Improvements of LaSalle's Theorem

In this paper, we improve LaSalle＇s invariance theorem based on Li＇s work （Li Yong, Asymptotic stability and ultimate boundedness, Northeast. Math. J., 6（1）（1990）, 53-59） by relaxing the restrictions, which make the theorem more easy to apply. In addition, we also improve LaSalle＇s theorem for stochastic differential equation established by Mao （Mao Xuerong, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153（1999）, 175-195）.

一种非线性金融系统的力学分析及基于有限时间LaSalle不变集的混沌控制

从力学角度,将一种三维非线性金融系统分解成Kolmogorov系统的保守力矩、耗散力矩以及外力矩,并分析不同力矩作用于该金融系统时的动态特性,剖析该金融系统产生混沌行为的原因,并结合金融市场解释其物理意义.为控制该金融系统的混沌行为,基于有限时间的LaSalle不变集理论,设计一种自适应控制器.理论分析和数值仿真表明,此控制器作用于系统时会在有限时间内有效地将混沌的金融系统控制到稳定状态.

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.

Adaptive Proportional-Derivative Sliding Mode Control Law With Improved Transient Performance for Underactuated Overhead Crane Systems

In this paper, an adaptive proportional-derivative sliding mode control(APD-SMC) law, is proposed for 2D underactuated overhead crane systems. The proposed controller has the advantages of simple structure, easy to implement of PD control, strong robustness of SMC with respect to external disturbances and uncertain system parameters, and adaptation for unknown system dynamics associated with the feedforward parts. In the proposed APD-SMC law, the PD control part is used to stabilize the controlled system, the SMC part is used to compensate the external disturbances and system uncertainties,and the adaptive control part is utilized to estimate the unknown system parameters. The coupling behavior between the trolley movement and the payload swing is enhanced and, therefore, the transient performance of the proposed controller is improved.The Lyapunov techniques and the La Salle's invariance theorem are employed in to support the theoretical derivations. Experimental results are provided to validate the superior performance of the proposed control law.

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

Theoretical and experimental biology in one<br>—A symposium in honour of Professor Kuo-Chen Chou’s 50th anniversary and Professor Richard Giegé’s 40th anniversary of their scientific careers

It has been a dream that theoretical biology can be extensively applied in experimental biology to accelerate the understanding of the sophiscated movements in living organisms. A brave assay and an excellent example were represented by enzymology, in which the well-established physico-chemistry is used to describe, to fit, to predict and to improve enzyme reactions. Before the modern bioinformatics, the developments of the combination of theoretical biology and experimental biology have been mainly limited to various classic formulations. The systematic use of graphic rules by Prof. Kuo-Chen Chou and his co-workers has significantly facilitated to deal with complicated enzyme systems. With the recent fast progress of bioinformatics, prediction of protein structures and various protein attributes have been well established by Chou and co-workers, stimulating the experimental biology. For example, their recent method for predicting protein subcellular localization (one of the important attributes of proteins) has been extensively applied by scientific colleagues, yielding many new results with thousands of citations. The research by Prof. Chou is characterized by introducing novel physical concepts as well as powerful and elegant mathematical methods into important biomedical problems, a focus throughout his career, even when facing enormous difficulties. His efforts in 50 years have greatly helped us to realize the dream to make “theoretical and experimental biology in one”. Prof. Richard Giege is well known for his multi-disciplinary research combining physics, chemistry, enzymology and molecular biology. His major focus of study is on the identity of tRNAs and their interactions with aminoacyl-tRNA synthetases (aaRS), which are of critical importance to the fidelity of protein biosynthesis. He and his colleagues have carried out the first crystallization of a tRNA/aaRS complex, that between tRNAAsp and AspRS from yeast. The determination of the complex structure contributed significantly to under- stand the interaction of protein and RNA. From his fine research, they have also found other biological function of these small RNAs. He has developed in parallel appropriate methods for his research, of which the protein crystallogenesis, a name he has coined, is an excellent example. Now macromolecular crystallogenesis has become a developed science. In fact, such contribution has accelerated the development of protein crystallography, stimulating the study of macromolecular structure and function.

