=0 with terminal datau(T, x)=g(x) assumed to be merely quasiconvex, i.e., having convex level sets. Using a new quasiconvex conjugateg *γ,p), the formula is given byu(t,x)=(g *(γ,p)-(T−t)H(γp))*. We give a direct proof thatu is a Subbotin minimax solution of the problem. The first-order obstacle problem associated with optimal control inL ∞ is also studied and an explicit solution given.
TL;DR: An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems, is proved in this paper, where the authors 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.
Abstract: An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems
$$\dot x = X_0 (x) + \sum
olimits_{j = 1}^m u _j X_j (x)$$
, 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
TL;DR: In this article, the central connection problem for linear systems of equations with two singularities is studied, 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).
Abstract: 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).
TL;DR: The mathematical aspects of the theory of nonabelian statistics and anyons are discussed in this article, where 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.
Abstract: 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.