Perturbation theorems for systems with impulse effect

In this paper two systems of ordinary differential equations with an impulse effect and their respective perturbed systems are considered. Sufficient conditions are found for which some solutions of the perturbed systems tend to zero as t→∞. Assuming that these systems have zero solutions, their stability is proved.     

p-Moment stability of functional differential equations with random impulses

By means of Liapunov's direct method coupled with Razumikhin technique some sufficient conditions for (uniform, uniform and globally asymptotic, uniformly globally asymptotic, and globally exponential) p-moment stability of the zero solution of functional differential equations with random impulses are presented, where the parameter λ(t) in (dV(t, x(t)))/dt is not required to be negative definite. Thus the obtained results are convenient to apply.     

Exponential stability of nonlinear time-delay systems with delayed impulse effects

The problem of exponential stability for nonlinear time-delay systems with delayed impulses is addressed in this paper. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for two kinds of delayed impulses (i.e., destabilizing delayed impulses and stabilizing delayed impulses). It is shown that if a nonlinear impulsive time-delay system without impulse input delays is exponentially stable, then under some conditions, its stability is robust with respect to small impulse input delays. Moreover, it is also shown that for a stable nonlinear impulsive time-delay system, if the magnitude of the delayed impulses is sufficiently small, then under some conditions, the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays. The efficiency of the proposed results is illustrated by three numerical examples.     

On partial detectability of the nonlinear dynamic systems

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the “partial” (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.     

Stability analysis of systems with impulse effects

The authors establish conditions for the uniform stability and the uniform asymptotic stability of equilibria of systems with impulse effects described by systems of nonlinear, time-varying ordinary differential equations. Under mild additional assumptions, these sufficient conditions turn out to be necessary as well. For the case of time-invariant systems with impulse effects described by nonlinear ordinary differential equations, the above results are used to establish conditions under which the uniform asymptotic stability of equilibria can be deduced from the linearizations of such systems     

Investigation of stability-similar properties of solutions to nonlinear systems

Consideration is given to the system of differential equations with the “partial” equillibrium position that is a generalization of systems of differential equations describing critical cases of k zero and 2h purely imaginary roots of characteristic equations, which correspond to simple elementary divisors.Here stability-similarity is stability, asymptotic stability, and different types of boundedness of solutions to systems of differential equations. In particular, theorems of asymptotic stability with respect to one part of phase variables are proved regarding the indicated system of differential equations; theorems of uniform boundedness of solutions, regarding the other part. Statements of the indicated theorem are also proved regarding a part of variables.     

有界随机噪声参数激励下碰撞系统的矩稳定性

研究了单自由度线性单边碰撞系统在有界随机噪声参数激励下系统的矩稳定性问题. 用 Zhuravlev 变换将碰撞系统转化为连续的非碰撞系统,然后用随机平均法得到了关于慢变量的随机微分方程. 利用伊藤法则给出了系统一、二阶矩满足的常微分方程,根据微分方程的稳定性理论得到了系统一阶矩稳定充分必要条件的解析表达式和二阶矩稳定充分必要条件的数值算法,并对理论结果用数值方法进行了仿真计算.理论分析和数值仿真表明,无论是相对于一阶矩还是二阶矩的稳定性,随着随机激励振幅变大,系统的稳定性区域变小从而使得系统变得不稳定. 而当调谐参数趋于零系统达到参数主共振情形时,系统的稳定性区域变得最小. 当随机噪声强度逐渐变小趋于零时,由二种矩稳定性给出的稳定性区域变得一致. 在一定的参数区域内,随机噪声使得系统稳定化.     

Research note System estimation by invariant imbedding with unknown inputs †

An equivalence relation, rim–similarity, is defined on the set of all sequences of square matrices of a given fixed dimension. For linear discrete – time systems, theorems are presented which show that certain stability properties are invariant under n∞ similarity. General linear systems as well as those of variational type are considered. Also, sufficient conditions are given which guarantee that the uniform stability of the null solution of a non– linear system carries over to the linear variational equation with respect to this solution. All of these results are analogous to known ones for ordinary differential equations.     

Set stability of differential equations with time delay

Set stability and uniform set stability involve known specific bounds on the solutions of differential equations. These concepts are extended to include the case of differential equations with time delay. Liapunov-like theorems are presented which yield sufficient conditions for the set stability of such systems. Examples are given to demonstrate the method.     

A new existence theory for positive periodic solutions to functional differential equations with impulse effects

The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results.     

On strong stability for linear integral equations

Summary The notion of strong or adjoint stability for linear ordinary differential equations is generalized to the theory of Volterra integral equations. It is found that this generalization is not unique in that equivalent definitions for differential equations lead to different stabilities for integral equations in general. Three types of stabilities arising naturally are introduced: strong stability, adjoint stability, and uniform adjoint stability. Necessary and sufficient conditions relative to the fundamental matrix for these stabilities are proved. Some lemmas dealing with non-oscillation of solutions and a semi-group property of the fundamental matrix are also given.     

Stability criterion of linear delayed impulsive differential systems with impulse time windows

In this paper, we study the uniform stability of linear delayed differential equationswith impulse time windows. By means of Lyapunov functions and Razumikhin technique combined with classification discussion technique, the criterion of uniform stability is obtained, which may be used to discuss others stability of delayed differential equations with impulse time win-dows. Two examples are given to illustrate the effectiveness of the theoretic result.     

Lyapunov conditions for input-to-state stability of impulsive systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.     

Stabilization of hyperbolic systems using concentrated sensors and actuators

Certain hyperbolic systems of partial differential equations which are known to be uniformly asymptotically stabilizable using point sensors/actuators (S/A) are considered. The issue to be investigated is the effect on stability when point S/A's are replaced by "concentrated" S/ A's, that is, S/A's which average over small regions of the spatial domain. Although it is known that passing from point to concentrated S/ A's necessarily destroys uniform stability, a necessary and sufficient condition for strong stability is obtained in terms of the S/A weighting functions. In addition, in the special case of a cantilevered beam controlled by a single sensor/actuator pair concentrated at the free end, another, more robust type of stability is shown to hold, even when strong stability does not. The latter result shows that the system energy is bounded by a part which goes uniformly to zero at infinity and a residual which can be explicitly estimated in terms of the support of the weight functions and the initial energy. Furthermore, the residual energy converges to zero as the support reduces to the point at the free end of the beam.     

Stability of the solutions of impulsive integro-differential equations in terms of two measures

In the present paper the stability of the solutions of impulsive systems of integro-differential equations of Volterra type with fixed moments of impulse effect in terms of two piecewise continuous measures is investigated. The investigations are carried out by means of piecewise continuous functions of the same type as the Lyapunov functions using differential inequalities for piecewise continuous functions.     

Some results of the degenerate fractional differential system with delay

An analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented. We discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems. The exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained. Finally, an example is provided to illustrate the effectiveness of the presented analytical approaches.     

Comparison theorems for set stability of differential equations

Set stability and uniform set stability of differential equations involve known specific bounds on solutions of the differential equations under consideration. Comparison theorems are presented giving sufficient conditions for these forms of stability. The conditions include the existence of a Liapunov-Like function and the set stability of a scalar differential equation. Examples are given to demonstrate the method.     

Determination of firing times for the stochastic Fitzhugh-Nagumo neuronal model

We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.     

Approximate controllability of stochastic delay differential systems driven by Poisson jumps with instantaneous and noninstantaneous impulses

Control systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed-point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results.     

Uniform asymptotic stability of impulsive delay differential equations

In this paper, criteria on uniform asymptotic stability are established for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior. Some examples are also discussed to illustrate the theorems.