Showing papers in "Journal of Dynamical and Control Systems in 1999"
TL;DR: In this article, the generalized De Casteljau algorithm for cubic splines with Hermite boundary conditions and more general boundary conditions for arbitrary m. Lie groups are the most simple symmetric spaces, and for these spaces the expressions for the first and second order derivatives of curves of arbitrary order obtained from the algorithm.
Abstract: We examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in {\Bbb R}^n, was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie groups are the most simple symmetric spaces, and for these spaces we develop expressions for the first and second order derivatives of curves of arbitrary order obtained from the algorithm. As an application of this theory we consider the problem of implementing the generalized De Casteljau algorithm on an m-dimensional sphere. We are able to fully develop the algorithm for cubic splines with Hermite boundary conditions and more general boundary conditions for arbitrary m.
92 citations
TL;DR: The relation between the reachable set of the semilinear system and that of its corresponding linear system by using degree theory is shown and the regularity of the solution of the given system is established.
Abstract: This paper deals with the approximate controllability of the semilinear functional differential equations with unbounded delays. We will also establish the regularity of the solution of the given system. It is shown the relation between the reachable set of the semilinear system and that of its corresponding linear system by using degree theory. Finally, a simple example to which our main result can be applied is given.
65 citations
TL;DR: In this paper, a topological analogue of the Abramov-Rokhlin formula for a free semigroup action is proposed, in which a skew-product transformation whose fiber entropy is taken to be the entropy of the initial action is considered.
Abstract: A definition of topological entropy for a free semigroup action is suggested. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action, we assign a skew-product transformation whose fiber entropy is taken to be the entropy of the initial action. The main result is Theorem 1, a topological analogue of the Abramov–Rokhlin formula.
56 citations
TL;DR: In this paper, a survey of the spectral theory of ergodic dynamical systems is presented, including the recent solution to Rokhlin's problem concerning ergodics having a homogeneous spectrum of multiplicity two.
Abstract: The purpose of this paper is to survey recent results in the spectral theory of ergodic dynamical systems. In addition we prove some known results using new methods and mention some new results, including the recent solution to Rokhlin's problem concerning ergodic transformations having a homogeneous spectrum of multiplicity two. We emphasize applications of ideas arising from the theory of joinings and Markov intertwinings.
55 citations
TL;DR: A short history of the discovery of the maximum principle in optimal control theory by L. S. Pontryagin and his associates is presented in this article, where the authors also present a model of optimal control.
Abstract: A short history of the discovery of the maximum principle in optimal control theory by L. S. Pontryagin and his associates is presented.
55 citations
TL;DR: Rokhlin's problem on the existence of an ergodic transformation having a homogeneous spectrum of a finite multiplicity is solved in this paper, and the question about the spectral multiplicity function of Cartesian powers for a generic transformation is also answered.
Abstract: Rokhlin's problem on the existence of an ergodic transformation having a homogeneous spectrum of a finite multiplicity is solved. Katok's question about the spectral multiplicity function of Cartesian powers for a generic transformations is also answered.
52 citations
Abstract: We show that for a generic automorphism T, the Cartesian product T × T has homogeneous spectrum of multiplicity two. New examples of automorphisms with the property \sigma \ \bot \ \sigma \ast \sigma are presented.
43 citations
TL;DR: In this paper, the Pontryagin maximum principle provides optimal control of the extremal trajectories associated with the problem, and an explicit integration of extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.
