Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.

LARGE TIME STABILITY OF THE S0LUTION OF A FILTRATION PROBLEM

We discuss a filtration problem in a bounded one-dimensional porous medium. Suppose that the.volumetric moisture content at the surface is constant, and the bottom is impermeable. We prove thatthe solution will tend uniformly to a stable solution of the filtration equation as time tends to infinity.An explicit expression of the limiting profile is given.

The Variation and Stability Analysis of Wheat Dough Stability Time

Farinograph dough stability time is an important index for classifying wheat, and it often indicates the most appropriate end use for the wheat cultivars. This study aimed at the problem of large fluctuations in dough stability time that occurs during the commercial wheat production. The variations in the dough stability time and its consistency across locations and years were analyzed using 12 principal high-quality wheat cultivars （varieties） obtained from Shandong Province, China, which were grown at nine different locations for three successive years. The results showed that the coefficient of variation for the dough stability time ranged from 24.29 to 49.60% across different varieties, locations, and years. Additive main effects and multiplicative interaction （AMMI） analysis indicated that there were significant interactions for the dough stability time between the varieties, the growth locations, and the years. The genotype effect was the most noticeable, followed by the interaction of the genotype and the environment. The environmental effect was the least significant. The interactions between the varieties and the locations differ considerably, however, each cultivar （variety） apparently has a specific adaptability to the growth location. Therefore, for the successful commercial scale production of the high-quality wheat varieties, both the selection of proper cultivars and its most suitable growth locations to meet the desired requirements for the dough mixing stability time are important.

Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

The problem of linear time-varying（LTV） system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the ＂frozen-time＂ assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stabilitypreserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system＇s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam exper- imental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides anew way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

Fixed time integral sliding mode controller and its application to the suppression of chaotic oscillation in power system

Chattering phenomenon and singularity are still the main problems that hinder the practical application of sliding mode control. In this paper, a fixed time integral sliding mode controller is designed based on fixed time stability theory, which ensures precise convergence of the state variables of controlled system, and overcomes the drawback of convergence time growing unboundedly as the initial value increases in finite time controller. It makes the controlled system converge to the control objective within a fixed time bounded by a constant as the initial value grows, and convergence time can be changed by adjusting parameters of controllers properly. Compared with other fixed time controllers, the fixed time integral sliding mode controller proposed in this paper achieves chattering-free control, and integral expression is used to avoid singularity generated by derivation. Finally, the controller is used to stabilize four-order chaotic power system. The results demonstrate that the controller realizes the non-singular chattering-free control of chaotic oscillation in the power system and guarantees the fixed time convergence of state variables, which shows its higher superiority than other finite time controllers.

Finite-time control of linear parameter-varying systems with norm-bounded exogenous disturbance

In this paper, a new approach is presented for finite-time control problems for linear systems subject to time-varying parametric uncertainties and exogenous disturbance. The disturbance is assumed to be time varying and bounded. Sufficient conditions are obtained for the existence of a linear parameter-dependent state feedback gain, which can ensure that the closed-loop system is finite-time bounded （FTB）. The conditions can be reduced to feasibility problems involving LMIs. Numerical examples show the validity of the proposed methodology.

Time-Varying Asymmetrical BLFs Based Adaptive Finite-Time Neural Control of Nonlinear Systems With Full State Constraints

This paper concentrates on asymmetric barrier Lyapunov functions(ABLFs)based on finite-time adaptive neural network(NN)control methods for a class of nonlinear strict feedback systems with time-varying full state constraints.During the process of backstepping recursion,the approximation properties of NNs are exploited to address the problem of unknown internal dynamics.The ABLFs are constructed to make sure that the time-varying asymmetrical full state constraints are always satisfied.According to the Lyapunov stability and finitetime stability theory,it is proven that all the signals in the closedloop systems are uniformly ultimately bounded(UUB)and the system output is driven to track the desired signal as quickly as possible near the origin.In the meantime,in the scope of finitetime,all states are guaranteed to stay in the pre-given range.Finally,a simulation example is proposed to verify the feasibility of the developed finite time control algorithm.

Dynamics and Long Time Convergence of the Extended Fisher-Kolmogorov Equation under Numerical Discretization

We present a numerical study of the long time behavior of approxima- tion solution to the Extended Fisher-Kolmogorov equation with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Furthermore, we obtain the long-time stability and convergence of the difference scheme and the upper semicontinuity d（Ah,τ, .A） → O. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.

