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双语推荐：Gronwall

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本文主要介绍了Gronwall-Bellman积分不等式及其推广形式在分数阶微分方程中的应用。利用Gronwall-Bellman积分不等式及其推广形式证明了分数阶微分方程解的唯一性,获得了一类分数阶时滞微分方程有限时间稳定的充分条件。
The applications of the Gronwall-Bellman integral inequality and its extended form in the fractional-order differential equation are mainly introduced in this paper. Taking advantage of Gronwall-Bellman integral inequality and its extended form, we prove the uniqueness of the fractional-order differential equations and obtain the sufficient conditions of finite-time stability for a class of fractional-order differential equation with delay.

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对随机Navier-Stokes方程的讨论,通常没有考虑Poisson跳对系统影响.在假设随机的外界环境对系统产生影响的条件下,给出了带Poisson跳的随机Navier-Stokes方程,利用连续鞅的性质,通过公式,Gronwall引理及广义的Gronwall扩展引理讨论了其解的指数稳定性,并给出了指数稳定性的充分条件。
The influence of the P oisson jumps upon the stochastic Navier-Stokes equations has never been consid-ered. This paper discusses the exponential stability of solutions to the stochastic Navier-Stokes equations with Pois-son jumps, on the condition that the system is perturbed by random external environment introducing the definition of solution to the stochastic Navier-Stokes equations and using continuous martingale property, By using Ito^formu-la, Gronwall lemma and extended Gronwall lemma, a sufficient condition of exponential stability is established. This result is an improvement and extension of existing results.

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利用压缩映射原理及时标上的Gronwall不等式理论得到了时标T上带有时滞的BAM神经网络的反周期解的存在性和指数稳定性的充分条件．
By using the method of contraction mapping principle and Gronwall inequality theory , several sufficient conditions are established for the existence and global exponential stability of anti-periodic solutions for BAM neural network with delays on time scale T .

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利用比较定理,研究了时间尺度上的几个Gronwall型积分不等式,并举例说明了它们的一些应用。这些不等式是一些已知的积分不等式及其离散形式的统一和延伸。
In this paper,using comparison theorem,we investigate some Gronwall type inequalities on time scales,which unify and extend some continuous inequalities and their corresponding discrete ana-logues.Two applications of the main results are given in the end of this paper.

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研究了一类四阶非线性波动方程的初边值问题,利用Faedo-Galerkin方法和Gronwall引理,证明了该初边值问题古典解的存在性。
In this paper we study a class of four order nonlinear wave equation with initial boundary value problem. By using the method of Faedo-Galerkin and Gronwall lemma, we proved that the existence of classical solutions of the initial boundary value problem. The results provided a good mathematical basis for the dynamics of beam structure stability.

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通过讨论Riemann-Liouville分数阶非线性系统的稳定性,特别地分析了扰动系统的稳定性.基于分数阶线性微分方程的稳定性理论,利用拉普拉斯变换、Mittag-Leffler函数和Gronwall不等式,给出了一些稳定性定理.
In this paper, we discuss the stability of fractional-order nonlinear systems with Riemann-Liouville derivate, in particular, we analysis the perturbed systems. On the stability theory fractional-order linear differential equation, use Laplace transform, Mittag-leffler function and the Gronwall inequality, we propose some theorems.

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推广了一类新的Gronwall-Bellman-Pachpatte型积分不等式,建立了一类新型的提供未知函数显式边界条件的时滞积分不等式.这些不等式可以用于特定的时滞微分方程和时滞积分方程的定性性质的研究.
In this paper,a new class of Gronwall-Bellman-Pachpatte integral inequalities and some new delay integral inequality with explicit bounds on unknown functions are established.The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral e-quations.

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针对时滞离散区间系统的时滞相关稳定性问题，利用离散Gronwall不等式，结合矩阵范数，导出了时滞离散区间系统的时滞相关性准则．为了证实所提方法的有效性，通过实例进行了仿真比较．仿真结果表明，该方法与其他方法相比，具有计算量小和时滞上界更大的优点．
Focusing on the time-lag correlation stability of the discrete interval system with time-lag, this paper deduces the time-lag correlation rule of the system, by taking Gronwall inequality and combining the matrix norm. To prove the effectiveness of this proposed method in this paper, simulation and comparison are performed through examples. The results show that the proposed method has some advantages over other methods, such as less computation and greater upper bound of time-lag.

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本文提出一种PIα控制器设计方法，实现了分数阶超混沌Chen系统的混沌同步．首先针对分数阶超混沌Chen系统设计了PIα控制器，然后利用改进的双参数Mittag-Leffler函数估计定理和扩展的Gronwall引理证明了系统的稳定性，并指出了使分数阶超混沌Chen系统同步的控制器参数需满足的条件．仿真结果证实了所设计方法的有效性．
In this paper, a kind of PIα controller is designed for the synchronization of fractional order hyperchaotic Chen system. The controller is designed according to the fractional order hyperchaotic Chen system. The stability of the system is proved by the im- proved double parameter Mittag-Leffler function estimate theorem and the extended Gronwall lemma. Furthermore, the condition of controller parameters is pointed out which makes the fractional order hyperchaotic Chen system synchronous. Numerical simulations are presented to verify the effectiveness of the method.

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为提高航空发动机某工作点的模型精度,并拓宽航空发动机在该工作点控制包线的范围,可应用非线性模型来描述该工作点的动态过程。基于该非线性模型,首先应用Lyapunov稳定性定理设计出一组控制器,然后应用广义Gronwall-Bellman引理的方法完成该控制器性能验证。仿真研究表明:系统响应速度快,能有效抑制干扰,具有良好的跟踪性和鲁棒性,验证了该设计方法的有效性。
In order to improve the model accuracy of the aero-engine in an operating point and extend the control envelope around the operating point, one strategy can be sought to apply nonlinear model to describe the dynamic characteristics. Firstly, a nonlinear controller can be obtained for the aero-engine using Lyapu-nov stability theorem. Then Generalized Gronwall-Bellman lemma is used to validate the controller perfor-mance. Finally, the simulation study shows that the proposed design can provide fast transient response, good tracking performance and robustness, and reject the disturbance effectively to validate the effective-ness of the control design approach.

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