Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.     

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

广义系统渐近稳定性的一种新的Lyapunov判别方法

本文中，对广义系统构造了一种新的Lyapunov函数，并给出相应的渐近稳定性的Laypunov判别方法。另外，我们证明了当广义系数为R－能稳时，必存在一反馈控制使得闭环系统为渐近稳定性的。     

Approximation on attraction domain of Cohen-Grossberg neural networks

In this paper, approximations of attraction domains of the asymptotically stable equilibrium points of some typical Cohen-Grossberg neural networks are achieved. Most Cohen-Grossberg neural networks are highly nonlinear systems which makes it difficult to approximate their attraction domain. Under some weak assumptions, we are allowed to employ the optimal Lyapunov method to obtain a Lyapunov function for asymptotically stable equilibrium points of a given Cohen-Grossberg neural network. With the help of this Lyapunov function, we approximate the corresponding attraction domain by the iterative expansion approach. Numerical simulations also illustrate that the approximation obtained is really part of the attraction domain.     

脉冲泛函微分方程关于两个测度的实际稳定性

利用Lyapunov,函数方法,建立了脉冲时滞微分方程关于两个测度的实际稳定的充分条件。     

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

Geometric Integration Methods that Preserve Lyapunov Functions

We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions. In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function. AMS subject classification (2000) 65L05, 65L06, 65L20, 65P40     

利用分数维微积分推广Lyapunov第二方法

利用分数维微积分(Fractional Calculus,简记为FC)理论,推广了Lyapunov第二方法,得到了类Lyapunov判据,给出了一种新的构造Lyapunov函数的方法和途径,并且把此判据推广到分数维系统,给出了一种分数维系统的Lyapunov稳定性问题的判别方法.     

Exponential dichotomy of linear systems with almost periodic matrices

Using the Lyapunov direct method, we establish a condition of exponential dichotomy for the class of dynamical systems under weaker assumptions as compared to the case of an arbitrary continuous matrix on the derivative of the Lyapunov function along the trajectories. By way of application, we obtain a sufficient condition for the dichotomy of a second order almost periodic vector equation in terms of coefficients.     

基于符号数值混合计算的混成系统Lyapunov函数构造

基于平方和松弛和有理向量恢复,提出了一种符号数值混合计算方法来构造多项式Lyapunov函数以判定非线性混成系统的稳定性,首先,为Lyapunov函数预定一个给定次数的多项式模板,则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题,然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.     

一类数学生物经济模型的概周期解的存在性

通过构造Lyapunov函数研究了一类非自治的数学生物经济模型的概周期解的存在性,得到了保证概周期解存在的充分性判椐.     

标量Lyapunov函数法的改良

本文针对标量Lyapunov函数法所存在的保守性较为严重的问题，在原标量Lyapunov函数理论的基础上，对其进行旨在减少保守性，增强理论的实用性改良，数据实验反映了我们的结果较好。     

广义大系统的Lyapunov稳定性分析

广义大系统的稳定性是一个非常重要的问题 ,由于广义大系统的复杂性 ,对其稳定性的研究也是一件相当困难的事情 .本文利用 Lyapunov方程 ,应用 Lyapunov函数法 ,研究了广义线性大系统和广义非线性大系统的稳定性和不稳定性 ,得到了系统的关联稳定参数域和不稳定域 .给出例子说明该方法的可行性 .     

Quenching rate for a nonlocal problem arising in the micro-electro mechanical system

In this paper, we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro-electro mechanical system. This question is equivalent to the stabilization of the solution to the transformed problem in self-similar variables. First, some a priori estimates are provided. In order to construct a Lyapunov function, due to the lack of time monotonicity property, we then derive some very useful and challenging estimates by a delicate analysis. Finally, with this Lyapunov function, we prove that the quenching rate is self-similar which is the same as the problem without the nonlocal term, except the constant limit depends on the solution itself.     

Stability analysis of positive switched systems via joint linear copositive Lyapunov functions

In this paper, we study the asymptotic stability of continuous-time positive switched linear systems for the case when each subsystem is only stable. By using the so-called “joint linear copositive Lyapunov function” (JLCLF) generalizing the common linear copositive Lyapunov function, we show that the system remains asymptotically stable under appropriate switching if it has a JLCLF. Then, the main result is extended to positive switched linear systems with time delay.     

关于三个未解决的不等式问题

通过求二阶导数或列举反例的方法可解决《常用不等式（第四版）》末尾提出的问题3、问题13和问题62,另外,利用Lyapunov不等式及Moran不等式可给出其问题190中一个不等式的下界.     

一类三阶时滞微分方程解的渐进稳定性

本文研究了三阶非线性时滞微分方程解的渐近稳定性. 利用Lyapunov 泛函, 得到了微分方程的零解是渐进稳定的, 这一结果推广了文献[2] 的结果.     

On the boundedness of solutions of the Chen system

By constructing a suitable Lyapunov function, we show that for the system parameters in some specified regions, the solutions of the Chen system are globally bounded.     

The variational Lyapunov function and strict stability theory for differential systems

In this paper, we investigate strict stability of differential systems by variational Lyapunov function. We obtain some sufficient conditions and comparison theorems.     

一类三阶时滞微分方程解的渐进稳定性

本文研究了三阶非线性时滞微分方程解的渐近稳定性.利用Lyapunov泛函,得到了微分方程的零解是渐进稳定的,这一结果推广了文献[2]的结果.