A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system?s poles in the complex plane. We include several examples to show the utility of this theory.     

Levi-Civita functional equations and the status of spectral synthesis on semigroups

We show, contrary to some published statements, that spectral synthesis does not generally hold for commutative semigroups that are not groups. On the positive side we prove that it holds if the semigroup is a monoid with no prime ideal. For semigroups with a prime ideal, the picture is not so clear. On the negative side we provide a variety of examples illustrating the failure of spectral synthesis for many semigroups with prime ideals, but we also give examples of semigroups with prime ideals on which spectral synthesis holds.

     

Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.     

Self-induced systems

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous or expansive case. Nevertheless, we exhibit a zero entropy self-induced system that is neither equicontinuous nor expansive. We also provide non-uniquely ergodic self-induced systems with infinite entropy. Moreover, we give a characterization of self-induced minimal Cantor systems in terms of substitutions on finite or infinite alphabets.     

时延混成系统的切换控制器合成

如何设计安全、可靠的信息物理融合系统是计算机科学和控制理论面临的一个重大挑战.时延现象在信息物理融合系统中普遍存在,时延对系统的稳定性、安全性和控制性能具有实质性影响.但是在已有时延系统验证和控制器合成的工作中往往忽略时延因素,这会导致在不考虑时延情况下能保证稳定和安全的系统在实际运行时因为时延原因而不再稳定和安全.因为时延使得系统的行为演化不仅与当前状态有关,还依赖于系统的历史状态,所以时延混成系统的验证和控制合成更加困难.本文研究信息物理融合系统在考虑时延情形下切换控制器合成问题,提出基于不变式生成技术的控制器合成方法.首先,利用谱分析和线性化技术将时延系统的微分不变式生成问题归结为有界时间的可达集计算问题;然后,提出基于抽象精化的算法计算时延系统有界时间可达集的上近似;最后,实现本文算法并使用实例验证该方法的有效性.     

Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

Generalized joint spectral radius and stability of switching systems

This paper extends the notion of generalized joint spectral radius with exponents, originally defined for a finite set of matrices, to probability distributions. We show that, under a certain invariance condition, the radius is calculated as the spectral radius of a matrix that can be easily computed, extending the classical counterpart. Using this result we investigate the mean stability of switching systems. In particular we establish the equivalence of mean square stability, simultaneous contractibility in square mean, and the existence of a quadratic Lyapunov function. Also the stabilization of positive switching systems is studied. Numerical examples are given to illustrate the results.     

Canonical systems

In this paper, we study canonical systems and pose the problem of constructing a quasihomogeneous canonical system, i.e., the canonical system whose spectral problem has the same solution as that of the spectral problem of some homogeneous canonical system. We present a rational algorithm for constructing a quasihomogeneous canonical systems.     

Soliton surfaces in the generalized symmetry approach

We investigate some features of generalized symmetries of integrable systems aiming to obtain the Fokas–Gel’fand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. We show that if there exists a common symmetry of the zero-curvature representation of an integrable partial differential equation and its linear spectral problem, then the Fokas–Gel’fand immersion formula is applicable in its original form. In the general case, we show that when the symmetry of the zero-curvature representation is not a symmetry of its linear spectral problem, then the immersion function of the two-dimensional surface is determined by an extended formula involving additional terms in the expression for the tangent vectors. We illustrate these results with examples including the elliptic ordinary differential equation and the CP^N?1 sigma-model equation.     

Finite Spectrum Assignment for Completely Regular Differential-Algebraic Systems with Aftereffect

For linear autonomous completely regular differential-algebraic systems with commensurable delays in the state and control, we study the problem of constructing a state feedback that ensures a finite spectrum for the closed-loop system. We propose criteria for spectral reducibility and weak spectral reducibility whose proofs contain the synthesis schemes of appropriate controllers. Several illustrative examples are given.     

A new block preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner.     

Twisted spectral triples and quantum statistical mechanical systems

Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of twisted spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.     

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer.     

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.     

Conditions for stability and stabilization of systems of neutral type with nonmonotone Lyapunov functionals

We suggest new tests for the stability and uniform asymptotic stability of an equilibrium in systems of neutral type. By using these tests, we prove conditions for optimal stabilization and derive new estimates for perturbations that can be countered by a system closed by an optimal control. We show that, by using nonmonotone sign-indefinite functionals as Lyapunov functionals, one can obtain conditions for uniform asymptotic stability that do not contain the a priori requirement of stability of the difference operator and do not imply the boundedness of the right-hand side of the system. When studying the action of perturbations on the stabilized systems, these conditions permit one to obtain new estimates of perturbations preserving the stabilizing properties of optimal controls. The obtained estimates do not imply any constraint on the value of perturbations in some domains of the phase space that are defined when constructing an optimal stabilizing control. Some examples are considered.     

Redundancy allocation at component level versus system level

In this paper, we treat the problem of stochastic comparison of standby [active] redundancy at component level versus system level. In the case of standby redundancy, we present some interesting comparison results of both series systems and parallel systems in the sense of various stochastic orderings for both the matching spares case and non-matching spares case, respectively. In the case of active redundancy, a likelihood ratio ordering result of series systems is presented for the matching spares case; and for the non-matching spares case, a counterexample is provided to show that there does not exist similar result even for the hazard rate ordering. The results established here strengthen and generalize some of those known in the literature. Some numerical examples are also provided to illustrate the theoretical results.     

QR methods and error analysis for computing Lyapunov and Sacker�CSell spectral intervals for linear differential-algebraic equations

In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker?CSell spectral intervals for linear differential-algebraic equations. Furthermore, a perturbation and error analysis for these methods is presented. We investigate how errors in the data and in the numerical integration affect the accuracy of the approximate spectral intervals. Although we need to integrate numerically some differential-algebraic systems on usually very long time-intervals, under certain assumptions, it is shown that the error of the computed spectral intervals can be controlled by the local error of numerical integration and the error in solving the algebraic constraint. Some numerical examples are presented to illustrate the theoretical results.     

Synchronization of nonlinear fractional order systems

This paper deals with the master-slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown state variables. For solving this problem we introduce a Fractional Algebraic Observability (FAO) property which is used as a main tool in the design of the master system. As numerical examples we consider a fractional order Rössler hyperchaotic system and a fractional order Lorenz chaotic system and by means of some simulations we show the effectiveness of the suggested approach.     

New results on the global asymptotic stability of?a?Lotka-Volterra system

In this paper, we revisit the famous Lotka-Volterra competitive system. By combining spectral matrix theory with Lyapunov function, some new sufficient conditions are obtained to guarantee the global asymptotic stability of a unique equilibrium for Lotka-Volterra competitive system. Our new results generalize and significantly improve the known results in the previous literature. The main purpose of this paper is to propose a new methodology to study the high-dimensional Lotka-Volterra system. And this method can be extensively used to study the global asymptotic stability of the equilibrium. Finally, some examples and their simulations show the feasibility and effectiveness of our results.