Multiparameter singular perturbation problems: Iterative expansions and asymptotic stability

Recursive formulae are derived which yield asymptotic expansions for the eigenvalues of multiparameter singular perturbation problems. These formulae follow readily from an exact expression for the eigenvalues which involves an implicit matrix function. The implicit function satisfies an algebraic matrix Riccati equation reminiscent of a similar equation of the single parameter theory. The results also explicate the ‘block D-stability’ criterion for asymptotic stability previously introduced by Khalil and Kokotovic.     

Global asymptotic stability for a linear system in a saturated mode

We study an ordinary differential equation which models the behaviour of a class of neural networks, which are similar to the Hopfield networks, and which can give convergence in finite time. For application to optimisation problems, and other applications, we are concerned with global asymptotic stability. This paper gives new results on this topic.     

On singular perturbations due to fast actuators in hybrid control systems

A stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators. If a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted, or equivalently, are infinitely fast, then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics. This result, which generalizes classical results for differential equations, justifies using a simplified plant model that ignores fast but continuous actuator dynamics, even when using a hybrid feedback control algorithm.     

Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production

This paper deals with the following quasilinear chemotaxis-growth system

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n(n\ge 3)$ under zero-flux boundary conditions. The parameters $\mu ,\delta$ and $\tau$ are positive and the diffusion function $D\left(u\right)$ is supposed to generalize the prototype $D\left(u\right)\ge D_0{u}^{\theta }$ with $D_0>0$ and $\theta \in \mathbb{R}$. Under the assumption $\theta >1?\frac{4}{n}$, it is proved that whenever $\mu >0$, $\tau >0$ and $\delta >0$, for any given nonnegative and suitably smooth initial data ($u_0$, $v_0$, $w_0$) satisfying $u_0?0$, the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the $\left|\right|v(?,t)|{|}_{W^{1,s}\left(\Omega \right)}$ with $s\in [1,\frac{2n}{n?2})$ to estimate the boundedness of $\left|\right|?v(?,t)|{|}_{L^{2q}\left(\Omega \right)}(q>1)$. Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that $D\left(u\right)\equiv 1$ and $n=3$.     

Global asymptotic stability in a generalized Lotka-Volterra system

Sufficient conditions are obtained for the global asymptotic stability of a positive steady state of Lotka-Volterra systems of the form

$

when such a steady state exists and a corresponding result for an integrodifferential system is indicated.     

Robustness of exponential stability to singular perturbations and delays

Singularly perturbed nonlinear differential equations with small time delays in the slow variables are considered. Averages of the fast variables are used in order to obtain a sufficient condition under which the exponential stability of the slow subsystem is robust to singular perturbations and delays.     

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem

in a bounded domain $\Omega ?{\mathbb{R}}^n,n\ge 1$, with smooth boundary, where $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,a_1,a_2$ are positive constants.When $a_1\in (0,1)$ and $a_2\in (0,1)$, it is shown that under some explicit largeness assumptions on the logistic growth coefficients ${\mu }_1$ and ${\mu }_2$, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state $(u^1,v^1,w^1)$ of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When $a_1\ge 1$ and $a_2\in (0,1)$, if ${\mu }_2$ is suitable large, for all sufficiently regular nonnegative initial data $u_0$ and $v_0$ with $u_0?0$ and $v_0?0$, the globally bounded solution of above system will stabilize toward $(0,1,1)$ as $t\to \infty$ in algebraic.     

On the stability of a reduced-order filter based on dominant singular value decomposition of the system dynamics

The stability of a reduced-order filter (ROF), the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics, is examined. A definition of s-detectability is introduced. It is found that the observability of all unstable and neutral singular vectors (s-detectability) is a sufficient condition for the existence of a stable filter.     

A new parameter estimate in singular perturbations

A new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system. General time-invariant systems with a single small parameter are considered. The paper employs a Riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics. An application of the Brouwer fixed point theorem to the Riccati equation and of Liapunov's direct method to the fast and slow subsystems results in the desired upper bound. Computation of the estimate requires only the solution of two Liapunov matrix equations.     

