Iterative methods for solving extended general mixed variational inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results.     

On the qualitative behaviors of solutions of third order nonlinear differential equations

We present some new results about oscillation and asymptotic behavior of solutions of third order nonlinear differential equations of the form

(r₂(t)(r₁(t)y^′)^′)^′+p(t)y^′+q(t)f(y(g(t)))=0.     

Some new error inequalities for a generalized quadrature rule of open type

Some new error inequalities for a generalized quadrature rule of open type are established. Especially, two sharp inequalities are derived when n is an odd and an even integer, respectively.     

State estimation for fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays

This paper deals with the problem of state estimation for fuzzy cellular neural networks (FCNNs) with time delay in the leakage term, discrete and unbounded distributed delays. In this paper, leakage delay in the leakage term is used to unstable the neuron states. It is challenging to develop a delay dependent condition to estimate the unstable neuron states through available output measurements such that the error-state system is globally asymptotically stable. By constructing the Lyapunov-Krasovskii functional which contains a triple-integral term, an improved delay-dependent stability criterion is derived in terms of linear matrix inequalities (LMIs). However, by using the free-weighting matrices method, a simple and efficient criterion is derived in terms of LMIs for estimation. The restriction such as the time-varying delay which was required to be differentiable or even its time-derivative which was assumed to be smaller than one, are removed. Instead, the time-varying delay is only assumed to be bounded. Finally, numerical examples and its simulations are given to demonstrate the effectiveness of the derived results.     

一类带随机跳跃的完全耦合的线性二次随机控制问题

研究一类带随机跳跃的完全耦合的线性二次随机控制问题. 得到了最优控制的显式解, 并可以证明最优控制是唯一的. 引入了一类推广的黎卡提方程并讨论了其可解性. 利用这一类推广的黎卡提方程的解, 得到了上述带随机跳跃的最优控制问题的线性状态反馈调节器.     

随机观测时滞系统的最优估计

研究了在观测中存在Markov跳跃时滞的离散时间系统的线性最小方差状态估计问题. 首先, 通过引入跳跃时滞的示性函数, 将带有跳跃时滞的观测方程转化为带有乘性噪声的定常时滞系统. 进一步采用状态扩维的方法, 将定常时滞系统转化为无时滞的Markov跳跃系统. 最后, 基于得到的无时滞系统, 采用Hilbert空间已有的几何论知识, 设计线性最优状态估计器, 得出基于Riccati方程的滤波器的表达式, 并证明了所得滤波器的渐渐收敛性.     

一类基于观测器的非线性网络化控制系统的绝对稳定性

主要考虑了基于观测器的Lurie网络化控制系统的绝对稳定性问题. 由于采用了基于观测器的反馈控制器, 传感器到控制器的网络诱导时延和控制器到执行器的网络诱导时延不再能合并到一起处理. 首先通过状态增广方法将Lurie网络化控制系统建模为一个多时滞的Lurie系统, 然后利用Newton-Leibniz公式和添加自由权矩阵的方法给出了时滞依赖的稳定性条件. 在此基础上, 给出三种求解控制器和观测器增益矩阵的方法. 此外, 还分别给出了被控对象存在范数有界不确定性和结构不确定性时系统的鲁棒稳定性条件及鲁棒控制器设计方法, 所有得到的结果都是以线性矩阵不等式的形式给出的. 便于利用线性矩阵不等式工具包进行求解. 最后, 通过两个仿真算例说明了方法的可行性和有效性.     

Detectability and observability of discrete-time stochastic systems and their applications

This paper studies detectability and observability of discrete-time stochastic linear systems. Based on the standard notions of detectability and observability for time-varying linear systems, corresponding definitions for discrete-time stochastic systems are proposed which unify some recently reported detectability and exact observability concepts for stochastic linear systems. The notion of observability leads to the stochastic version of the well-known rank criterion for observability of deterministic linear systems. By using these two concepts, the discrete-time stochastic Lyapunov equation and Riccati equations are studied. The results not only extend some of the existing results on these two types of equation but also indicate that the notions of detectability and observability studied in this paper take analogous functions as the usual concepts of detectability and observability in deterministic linear systems. It is expected that the results presented may play important roles in many design problems in stochastic linear systems.     

Perturbation analysis and condition numbers of symmetric algebraic Riccati equations

This paper is devoted to the perturbation analysis of symmetric algebraic Riccati equations. Based on our perturbation analysis, the upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are demonstrated by our preliminary numerical experiments.     

An alternative Kalman-Yakubovich-Popov lemma and some extensions

This paper introduces an alternative formulation of the Kalman-Yakubovich-Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this new formulation encompasses previous generalizations of the KYP Lemma which hold in the case the coefficient matrix of the FDI does not depend on frequency. In addition, it allows the coefficient matrix of the frequency domain inequality to vary affinely with the frequency parameter. One application of this results is illustrated in an example of computing upper bounds to the structured singular value with frequency-dependent scalings.