Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion

In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges to the unique stationary solution of the original system as the stepsize of the discretization decreases. Partially supported by the DAAD, Ministerio de Educación y Ciencia (Spain) and FEDER (European Community) under grants MTM2005-01412 and HA2005-0082, by Junta de Andalucía under the Proyecto de Excelencia P07-FQM-02468, and the DFG-project “Pathwise numerics and dynamics of stochastic evolution equations”.     

Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.     

周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

Identification for Linear Stochastic Systems Driven by Fractional Brownian Motion

Abstract

We apply Grenander's method of sieves to the problem of identification or estimation of the “drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function.     

Clustering Motion

Given a set of moving points in ^d, we show how to cluster them in advance, using a small number of clusters, so that at any time this static clustering is competitive with the optimal k-center clustering at that time. The advantage of this approach is that it avoids updating the clustering as time passes. We also show how to maintain this static clustering efficiently under insertions and deletions. To implement this static clustering efficiently, we describe a simple technique for speeding up clustering algorithms and apply it to achieve faster clustering algorithms for several problems. In particular, we present a linear time algorithm for computing a 2-approximation to the k-center clustering of a set of n points in ^d. This slightly improves the algorithm of Feder and Greene, that runs in (n log k) time (which is optimal in the algebraic decision tree model).     

Motion Control Algorithms for Simple Mechanical Systems with Symmetry

We treat underactuated mechanical control systems with symmetry, taking the viewpoint of the affine connection formalism. We first review the appropriate notions and tests of controllability associated with these systems, including that of fiber controllability. Secondly, we present a series expansion describing the evolution of the trajectories of general mechanical control systems starting from nonzero velocity. This series is then used to investigate the behavior of the system under small-amplitude periodic forcing. On this basis, motion control algorithms are designed for systems with symmetry to solve the tasks of point-to-point reconfiguration, static interpolation and stabilization problems. Several examples are given and the performance of the algorithms is illustrated in the blimp system.     

Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion

Newton systems     , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type , then it has in fact n quadratic integrals of motion and can be embedded into a  (2 n + 1)  -dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations ), and use this to recursively construct infinite families of cofactor pair systems.     

Using Dynamic Geometry Software to Simulate Physical Motion

In this paper we analyse to what extent the computational model of the geometry implemented in a dynamic geometry environment provides models for physical motion, focusing on the continuity issues related to motion. In particular, we go over the utility of dynamic geometry environments to simulate the motion of mechanical linkages, as this activity allows us to compare, by means of dynamic drawings, the computable representation of geometric properties with the real motion of a mechanism. Analysing a simple example, we provide theoretical foundations for particular behaviours observed in the motion of a picture on the screen, which require a subtle interpretation to be understood in a purely physical context. In this way, we reflect on some requirements imposed by the computable representation of knowledge. We consider this work to be a necessary step to determine didactic consequences related to students' perceptions of the moving displays; in particular those concerning the uses of the dragging mode as a tool not only for automatic drawing of many instances of a construction,but also to produce continuous motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Equations of Motion for Constrained Multibody Systems and their Control

This paper presents some recent advances in the dynamics and control of constrained multibody systems. The constraints considered need not satisfy the D’Alembert principle and therefore the results are of general applicability. They show that, in the presence of constraints, the constraint force acting on the multibody system can always be viewed as made up of the sum of two components whose explicit form is provided. The first of these components consists of the constraint force that would have existed were all the constraints ideal; the second is caused by the nonideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They point toward the development of new and novel approaches for the exact control of complex multibody nonlinear systems. In honor of Bob kalaba, friend, colleague, and mentor.     

Hybrid Systems and Hybrid Computation 1st Part: Hybrid Systems

In the first part of this paper we will give a short historical survey of the field of hybrid systems, a precise definition of a hybrid system and some comments on the definition. In a second paper (Hybrid systems and hybrid computation – 2nd part: Hybrid computation) we will concentrate on a particular aspect of the theory closely related to scientific computation, that we have called hybrid computation.     

受单面约束的非完整力学系统的运动方程

给出同时受有单面完整约束和单面非完整约束的非完整力学系统的运动方程,并举例说明其应用     

Geometry of Discrete-Time Spin Systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space \((S^2)^n\). In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on \(T^*\mathbf {R}^{2n}\) for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.     

Motion of vortex-filaments for superconductivity

In this paper, we study the asymptotic behavior of solutions to the simplified Ginzburg–Landau model for superconductivity. We prove that, asymptotically, vortex-filaments evolves according to the mean curvature flow in the sense of weak formulation. This can be seen as a first attempt to understand the nature of the motion of vortex filaments in three dimensions with magnetic field. On the other hand, this paper revisits the pioneering work of Bethuel–Orlandi–Smets [F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. 163 (2006) 37–163] in a slightly relaxed setting.     

Topological Complexity of Motion Planning

   Abstract. In this paper we study a notion of topological complexity TC (X) for the motion planning problem. TC (X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC (X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC (X) . We give an upper bound for TC (X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.     

Brownian Motion Indexed by a Time Scale

In this article, we generalize Wiener's existence result for one-dimensional Brownian motion by constructing a suitable continuous stochastic process where the index set is a time scale. We construct a countable dense subset of a time scale and use it to prove a generalized version of the Kolmogorov–?entsov theorem. As a corollary, we obtain a local Hölder-continuity result for the sample paths of generalized Brownian motion indexed by a time scale.     

Topological Complexity of Motion Planning

Abstract. In this paper we study a notion of topological complexity TC (X) for the motion planning problem. TC (X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC (X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC (X) . We give an upper bound for TC (X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.     

The Dissipation Inequality for Differential-Algebraic Systems

We discuss the infinite time-horizon linear-quadratic optimal control problem for differential-algebraic equations. In contrast to previous approaches we do not impose any assumptions on the system except for impulse controllability. In particular, we show that the optimal control problem is feasible if and only if a dissipation inequality has a solution. Moreover, we discuss conditions under which the problem has a minimizing (instead of only an infimizing) solution and furthermore, when this minimizing solution is unique. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unavoidable Traces Of Set Systems

Sauer, Shelah, Vapnik and Chervonenkis proved that if a set system on n vertices contains many sets, then the set system has full trace on a large set. Although the restriction on the size of the groundset cannot be lifted, Frankl and Pach found a trace structure that is guaranteed to occur in uniform set systems even if we do not bound the size of the groundset. In this note we shall give three sequences of structures such that every set system consisting of sufficiently many sets contains at least one of these structures with many sets.     

The Generalized Multifractional Brownian Motion

It is well known that the fractional Brownian motion (FBM) is of great interest in modeling. However, its Hölder is the same all along its path and this restricts its field of application. Therefore, it would be useful to construct a Gaussian process extending the FBM and having a Hölder that is allowed to change. A partial answer to this problem is supplied by the multifractional Brownian motion (MBM); but the Hölder of the MBM must necessarily be continuous and this may be a drawback in some situations. In this paper we construct a Gaussian process generalizing the MBM and having a Hölder that can be a very irregular function.     

一类变系数动力系统的运动稳定性

本文通过引入线性算子L_A,得到无穷矩阵序列{A_k(t)},基于此,讨论由“平均”系统稳定性确定原系统稳定性的条件,得到一些较为广泛的结果,包含了文[1]、[2]的相应结论。特别,对一些特殊类型的二阶系统,得到一些易于验证的直接判据。