Abstract: Let H denote either the Heisenberg group \Bbb R^{2n+1}, or the Cartesian product of n copies of the three-dimensional Heisenberg group \Bbb R^3. Let lX1, Y1, …, Xn, Ynr be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics \dot{q}(t)=\sum\limits^n_{i=1}u_i(t)X_i(q(t))+v_i(t)Y(q(t)), the cost functional \Lambda(q,u)={1 \over 2}\int \sum\limits^n_{i=1}\mu_i(u^2_i+v^2_i), with arbitrary positive parameters μ1, …, μn, and admissible controls taken from the set of measurable functions t\mapsto u(t)=(u_1(t),v_1(t),\ldots , u_n(t), v_n(t)). The above control system is encoded either in the kernel of a contact 1-form (for \Bbb R^{2n+1}), or in the kernel of a Pfaffian system (for \Bbb R^{3n}). In both cases, the action of the semi-direct product of the torus Tn with H describe the symmetries of the problem.
The Pontryagin maximum principle provides optimal controlss extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant.
An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.
39 citations
TL;DR: In this paper, the existence and rigidity of non-regular abnormal extremals of completely nonholonomic 2-distribution was studied and a Jacobi curve of Lagrangian subspaces was constructed.
Abstract: We study existence and rigidity (W^1_\infty-isolatedness) of nonregular abnormal extremals of completely nonholonomic 2-distribution (nonregularity means that such extremals do not satisfy the strong generalized Legendre–Clebsch condition). Introducing the notion of diagonal form of the second variation, we generalize some results of A. Agrachev and A. Sarychev about rigidity of regular abnormal extremals to the nonregular case. In order to reduce the second variation to the diagonal form, we construct a special curve of Lagrangian subspaces, a Jacobi curve. We show that certain geometric properties of this curve (like simplicity) imply the rigidity of the corresponding abnormal extremal.
36 citations
TL;DR: In this article, it was shown that for a generic deformation of a linear analytic differential equation with an irregular singularity of order 1 of a generic (nonresonant) type, Stokes operators of the nonperturbed equation are limits of transition operators between appropriate eigenbases of the monodromy operators of perturbed equation.
Abstract: We show that for a generic deformation of a linear analytic differential equation with an irregular singularity of order 1 of a generic (nonresonant) type, Stokes operators of the nonperturbed equation are limits of transition operators between appropriate eigenbases of the monodromy operators of the perturbed equation. We prove a generalization of this statement for arbitrary degree nonresonant irregular singularity.
36 citations
TL;DR: In this article, a polynomial vector field with algebraic foliation admits a Liouvillian first integral on a P(2)-constrained algebraic invariant curve is defined.
Abstract: Let X be a polynomial vector field in {\Bbb C}^2s then it defines an algebraic foliation {\cal F} on {\Bbb C}P(2). If {\cal F} admits a Liouvillian first integral on {\Bbb C}P(2), then it is transversely affine outside some algebraic invariant curve S\subset {\Bbb C}P(2). If, moreover, for some irreducible component S_0 \subset S, the singularities q ∈ Sing {\cal F} \cup S are generic, then either {\cal F} is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation {\cal R}: p(x)\thinspace dy-(y^2a(x)+yb(x))\thinspace dx=0 on \overline {\Bbb C} \times \overline {\Bbb C}. This result has several applications such as the study of foliations with algebraic limit sets on {\Bbb C}P(2)(2), the classification polynomial complete vector fields over {\Bbb C}^2, and topological rigidity of foliations on {\Bbb C}P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.
TL;DR: In this paper, the authors studied the subanalyticity of the distance function d defined by a sub-Riemannian structure (Δ, g) and proved that d is subanalytic in a neighborhood of x 0.
Abstract: The aim of this paper is to study the subanalyticity of the distance function d defined by a sub-Riemannian structure (Δ, g). If the distribution is of degree 2, we prove that d is subanalytic and if Δ2 is fat d is subanalytic far away from the diagonal. In this last case we prove in fact that the function d(x0,·) is subanalytic in a neighborhood of x0.
TL;DR: In this paper, a singularly perturbed linear time-invariant system with delays in state and control variables is considered, and connections between properties of open-loop stabilizability (detectability) of the reduced-order and boundary-layer systems associated with the original system are analyzed.