具有未知函数非线性系统的全局渐近稳定控制

本文讨论了一类具有未知函数和未知控制方向非线性系统的全局渐近稳定问题.通过提出一个引理处理未知函数问题,从而得到了一种基于反步法和Nussbaum增益技术的全局渐近稳定控制算法.与逼近方法处理未知函数的算法相比,本文提出的算法解决了非线性系统的全局渐近稳定问题;与现存解决非线性系统的全局渐近稳定控制算法相比,本文避免了使用未知函数的假设条件,因此降低了保守性.值得一提的是本文的算法也解决了反步法的“微分爆炸”问题,因此所提出的控制方案不仅仅得到了全局渐近稳定控制方案,而且降低了计算的复杂性.最后,将该方案应用到刚性单链杆机械手系统中,仿真结果验证了其有效性.     

On the asymptotic stability of switched homogeneous systems

The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L>0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.     

New results on stability analysis of singular time-delay systems

This paper deals with the problem of delay-dependent stability analysis for singular systems with a constant time delay. By employing the Jensen inequality and the Wirtinger-based inequality, new delay-dependent stability criteria are developed. Furthermore, in order to obtain less conservative results, a new method of constructing Lyapunov--Krasovskii functionals (LKFs) is used. It should be pointed out that the positive-definiteness restrictions on some symmetric matrices of the LKFs are removed. All the criteria are proposed in terms of strict linear matrix inequalities. Finally, numerical examples are given to demonstrate the less conservatism of the results.     

一类中立型时滞神经网络的全局渐近稳定性

讨论了一类带有时滞的中立型神经网络的稳定性问题。通过构造Lyapunov-Krasovskii泛函,利用矩阵Schur补性质研究了此类中立型时滞神经网络模型的全局渐近稳定性,得出基于矩阵特征值的稳定性的充分判据,并给出基于矩阵特征值的时滞Hopfield神经网络全局渐近稳定性的充分条件;数值仿真检验了结果的有效性。     

[n]种群Gilpin-Ayala型竞争系统的全局渐近稳定性

研究了一类连续型时滞与离散型变时滞的非自治[n]种群Gilpin-Ayala型竞争系统,分别利用比较原理和构造Lyapunov函数得到了系统持久生存与全局渐近稳定性的充分条件,通过数值举例验证了定理的可行性。     

Global asymptotic stability of delayed neural networks with discontinuous neuron activations

This paper investigates a class of delayed neural networks whose neuron activations are modeled by discontinuous functions. By utilizing the Leray–Schauder fixed point theorem of multivalued version, the properties of M-matrix and generalized Lyapunov approach, we present some sufficient conditions to ensure the existence and global asymptotic stability of the state equilibrium point. Furthermore, the global convergence of the output solutions are also discussed. The assumptive conditions imposed on activation functions are allowed to be unbounded and nonmonotonic, which are less restrictive than previews works on the discontinuous or continuous neural networks. Hence, we improve and extend some existing results of other researchers. Finally, one numerical example is given to illustrate the effectiveness of the criteria proposed in this paper.     

Delay dependent robust stability of singular systems with additive time-varying delays

This paper considers the problem of delay-dependent robust stability for uncertain singular systems with additive time-varying delays. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties. The results are expressed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.     

On asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks

In this paper, we obtain some sufficient conditions for determining the asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks by utilizing the Lyapunov functional method. An example is given to show the validity of the results.     

A stability criterion for singular systems with two additive time-varying delay components

The problem of stability for singular systems with two additive time-varying delay components is investigated. By constructing a simple type of Lyapunov-Krasovskii functional and utilizing free matrix variables in approximating certain integral quadratic terms, a delay-dependent stability criterion is established for the considered systems to be regular, impulse free, and stable in terms of linear matrix inequalities (LMIs). Based on this criterion, some new stability conditions for singular systems with a single delay in a range and regular systems with two additive time-varying delay components are proposed. These developed results have advantages over some previous ones in that they have fewer matrix variables yet less conservatism. Finally, two numerical examples are employed to illustrate the effectiveness of the obtained theoretical results.