Two notes on measure-theoretic entropy of random dynamical systems

In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.     

非自治动力系统Bowen估计熵的变分原理

推广了经典动力系统的Bowen拓扑熵,给出了非自治动力系统的Bowen估计熵的定义,讨论了非自治动力系统的Bowen估计熵和局部估计熵的关系,推广了自治动力系统Bowen拓扑熵的Billingsley型定理,研究了非自治动力系统的Bowen α-估计熵的Billingsley型定理.此外,给出了非自治动力系统Bowen...     

Lower bounds of the topological entropy for nonautonomous dynamical systems

In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.     

赋予hit—or-miss拓扑超空间动力系统的拓扑熵

设（X,d,f）为拓扑动力系统,其中X为局部紧第二可数Hausdorff空间,d为紧型度量,f为完备映射,用2^x和f分别表示由X的所有非空闭子集和所有闭子集构成的集族,（2^x,ρ,2^f）和（f,ρ,2^f）为由（X,d,f）诱导的赋予hit—or—miss拓扑的超空间动力系统．本文研究了h（X,d,f）和h（2^...     

Spectral Components of Operators with Spectrum on a Curve

Trace class perturbations of normal operators with spectrum on a curve and spectral components of such operators are studied. We establish duality relations for the spectral components of an operator and its adjoint. The generalized Sz.-Nagy–Foia–Naboko functional model introduced in the paper is a basic tool for this theorem. The results have applications in nonself-adjoint scattering theory and to extreme factorizations of J-contraction-valued functions (J-inner-outer and A-regular-singular factorizations).     

A concept of local metric entropy for finite-time nonautonomous dynamical systems

We introduce a concept of entropy for difference and differential equations which is a local-in-space and transient-in-time version of the classical concept of metric entropy. Based on that, a finite-time (or transient) version of Pesin’s entropy theorem and also an explicit formula of finite-time entropy for 2-D systems are derived.     

一类线性非自伴算子的谱

近年来国际上出现了一类以迁移理论为背景的所谓＂ｕ－标算子（ｕ－ｓｃａｌａｒｏｐｅｒａｔｏｒ）＂．本文研究ｕ－标算子与经典的标型谱算子（ｓｐｅｃｔｒａｌｏｐｅｒａｔｏｒｏｆｓｃａｌａｒｔｙｐｅ）的内在联系，给出了此类算子的谱与其无界投影族之间的关系，证明了该类算子的剩余谱是空集．     

Robust stabilization for a class of uncertain dynamical systems with time delay

In this paper, a new and simple approach whereby we derive several sufficient conditions on robust stabilizability for a class of uncertain dynamical systems with time delay is presented. Some analytical methods and the Bellman-Gronwall inequality are employed to investigate these sufficient conditions. The notable features of the results obtained are their simplicity in testing the stability of uncertain dynamical systems with time delay and their clarity in giving insight into system analysis. Finally, several numerical examples are given to demonstrate the utilization of the results.The authors would like to acknowledge the many helpful comments provided by the reviewer. Particularly, in the light of these comments, the proof of Theorem 3.1 has been considerably shortened.     

The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H _P of H to the subspace of states with given value P of the total momentum of the system. For the operators H _P we discover a) two-sided estimates for the counting function of the discrete spectrum _d (H _P ) of H _P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of _d (H _P ) near the lower bound inf _ess(H _P ) of the essential spectrum of H _P . The structure of the discrete spectrum of such systems was known earlier only for n=1.     

On the absence of invariant measures with locally maximal entropy for a class of shifts of finite type

We prove that for a class of shifts of finite type, , any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.

     

Large time behavior of entropy solutions to some hyperbolic system with dissipative structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t →∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.     

Sliding mode control experiments of uncertain dynamical systems with time delay

This article presents the simulation and experimental studies for the nonlinear time-delayed dynamical systems with uncertainties. A rotary flexible joint made by Quanser is chosen as the model system to investigate the method for sliding mode control design. We considered the geometric nonlinearity of the flexible joint consisting of two linear springs. The system is assumed to have constant delay time and uncertain parameters with known upper and lower bounds. We also design an optimal sliding surface for the sliding mode control. Simulations and experiments are carried out to demonstrate the utility of the control method. Finally, the results from the simulations and experiments are in excellent agreement.     

有缺失数据的条件独立正态母体中参数的最优同变估计

在缺失数据机制是可忽略的假设下,导出了有单调缺失数据的条件独立正态模型中协方差阵和精度阵的Cholesky分解的最大似然估计和无偏估计.通过引入一类特殊的变换群并在更广义的损失下,获得了其最优同变估计.这表明最大似然估计和无偏估计是非容许的.最后,通过数值模拟验证了相关结果的有效性.     

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

We present sharp bounds on the Kolmogorov probabilistic (N,δ)-width and p-average N-width of multivariate Sobolev space with mixed derivative

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, equipped with a Gaussian measure μ in , that is where 1<q<∞,0<p<∞, and ρ>1 is depending only on the eigenvalues of the correlation operator of the measure μ (see (4)).     

Polynomial stability of positive switched homogeneous systems with time delay

Based on the logarithm contraction average dwell-time method, this paper investigates the polynomial stability of positive switched homogeneous time-delay systems whose vector fields are of different degrees with respect to a dilation map. Using the analytical skills developed in positive systems, an explicit polynomial stability criterion is established for the first time for the involved system under the logarithm contraction average dwell-time switching. Moreover, the main result is applied to the polynomial stability of Persidskii-type switched systems.     

Global dynamics for a model of a class of continuous‐time dynamical systems

In this paper, the global asympotic behavior of solutions of a class of continuous‐time dynamical system is studied. Not only do we obtain the ultimate boundedness of solutions of the system but we also obtain the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set, which can be applied to study chaotic control and chaotic synchronization of the system. Copyright © 2015 John Wiley & Sons, Ltd.     

A note on the existence of a largest topological factor with zero entropy

Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, .