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.

THE STRUCTURE OF LASALLE’S INVARIANT SET FOR LOTKA-VOLTERRA SYSTEMS

By developing the mechanical proving method, we determine the structure of LaSalle’sinvariant set for 7-dimensional Lotka-Volterra food chain systems, based on Ritt-Wu’s prin-ciple. We then further show that the locally asymptotical stable equilibrium point of Lotka-Volterra food chain systems for 7 dimension must be globally stable.

Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables

Although frequently encountered in many practical applications, singular nonlinear optimization has been always recognized as a difficult problem. In the last decades, classical numerical techniques have been proposed to deal with the singular problem. However, the issue of numerical instability and high computational complexity has not found a satisfactory solution so far. In this paper, we consider the singular optimization problem with bounded variables constraint rather than the common unconstraint model. A novel neural network model was proposed for solving the problem of singular convex optimization with bounded variables. Under the assumption of rank one defect, the original difficult problem is transformed into nonsingular constrained optimization problem by enforcing a tensor term. By using the augmented Lagrangian method and the projection technique, it is proven that the proposed continuous model is convergent to the solution of the singular optimization problem. Numerical simulation further confirmed the effectiveness of the proposed neural network approach.

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.

Generalized Noether Theorem and Poincaré-Cartan Integral Invariant for Singular High-order Lagrangian in Fields Theories

A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identities (GNI)) for variant system under the infinite continuous group of field theory in canonical formalism are derived. The strong and weak conservation laws in canonical formalism are also obtained. It is pointed out that some variant systems also have Dirac constraint. Based on the canonical action, the generalized Poincaré-Cartan integral invariant (GPCⅡ) for singular high-order Lagrangian in the field theory is deduced. Some confusions in literafure are clarified. The GPCⅡ connected with canonical equations and canonical transformation are discussed.

结合快速鲁棒性特征改进ORB的特征点匹配算法

针对定向二进制简单描述符(ORB)算法不具备尺度不变性的问题,提出一种结合快速鲁棒性特征(SURF)算法和ORB的改进算法。首先,利用Hessian矩阵检测特征点的方法,使得提取出的特征点具有尺度不变性;然后,用ORB生成特征描述子;接着采用K-近邻算法进行粗匹配;最后,通过比率测试、对称测试、最小平方中值(LMed S)定理进行提纯。尺度变化时,该算法比ORB的匹配精度提高了74.3个百分点,比SURF的匹配精度提高了4.8个百分点;旋转变化时,该算法比ORB的匹配精度提高了6.6个百分点;匹配时间高于SURF低于ORB。实验结果表明,改进算法不仅保持了ORB的旋转不变性,而且具备了尺度不变性,在不失速度的前提下,匹配精度得到较大提高。

基于能量分析的欠驱动飞行吊运系统协同控制

旋翼无人飞行器具有灵活性强、机动性高的特点,能够在复杂环境下运送由于地形限制地面机器人所无法处理的物品.然而,当单个飞行器难以完成任务时,需要多飞行器协同运作以提高系统的运送能力.相比单飞行器吊运系统而言,多飞行器协同系统具有更高的自由度,更为复杂的系统动态耦合;这就为实际控制带来了巨大挑战.在本文中,作者提出了一种基于能量分析的控制方法.首先构造储能函数,在此基础上设计了控制输入,可依据实际需求选择控制增益,确保控制量在约束范围内;同时避免期望姿态的奇异性问题.随后,对闭环系统稳定性进行了分析.在与现有方法的仿真对比中可以看出,该控制律具有良好的控制性能,对不确定参数和外界干扰表现出了较好的鲁棒性.

