Global asymptotical stability of a unique almost periodic solution for enterprise clusters based on ecology theory with time-varying delays and feedback controls

This paper is concerned with some dynamic behavior of enterprises cluster constituted by n satellite enterprises and a dominant enterprise. We present a model involving time-varying delays and feedback controls based on ecology theory, which effectively describe the competition and cooperation of enterprises cluster in real economic environments. Applying the comparison theorem of differential equations and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are obtained. Finally, we present an example to explain the economical significance of mathematical results.     

不确定离散模糊随机系统的鲁棒方差约束输出反馈控制

对一类具有范数有界不确定性的离散T-S模糊随机系统。研究不仅使整个闭环模糊系统全局渐近稳定。而且每个模糊子系统的稳态状态方差满足给定上界性能指标约束的输出反馈鲁棒方差控制律的设计问题。利用线性矩阵不等式(LMI)技术,导出输出反馈鲁棒方差控制律的存在条件,并基于矩阵相似变换给出其可解性条件,同时用一组线性矩阵不等式的可行解。给出输出反馈鲁棒方差控制律的一个参数化表达形式。     

Synchronization of a growing chaotic network model

Synchronization in large ensembles of coupled interacting units is a fundamental phenomenon, which is helpful for the understanding of working mechanisms in neuronal networks, social network, etc. In this paper, we will investigate the synchronization phenomenon in a network model. A feedback control scheme is proposed for the synchronization of the given complex networks. The obtained result indicates that synchronization can be achieved for growing chaotic network model. Method enhance the synchronizability of the given model are given at the same time. Finally, numerical simulations are given to show the effectiveness of obtained results.     

Examining students’ variational reasoning in differential equations

In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.     

On feedback and stable price adjustment mechanisms

Given an excess demand function of an economy, say Z(p)$Z(p)$, a stable price adjustment mechanism (SPAM) guarantees convergence of solution path p(t,_p0)$p(t,p_0)$ to an equilibrium _peq$p_{\mathit{eq}}$ solution of Z(p)=0$Z(p)=0$. Besides, all equilibrium points of Z(p)$Z(p)$ are asymptotically stable. Some SPAMs have been proposed, including Newton and transpose Jacobian methods. Despite this powerful stability property of SPAMs, their acceptation in the economics community has been limited by a lack of interpretation. This paper focuses on this issue. Specifically, feedback control theory is used to link SPAMs and price dynamics models with control inputs, which match the economically intuitive Walrasian Hypothesis (i.e., prices change with excess demand sign). Under mild conditions, it is shown the existence of a feedback function that transforms the price dynamics into a desired SPAM. Hence, a SPAM is interpreted as a fundamental (e.g., Walrasian) price dynamics under the action of a feedback function aimed to stabilize the equilibrium set of the excess demand function.     

Fluorescence feedback control of a diode pumped solid state laser

We implement active feedback control in a Q-switched diode-pumped Nd:YVO₄ laser by monitoring the fluorescence intensity from the laser crystal. When the initial inversion level indicated by the detected fluorescence has reached a predetermined value, Q-switching is initiated. The Q-switched pulse energy is stabler with our feedback scheme based on the fluorescence intensity than that with the conventional Q-switching when pumping source is not stable.     

‘Best-case’ noise performance of feedback interferometers, potential for micromachine applications

We present a theoretical analysis of the noise performance of phase measuring feedback interferometers. We first analyse the operation of this kind of instrument and note that under certain circumstances bistability can occur. The bistable region should be avoided if possible when using feedback interferometry for high-accuracy measurement with low-input powers, as it can give rise to very high-noise levels. We then go on to investigate the effects of shot noise (on the interferometer output signal) and thermal noise (in the feedback loop), and relate these to the phase measurement accuracy of the interferometer. A ‘best-case’ calculation indicates that phase noise of about 0.005λ is possible at input powers in the nanowatt region. In practice, we expect that noise levels will be higher than predicted (particularly at high-input powers) due to the effects of vibration and air turbulence.     

Imaging of enzyme activity by scanning electrochemical microscope equipped with a feedback control for substrate-probe distance

The enzymatic activity of diaphorase (Dp) immobilized on a solid substrate was characterized using a scanning electrochemical microscope (SECM) with shear force feedback to control the substrate-probe distance. The shear force between the substrate and the probe was monitored with a tuning fork-type quartz crystal and used as the feedback control to set the microelectrode probe close to the substrate surface. The sensitivity and the contrast of the SECM image were improved in the constant distance mode (distance, 50 nm) with the shear force feedback compared to the image in the constant height mode without the feedback. By using this system, the SECM and topographic images of the immobilized diaphorase were simultaneously measured. The microelectrode tip used in this study was ground aslant like a syringe needle in order to obtain the shaper topographic images. This shape was also effective for avoiding the interference during the diffusion of the enzyme substrates.