Abstract: A singularly perturbed linear time-invariant system with delays in state and control variables is considered. Connection between properties of open-loop stabilizability (detectability) of the reduced-order and boundary-layer systems, associated with the original system, and such properties of the original system itself are analyzed.
TL;DR: In this article, sufficient conditions for positive solvability of the Riemann-Hilbert problem are presented, which are known and new conditions for the Hilbert problem, as well.
Abstract: Recent sufficient conditions for positive solvability of the Riemann–Hilbert problem, known and new, are presented.
TL;DR: In this article, a generalization of the Kohno theorem on the restricted Riemann-Hilbert problem for the generalized KZ equation of the Bn type is given.
Abstract: A generalization of the Kohno theorem on the restricted Riemann–Hilbert problem for the KZ equation of the Bn type is given. The representation of the the braid group of the Bn type in the algebra of symmetrical chord diagrams is constructed and its connection with the Bn type quasi-bialgebra structure and with the monodromy of the generalized KZ equation of the Bn type is discussed.
TL;DR: In this article, a singularly perturbed system with a small parameter e at the velocity of the slow variable y and with the fast variable x is considered and the main hypothesis is that for all y from some bounded domain D, the fast subsystem has a stable invariant or overflowing manifold M 0(y) and that the motions in this system going in the directions transversal to M 0 (y) are more fast than the mutual approaching of trajectories on M0(y).
Abstract: A singularly perturbed system with a small parameter e at the velocity of the slow variable y and with the fast variable x is considered. The main hypothesis is that for all y from some bounded domain D, the fast subsystem has a stable invariant or overflowing manifold M0(y) and that the motions in this system going in the directions transversal to M0(y) are more fast than the mutual approaching of trajectories on M0(y) (a precise statement is given in terms of appropriate Lyapunov-type characteristic numbers). It is proved that for a sufficiently small e, the whole system has an invariant manifold close to \bigcup\limits_{y\in D}M_0(y)\times \{y\}; the degree of its smoothness is specifed.
TL;DR: In this article, it was shown that infinitely many a priori possible choices are impossible if and only if the fuchsian system is obtained from another fuchsians system with a smaller number of poles and with a pole at 0 by the change of time.
Abstract: Any irreducible finitely generated matrix group (with generators M1,…,Mp+1 satisfying the only relation M1…Mp+1 e I) is the monodromy group of some fuchsian linear system on Riemann's sphere. The eigenvalues of the matrices Mj define λk,j, the eigenvalues of the matrices-residues of the system only up to integers. There are always infinitely many possible choices of λk,j, a priori they must satisfy the only condition that their sum is 0. However, not always all a priori possible choices can be made. Some of them can be impossible due to the positions of the poles.
Consider the a priori possible choices when the eigenvalues of only one matrix-residuum change (we presume that its pole is at 0). We show that infinitely many a priori possible choices are impossible if and only if the fuchsian system is obtained from another fuchsian system with a smaller number of poles and with a pole at 0 by the change of time t\mapsto t^k / (p_kt^k+p_{k-1}t^{k-1}+\ldots + p_0), p_i \in {\Bbb C}, p_0
eq 0, k > 1. The result is applied to the Riemann–Hilbert problem.
TL;DR: In this paper, it is shown that there are parameter values such that endomorphisms built by coupling of two identical 1-dimensional quadratic mappings have two kinds of trapping regions in the phase space: a large simply-connected domain inside of which there is a smaller trapping subregion consisting of two disjoint domainss (b) restrictions of the main diagonal y e x of their nonwandering sets are invariant subsets, which may not belong to attractors of the given endomorphism, but in any way, can be nonisolated in their non
Abstract: In this paper, it is shown that there are parameter values such that endomorphisms built by coupling of two identical 1-dimensional quadratic mappings (a) have two kinds of trapping regions in the phase space: a large simply-connected domain inside of which there is a smaller trapping subregion consisting of two disjoint domainss (b) restrictions of the main diagonal y e x of their nonwandering sets are invariant subsets, which may not belong to attractors of the given endomorphisms, but in any way, can be nonisolated in their nonwandering sets. Numerical investigation results represented in the Appendix display the existence of a couple of bifurcation cascades and this leads to a couple of nontrivial symmetrically disposed chaotic strange attractors each of which consists of four disjoint simply connected regions. As parameters vary, these attractors merge into the one consisting at first of two and then of one such region as mentioned above.
