Strong invariance principle for a counterbalanced random walk

We study a counterbalanced random walkS_(n)=X_(1)+…+X_(n),which is a discrete time non-Markovian process andX_(n) are given recursively as follows.For n≥2,X_(n) is a new independent sample from some fixed law̸=0 with a fixed probability p,andX_(n)=−X_(v(n))with probability 1−p,where v(n)is a uniform random variable on{1;…;n−1}.We apply martingale method to obtain a strong invariance principle forS_(n).

Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm

We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained.

Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.

Stability of switched nonlinear systems via extensions of LaSalle's invariance principle

This paper studies the extension of LaSalle＇s invariance principle for switched nonlinear systems. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle＇s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.

On a Conjecture of an Invariance Principle for Sequences of Associated Random Variables

A finite collection of random variables, X1} —, Xm, is said to be associated if any two coordinatewise nondecreasing functions /i and /a on Rm such that Jt = fj(Xi} , Xm~) has finite variance for j = l, 2, Oov(/i,fa)>0; an infinite collection is said to be associated if every finite subcollection is. associated. Thus the concept of "association" is introduced as dependence in probability (Esary, Proschan and Walkup (1967)). It is relative to a lot of practical models, such as the percolation models, the Ising models of statistical mechanics. Therefore, some people are

The Invariance Principle for p-■Chain

There are two parts in this paper. In the first part we construct the Markov chain in random environment（MCRE）, the skew product Markov chain and p-θ^→ chain from a random transition matrix and a two-dimensional probability distribution, and in the second part we prove that the invarianee principle for p-θ^→ chain, a more complex non-homogeneous Markov chain, is true under some reasonable conditions. This result is more powerful.

Uniform Invariance Principle of Discontinuous Systems with Parameter Variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.

Cluster consensus of second-order multi-agent systems via pinning control

This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.

ON THE ASYMPTOTICAL BEHAVIOUR OF SOLUTIONS OF A CLASS OF NONLINEAR CONTROL SYSTEMS

In this paper, the asymptotical behaviour solutions of a class of nonlinear control systems are studied. By establishing infinite integrals along solutions of the system and drawing support from a LaSalle's invariance principle of integral form, criteria of dichotomy and global asymptotical behaviour of solutions are obtained. This work is an improvement and further extension of research methods and results of A. S. Aisagaliev.

Stability and Hopf Bifurcation of a Virus Infection Model with a Delayed CTL Immune Response

In this paper, a virus infection model with standard incidence rate and delayed CTL immune response is investigated. By analyzing corresponding characteristic equations,the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the CTL-activated infection equilibrium are established, respectively. By means of comparison arguments, it is verified that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio is less than unity. By using suitable Lyapunov functional and LaSalle's invariance principle, it is shown that the CTL-inactivated infection equilibrium of the system is globally asymptotically stable if the immune response reproduction ratio is less than unity and the basic reproduction ratio is greater than unity. Numerical simulations are carried out to illustrate the theoretical result.

Global Stability of An Eco-epidemiological Model with Beddington-DeAngelis Functional Response and Delay

In this paper,an eco-epidemiological model with Beddington-DeAngelis functional response and a time delay representing the gestation period of the predator is studied.By means of Lyapunov functionals and Laselle’s invariance principle,sufficient conditions are obtained for the global stability of the interior equilibrium and the disease-free equilibrium of the system,respectively.

Central limit theolrem for linear processes generated byⅡD random variables under the sub-linear expectation

In this paper,we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed(IID)random variables for sub-linear expectations initiated by Peng[19].It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's central limit theorem and invariance principle to the case where probability measures are no longer additive.

Generalized control of quantum systems in the frame of vector treatment

For the state control problem in finite-dimensional quantum systems with any initial state and a goal eigenstate, this paper studies the design method of control laws via the Lyapunov technology and in the vector frame, which ensures the convergence of any initial state toward the goal state. The stability of the closed-loop system in the goal eigenstate is analyzed and proven via the invariance principle. The simulation experiment on a spin-1/2 system shows the effectiveness of the designed control laws.

CONVERGENCE OF SOLUTIONS FOR RLC-NONLINEAR NETWORKS WITH TIME-VARYING ELEMENTS

This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].

Necessary Conditions for Renewal Processes Approximating to Wiener Processes

Let {X, X i ; i≥1} be a sequence of i.i.d.r.v’.s, Z (t)= t i=1 X i,t≥0. Consider the renewal process N(t) =inf{x; Z(x) 】t}. In this paper we study the necessary momentconditions for { N(t) ; t≥0} approximating to Wiener processes.Our results have improved those of Csrg, Horváth and Stei, Horváth and Steinebach

Design of switched controllers for discrete singular bilinear systems

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems. The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems that are not open-loop stable.

Stability analysis of an SIR epidemic model with homestead-isolation on the susceptible and infectious,immunity,relapse and general incidence rate

In this paper,we propose the global dynamics of an SIR epidemic model with distributed latent period,immunity,relapse,homestead-isolation of the susceptible and infectious individuals and general incidence rate.The resulting model has a disease-free equilibrium and if Ro>1,then the SIR epidemic model admits a unique endemic equilibrium.By using suitable Lyapunov functionals and LaSalle's invariance principle,the global stability of the disease-free equilibrium and the endemic equilibrium is established,under suitable monotonicity conditions on the incidence function.

Convergence Rates for Probabilities of Moderate Deviations for Moving Average Processes

The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.

Testing for Changes in the Mean or Variance of Long Memory Processes

In this paper, we study the asymptotic CUSUM tests for detecting changes in the mean or variance of a moving-average process with long memory. When there is no change over [O,T], the asymptotic distribution of the test statistic is derived, which allows us to find asymptotic critical values. When there is a change, the behavior of the test statistic is discussed. Conditions for the consistency of these tests are also discussed. Based on the asymptotic results, simulation studies of testing for changes in the mean show that the CUSUM test proposed performs well.

Convergence to a self-normalized G-Brownian motion

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion.Its quadratic variation process is also a continuous process with independent and stationary increments.We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation.To prove the self-normalized central limit theorem,we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.