BOUNDEDNESS AND UNIFORM BOUNDEDNESS RESULTS FOR CERTAIN NON-AUTONOMOUS DIFFERENTIAL EQUATIONS OF FOURTH ORDER

Sufficient conditions have been obtained so that all solutions of a different fourth order differential equation are bounded and uniformly bounded.

Stability and Uniform Boundedness for a Class of Anaerobic Digestion Ecological Models

Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Method, the local stability of all equilibria in the system is got. According to the vector fields described by the system, the proof of the boundedness of the solution on the anaerobic digestion processes is completed in three steps. The method proposed in the discussion on the boundedness can be generalized to the similar problems. Results in this paper give information on how to run the ecological system well by adjusting the system parameters.

Uniform Boundedness for Neutral Equations with Unbouned Delay

We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.

UNIFORM BOUNDEDNESS AND STABILITY OF SOLUTIONS TO A CUBIC PREDATOR-PREY SYSTEM WITH CROSS-DIFFUSION

Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.

Uniformly ultimate boundedness for discontinuous systems with time-delay

The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.

Dynamical Model to Optimize Student’s Academic Performance

Excellent student’s academic performance is the uppermost priority and goal of educators and facilitators.The dubious marginal rate between admission and graduation rates unveils the rates of dropout and withdrawal from school.To improve the academic performance of students,we optimize the performance indices to the dynamics describing the academic performance in the form of nonlinear system ODE.We established the uniform boundedness of the model and the existence and uniqueness result.The independence and interdependence equilibria were found to be locally and globally asymptotically stable.The optimal control analysis was carried out,and lastly,numerical simulation was run to visualize the impact of the performance index in optimizing academic performance.

UNIFORM BOUNDEDNESS AND UNIFORM STABILITY OF A KIND OF DELAY──DIFFERENTIAL EQUATIONS

This paper presents a discussion of differential equations with two terms generated by delays. They are as follows x′(t) = a(t)x(t - p(t)) - b(t)x(t - r(t)).Sufficient conditions for its solution with uniform bound and for its zero solution with uniform stability are established.

Robust adaptive control for interval time-delay systems

This paper focuses on the robust adaptive control problems for a class of interval time-delay systems and a class of large-scale interconnected systems. The nonlinear uncertainties of the systems under study are bounded by high- order polynomial functions with unknown gains. Firstly, the adaptive feedback controller which can guarantee the stability of the closed-loop system in the sense of uniform ultimate boundedness is proposed. Then the proposed adaptive idea is extended to robust stabilizing designing method for a class of large-scale interconnected systems. Here, another problem we address is to design a decentralized feedback adaptive controller such that the closed-loop system is stable in the sense of uniform ultimate boundedness for all admissible uncertainties and time-delay. Finally, an illustrative example is given to show the validity of the proposed approach.

Tracking Control of Uncertain Nonlinear Systems With Unknown Constant Input Delay

A robust delay compensator has been developed for a class of uncertain nonlinear systems with an unknown constant input delay.The control law consists of feedback terms based on the integral of past control values and a novel filtered tracking error,capable of compensating for input delays.Suitable Lyapunov-Krasovskii functionals are used to prove global uniformly ultimately bounded(GUUB)tracking,provided certain sufficient gain conditions,dependent on the bound of the delay,are satisfied.Simulation results illustrate the performance and robustness of the controller for different values of input delay.

A New Robust Adaptive Neural Network Backstepping Control for Single Machine Infinite Power System With TCSC

For a single machine infinite power system with thyristor controlled series compensation(TCSC) device, which is affected by system model uncertainties, nonlinear time-delays and external unknown disturbances, we present a robust adaptive backstepping control scheme based on the radial basis function neural network(RBFNN). The RBFNN is introduced to approximate the complex nonlinear function involving uncertainties and external unknown disturbances, and meanwhile a new robust term is constructed to further estimate the system residual error,which removes the requirement of knowing the upper bound of the disturbances and uncertainty terms. The stability analysis of the power system is presented based on the Lyapunov function,which can guarantee the uniform ultimate boundedness(UUB) of all parameters and states of the whole closed-loop system. A comparison is made between the RBFNN-based robust adaptive control and the general backstepping control in the simulation part to verify the effectiveness of the proposed control scheme.

FLOATING QUANTIZATION EFFECTS ON MULTIRATE SAMPLED-DATA NONLINEAR CONTROL SYSTEMS

In this article, floating quantization effects on multirate sampled-data control systems are studied. It shows that the solutions of multirate digital feedback control systems with nonlinear plant and with floating quantization in the controller are uniformly ultimately bounded if the associated linear systems consisting of linearization of the plant and controller with no quantization are Schur stable. Moreover, it also shows that the difference between the response of multirate digital controllers without quantizers and the same plant with floating quantization in the controllers can be made as small as desired by selecting proper quantization level.