欠驱动RTAC系统的自适应耦合控制器设计

针对欠驱动(Rotational/translational actuator,RTAC)系统,设计了一种基于能量分析的自适应控制器.相比其他控制方法,该控制策略可根据系统响应情况对平移振荡器质量等系统参数进行在线估计.具体而言,首先分析了RTAC系统的总能量;随后,在此基础上构造了一个新颖的Lyapunov函数,从而得到了一种自适应耦合控制器,采用投影算子作为更新律以确保估计参数在预设的界内并保证系统的收敛性;最后,采用Lyapunov方法及La Salle不变性原理证明了闭环系统的稳定性.通过数值仿真验证了所提控制器的有效性,结果表明所提自适应控制器具有良好的控制性能.

欠驱动基准系统的约束控制

针对一种欠驱动基准系统,具有旋转激励的平移振荡器(translation oscillators with rotating actuator,TORA)系统,本文首次提出了一种具有约束的控制方法.该方法不仅可以保证闭环系统的稳定性,而且能够保证旋转小球在预设的范围内转动.相比已有控制方法,本文所提方法可以预设小球的转动范围以避免不理想的"循环"行为.具体而言,首先对系统的总机械能进行了详细分析;随后在其总机械能的基础上通过能量整形构造出一个新颖的能量函数;最后基于所构造的能量函数提出了一种具有约束的控制器,采用Lyapunov方法及La Salle不变性原理证明了相应闭环系统的稳定性.通过与已有方法进行仿真对比可知,本文方法在镇定控制与约束控制方面均表现出良好的控制性能.

一类具有时滞和Holling Ⅱ型功能反应的捕食模型的全局稳定性和分支（英文）

研究一类考虑捕食者妊娠产生的时滞和Holling II功能性反应的两种群捕食模型。通过分析特征方程,研究模型可行平衡点的局部稳定性及共存平衡点处Hopf分支的存在性。使用无穷维系统的持久性理论,证明当共存平衡点存在时,模型的持久性。通过构造恰当的李雅普诺夫函数以及使用La Salle不变性原理,证明当共存平衡点不存在时,捕食者灭绝平衡点是全局渐近稳定的;给出共存平衡点全局渐近稳定的充分条件。数值模拟例子验证了理论结果。

一个时滞微生物增长模型全局动力学性态的Lyapunov函数法

利用Lyapunov函数和LaSalle不变原理研究了一个时滞微生物增长模型的全局动力学性态,证明了微生物灭绝平衡点和持续生存平衡点的全局渐近稳定性,简化了现有研究方法,并将结论从已有文献中仅适用于微生物营养利用的Michaelis-Menten功能反应函数加强到适用于一般功能反应函数.

Boundedness of solutions and existence of invariant tori for generalized pendulum type equation

MOTIVATED by various significant applications to non-Newtonian fluid theory, diffusion offlows in porous media, nonlinear elasticity, and theory of capillary surfaces, several authors(see refs.[1,2] and references cited therein ) have recently studied the existence of periodicsolutions and other properties for the following differential equation:

Global stability and Hopf bifurcation of an eco-epidemiological model with time delay

In this paper,an eco-epidemiological model with time delay representing the gestation period of the predator is investigated.In the model,it is assumed that the predator population suffers a transmissible disease and the infected predators may recover from the disease and become susceptible again.By analyzing corresponding characteristic equations,the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free and coexistence equilibria are established,respectively.By means of Lyapunov functionals and LaSalle’s invariance principle,sufficient condi tions are obtained for the global stability of the coexistence equilibrium,the disease-free equilibrium and the predator-extinct equilibrium of the system,respectively.

Distributed cooperative formation of multiple mobile agents with preserved connectivity

This paper investigates distributed cooperative formation control of a group of multiple mobile agents with a virtual leader,where information exchange among agents is modeled by the group topology,and the states of the virtual leader are known only by parts of the agents.We develop a class of distributed formation control laws with similar form.The steered group is proved to achieve the desired formation objectives as long as the intersection of the initial communication topology and the formation goal topology is connected.This requirement of connectivity can be easily achieved by many practical applications;consequently,our developed distributed control laws are effective and feasible.Furthermore,for the developed control laws,we show the influence of different information flow graph of agents on the convergence rate and robustness to node and connection failures.