TL;DR: In this article, rational approximations of the transfer function of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end are studied.
Abstract: We study rational approximations of the transfer function \widehat {P} of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end. The approximation is to be carried out in a way that is appropriate, with respect to stability and performance, for the construction of suboptimal rational stabilizing compensators for the rod. The function \widehat {P} can be expressed as {\widehat P}(s)=s^{-2}g(\beta^2(s)), where g is an infinite product of fractional linear transformations and β is a (generally transcendental) function that characterizes a particular viscoelastic material. First, g(β2) is approximated by its partial products gN(β2). For relevant values of β2, convergence rates for gN are analyzed in detail. Convergence suitable for our problem requires the introduction of a new irrational convergence factor, which must be approximated separately. In addition, the fractional linear factors in β2(s) that appear in gN(β2(s)) must be replaced by something rational. When the damping is weak it is possible to do this by separating the oscillatory modes from the “creep” modes and ignoring the latters in general, this step remains incomplete. Some numerical data illustrating all the stages of the process as well as the final results for various viscoelastic constitutive relations are presented.
TL;DR: In this paper, it was shown that under the conditions of Theorems 1 and 2 in l2r, the Fuchsian systems realizing the reducible monodromy are in fact reducible systems.
Abstract: It is shown that under the conditions of Theorems 1 and 2 in l2r, the Fuchsian systems realizing the reducible monodromy are in fact reducible systems. On the other hand, when the reducible monodromy is realized by a Fuchsian system, sufficient conditions on the monodromy matrices under which the final Fuchsian system can be chosen reducible are given.
TL;DR: In this article, a quasilinear elliptic problem is considered for which conditions for existence and nonexistence of positive solutions are discussed and conditions for negative solutions are considered.
Abstract: A quasilinear elliptic problem is considered for which conditions for existence and nonexistence of positive solutions are discussed.
TL;DR: In this article, a calculus of exterior forms with values in an algebra A is developped of which the limit cases are the differential forms calculus (for A = Bbb R ) and gauge theory (for 1-forms) and moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in A) is introduced with elements giving representations of Πk in G e expA.
Abstract: Groups Πk(Xsσ) of “flagged homotopies” are introduced of which the usual (abelian for k > 1) homotopy groups πk(Xsp) is the limit case for flags σ contracted to a point p. The calculus of exterior forms with values in an algebra A is developped of which the limit cases are the differential forms calculus (for A = {\Bbb R}) and gauge theory (for 1-forms). Moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in A) is introduced with elements giving representations of Πk in G e expA.
TL;DR: In this article, it was shown that when two transformations satisfy the weak Pinsker property, those among their joinings which also satisfy this property are dense, and that the \bar {d}-bar distance can be arbitrarily well approached while staying in the class.
Abstract: We show that when two transformations satisfy the weak Pinsker property, those among their joinings which also satisfy this property are dense. In particular, the \bar {d}-bar distance can be arbitrarily well approached while staying in the class.
TL;DR: In this article, a classification of ergodic self-joinings on n-torus is given, based on a linear combination of the Haar measures on subgroups of special form.
Abstract: In this paper a classification of ergodic self-joinings of {\rm GL}(n, {\Bbb Z})-action on n-torus is given. Our study generalizes the description of 2-fold self-joining of {\rm GL}(2, {\Bbb Z}) action on {\Bbb T}^2 due to K. Park. We show that any joining is a linear combination of the Haar measures on subgroups of special form.