QUALITATIVE ANALYSIS OF A DICRETE POPULATION MODEL

In this paper, authors study the qualitative behavior of solutions of the discrete population model xn-xn-1=xn (a+bxn-k-cx2n-k),where a ∈ (0, 1), b ∈ (-∞, 0),c ∈ (0,∞ ), and k is a positive integer. They hot only obtain necessary as well as sufficient and necessary conditions for the oscillation of ail eventually positive solutions about the positive equilibrium, but also obtain some sufficient conditions for the convergence of eventually positive solutions. Furthermore, authors also show that such model is uniformly persistent, and that all its eventually positive solutions are bounded.

THEORETICAL ANALYSIS TO A REACTION-DIFFUSION SYSTEM ARISING IN A DIFFUSIVE EPIDEMIC MODEL

This paper developes a diffusive epidemic model and investigates the global existence, uniform bounds, stability, asymptotic behavior and decay rate for solution of related reaction-diffusion system.

Outlier-resistant distributed fusion filtering for nonlinear discrete-time singular systems under a dynamic event-triggered scheme

This paper investigates the problem of outlier-resistant distributed fusion filtering(DFF)for a class of multi-sensor nonlinear singular systems(MSNSSs)under a dynamic event-triggered scheme(DETS).To relieve the effect of measurement outliers in data transmission,a self-adaptive saturation function is used.Moreover,to further reduce the energy consumption of each sensor node and improve the efficiency of resource utilization,a DETS is adopted to regulate the frequency of data transmission.For the addressed MSNSSs,our purpose is to construct the local outlier-resistant filter under the effects of the measurement outliers and the DETS;the local upper bound(UB)on the filtering error covariance(FEC)is derived by solving the difference equations and minimized by designing proper filter gains.Furthermore,according to the local filters and their UBs,a DFF algorithm is presented in terms of the inverse covariance intersection fusion rule.As such,the proposed DFF algorithm has the advantages of reducing the frequency of data transmission and the impact of measurement outliers,thereby improving the estimation performance.Moreover,the uniform boundedness of the filtering error is discussed and a corresponding sufficient condition is presented.Finally,the validity of the developed algorithm is checked using a simulation example.

BOUNDEDNESS IN p-NFDES

This paper considers a class of p-NFDEs. By Liapunov functionals, we establish two boundedness criteria, which improve some boundedness conditions for FDEs in the literature. Some examples are also worked out to demonstrate the advantages of our results.

EXISTENCE OF THE UNIFORMLY ULTIMATE BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS

In this paper, we study the existence of the uniformly ultimately bounded solutions and periodic solutions for a class of third order nonlinear control systemx …+F(r)x ¨+G(r)x ·+H(r)x=e(t).(1)Using analytic method of Liapunov Function, we established the sufficient conditions of the existence of the uniformly ultimately bounded solutions and periodic solutions of the system(1).

Local and global well-posedness of entropy-bounded solutions to the compressible Navier-Stokes equations in multi-dimensions

Due to the high degeneracy and singularity of the entropy equation,the physical entropy for viscous and heat-conductive polytropic gases behaves singularly in the presence of vacuum,and it is thus a challenge to study its dynamics.It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat-conductive gases in R3,provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density.Precisely,it is proved that for any strong solution to the Cauchy problem of the heat-conductive compressible Navier-Stokes equations,the corresponding entropy remains uniformly bounded,and the L2 regularities of the velocity and temperature can be propagated,up to the existing time of the solution,as long as the initial density vanishes only at far fields with a rate not faster than O(1/|x|^(2)).The main tools are some singularly weighted energy estimates and an elaborate De Giorgi-type iteration technique.We apply the De Giorgi-type iterations to different equations in establishing the lower and upper bounds of the entropy.

A Framework of Convergence Analysis of Mini-batch Stochastic Projected Gradient Methods

In this paper,we establish a unified framework to study the almost sure global convergence and the expected convergencerates of a class ofmini-batch stochastic(projected)gradient(SG)methods,including two popular types of SG:stepsize diminished SG and batch size increased SG.We also show that the standard variance uniformly bounded assumption,which is frequently used in the literature to investigate the convergence of SG,is actually not required when the gradient of the objective function is Lipschitz continuous.Finally,we show that our framework can also be used for analyzing the convergence of a mini-batch stochastic extragradient method for stochastic variational inequality.

PERIODICITY IN FUNCTIONAL DIFFERENTIAL EQUATIONS

A criterion for the existence of T-periodic solutions of functional differential equations with finite delay is established, in which the uniform boundedness of solutions has been removed from the conditions. The obtained result is the counterpart of the one recently obtained for infinite delay equations, but the conditions in this work are simplified and much easier to verify. Also, the proof has been improved to be more clear and compact.

BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS OF VISCOUS POLYTROPIC GAS EQUATIONS

A piston problem of viscous polytropit gas equations is discussed.It is shown that the global solution is bounded uniformly in time if the piston motion is bounded and that if the piston motion is periodic in time,then there exists a periodic solution to the piston problem with the same period.