The Influence of Different Sterilization Techniques on the Time-Dependent Behavior of Polyamides

For this investigation conventional polyamide 6 with monomodal molecular mass distribution, and the newly developed bimodal one were used. Conventional polyamide 6 was used as a reference material in order to emphasize prospects of using bimodal material for medical applications from the point of view of sterilization resistance and improved creep behavior. Time-dependent mechanical properties of testing samples were characterized by torsional creep measurements in non-sterilized state and after sterilization with three different techniques: with autoclave, ethylene oxide, and hydrogen peroxide plasma. Results show that the two materials exhibit pronounced difference in morphology and consequently, mechanical properties. Both of them were not significantly affected by any of used sterilization techniques. However, bimodal material, originally being noticeably more time-stable in comparison to monomodal one, retains these preferences also post sterilization.

A new simple method of implicit time integration for dynamic problems of engineering structures

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai＇s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

An improved time integration scheme based on uniform cubic B-splines and its application in structural dynamics

A time integration algorithm for structural dynamic analysis is proposed by uniform cubic B-spline functions. The proposed algorithm is successfully used to solve the dynamic response of a single degree of freedom （SDOF） system, and then is generalized for a multiple-degree of freedom （MDOF） system. Stability analysis shows that, with an adjustable algorithmic parameter, the proposed method can achieve both conditional and unconditional stabilities. Validity of the method is shown with four numerical simulations. Comparison between the proposed method and other methods shows that the proposed method possesses high computation accuracy and desirable computation efficiency.

Time-stability of sintered Nd-Fe-B permanent magnet with lower oxygen content

Time-stability of sintered Nd-Fe-B magnet with lower content of oxygen at different temperatures and humidity conditions for 400 d was investigated.Results showed that the magnetic flux loss was-0.68% for the non-electroplating samples and-0.43% for the electroplating samples at room temperature and average humidity was 25% during 400 d,respectively.The magnetic flux loss of the plated samples was lower than non-plated samples under the conditions of room temperature and with the average humidity of 54%,but the magnetic flux loss of the plated samples was larger than that of the non-plated sample at 80 oC.The magnetic flux loss of electroplated and non-electroplated sintered Nd-Fe-B magnets was less than 1% for 400 d,which showed that the N40SH grade magnets have good time stability.

INDIRECT ACCELERATED ADAPTIVE FUZZY CONTROLLER

According to a type of normal nonlinear system, an indirect adaptive fuzzy （IAF） controller has been applied to those systems where no accurate mathematical models of the systems under control are available. To satisfy with system performance, an indirect accelerated adaptive fuzzy （IAAF） controller is proposed, and its general form is presented. The general form IAAF controller ensures necessary control criteria and system＇s global stability using Lyapunov Theorem. It has been proved that the close-loop system error converges to a small neighborhood of equilibrium point. The optimal IAAF controller is derived to guarantee the process＇s shortest settling time. Simulation results indicate the IAAF controller make the system more stable, accurate, and fast.

SCALE-TYPE STABILITY FOR NEURAL NETWORKS WITH UNBOUNDED TIME-VARYING DELAYS

This paper studies scale-type stability for neural networks with unbounded time-varying delays and Lipschitz continuous activation functions. Several sufficient conditions for the global exponential stability and global asymptotic stability of such neural networks on time scales are derived. The new results can extend the existing relevant stability results in the previous literatures to cover some general neural networks.

Stability of Time-varying Linear Systems with Time Delays

Easy ways to test the stability of systems involving time delays have been sought.In this paper,some unconditional stability and asymptotically stable with decay rate α criteria for time-varying linear systems with time delays are presented by matrix measure and comparisontheorem.

Birkhoff Normal Form for the Derivative Nonlinear Schrodinger Equation

This paper is concerned with the derivative nonlinear Schr?dinger equation with periodic boundary conditions.We obtain complete Birkhoff normal form of order six.As an application,the long time stability for solutions of small amplitude is proved.

Approach and landing guidance design for reusable launch vehicle using multiple sliding surfaces technique

An autonomous approach and landing（A＆L） guidance law is presented in this paper for landing an unpowered reusable launch vehicle（RLV） at the designated runway touchdown. Considering the full nonlinear point-mass dynamics, a guidance scheme is developed in threedimensional space. In order to guarantee a successful A＆L movement, the multiple sliding surfaces guidance（MSSG） technique is applied to derive the closed-loop guidance law, which stems from higher order sliding mode control theory and has advantage in the finite time reaching property.The global stability of the proposed guidance approach is proved by the Lyapunov-based method.The designed guidance law can generate new trajectories on-line without any specific requirement on off-line analysis except for the information on the boundary conditions of the A＆L phase and instantaneous states of the RLV. Therefore, the designed guidance law is flexible enough to target different touchdown points on the runway and is capable of dealing with large initial condition errors resulted from the previous flight phase. Finally, simulation results show the effectiveness of the proposed guidance law in different scenarios.