Linear stability of numerical methods for systems of functional differential equations with a proportional delay

The numerical solution of functional differential equations with a proportional delay is considered. The stability of general linear methods for linear systems of neutral type is investigated. It is shown that a general linear method with strict stability at infinity can preserve the asymptotic stability of the underlying system if we employ an appropriate equi-stage interpolation to approximate the delay argument.     

Analytic solution of phreatic surface in the slope of reservoir bank

In most cases, the slope stability of reservoir bank is analyzed on the premise that the location of phreatic surface is obtained. But many designers generalize a line as the phreatie surface through their experience to analyze the stability, which is unsafe in the project. To find a solution of the phreatic surface which is convenient to put into use and in accordance with the practice, the article, based on Boussinesq equation, infers analytic solutions suitable to the water level at different ratios and achieves an analytic solution equation through fitting curves. The correctness of the equation is also proved by the experiments of sand and sand-clay models and the inaccuracy of empirical generalization is analyzed quantitatively. The calculation results show that the inaccuracy through the method of experiential generalizing is so large that the designers should be awake to it.     

Stability Analysis of Runge-Kutta Methods for Delay Integro-Differential Equations

Considering a linear system of delay integro-differential equations with a constant delay whose zero solution is asympototically stable, this paper discusses the stability of numerical methods for the system. The adaptation of Runge-Kutta methods with a Lagrange interpolation procedure was focused on inheriting the asymptotic stability of underlying linear systems. The results show that an A-stable Runge-Kutta method preserves the asympototic stability of underlying linear systems whenever an unconstrained grid is used.     

Robust exponential control of a class of systems with uncertainties

The robust exponential stabilization problem for uncertain systems isstudied. Based on the solution for a nominal linear quadratic regulator problem with a prescribed degree of stability, the methods of constructing state feedback controllers are developed to ensure the robust stability of the closed loop system under the conditions weaker than the matching condition. Also, the cases where the matching condition is satisfied are considered in detail. Some examples are included to show the solution methods.     

Dynamic System for Solving Saddle Point Problems in Hilbert Spaces and Its Application to Neural Computing

This paper studies the existence and uniqueness of solutions and the stability and convergence of a dynamic system for solving saddle point problems (SPP) in Hilbert spaces. The analysis first converts the SPP into a problem of searching for equilibriums of a dynamic system using a criterion for solutions of the SPP, then shows the existence and uniqueness of the solutions by creating a positive function whose Fréchet derivative is decreasing along any solution. The construction of positively invariant subsets gives the global stability and convergence of this dynamic system, that is, the dynamic system globally converges to some exact solution of the SPP. Finally, the paper also shows that the obtained results can be applied to neural computing for solving SPP.     

The relationship between short-range motion of atmosphere and ocean and conservative and nonconservative scheme

Taking one-dimensional shallow water wave equation as an example, a comparative analysis on the computational stability related to the short-range motion of atmosphere and ocean is carried out for the conservative and nonconservative scheme, and it is pointed out that the computational stability of conservative scheme is absolutely different from that of the nonconservative scheme. The relationship between the short-range motion of atmosphere and ocean and the conservative and nonconservative scheme is further discussed. The constructed conservative scheme is proved to be stable by the numerical experiment for the short-range motion of atmosphere and ocean while the CTCS scheme is unstable. So that the conservative scheme for the solution of this kind of problem has more advantages.     

Microstructural evolution in the surface of Ti-10V-2Fe-3Al alloy by solution treatments

By solution treatments in vacuum and in air,the surface region of Ti-10V-2Fe-3Al exhibits different microstructures from the interior region of the alloy.When solution treated in vacuum,the microstructure of the specimen is composed of dominant β phase and a few of α " martensite in surface region,with a large amount of α" martensite and a little of β phase in interior.When solution treated in air,the specimen is characterized by dominant surficial a phase and interior β phase.In addition,the lattice parameter evolutions of a and β phases were observed in the surface region of the specimens by solution treatment in air.The mechanism for the surface effects is discussed in light of the structural stability and phase evolution in Ti-10V-2Fe-3Al alloy with solution treatments.     

Long Time Behaviors of the Solutions to Stochastic Wave Equations with Damping

The long time behavior of the solution to a stochastic wave equation with damping is considered. Under some appropriate conditions, the exponential stability of the solutions holds almost surely.Finally two examples to illustrate the results are given.     

Solitons and production of defects in flow-aligning nematic liquid crystals under simple shear flow

The production of defects in flow-aligning nematic liquid crystals under simple shear flow is analyzed by linear stability analysis based on Leslie-Ericksen theory. It is pointed out that the equation of motion of the nematic director under simple shear flow conforms to the driven over-damped sine-Gordon equation and has a soliton solution of amplitude π. It has also been shown that the stationary state with the director uniformly oriented at a Leslie angle is only a metastable state and that the potential, which governs the motion of the director, has infinite numbers of stable stationary states. Therefore, the defects, appearing as a stable solitary solution, can be nucleated from a uniformly aligned flow-aligning type of nematic liquid crystal by shear flow. On the other hand, the bands with long axis parallel to the vorticity axis, appearing as an unstable solution, can be observed as transient patterns at low shear rate and low shear strain value. The theoretical predictions are compared with previous experimental observations.     

A Parallel Algorithm for the Convection Diffusion Problem

Based on the second-order compact upwind scheme, a group explicit method for solving the two-dimensional time-independent convection-dominated diffusion problem is developed. The stability of the group explicit method is proven strictly. The method has second-order accuracy and good stability. This explicit scheme can be used to solve all Reynolds number convection-dominated diffusion problems. A numerical test using a parallel computer shows high efficiency. The numerical results conform closely to the analytic solution.     

Reaction of Silane Alkoxide with Acid Anhydride as a Novel Synthetic Method for Organic-Inorganic Hybrid Materials

Sol-gel method is a potent method to produce new inorganic and organic-inorganic hybrid materials. The key step of this methodology is the hydrolysis of a metal alkoxide or other metal substrates such as acetylacetonates to form hydroxyl metal species, followed by their condensation to metal-oxygen-metal （M - O - M） bonds. In this process, the utilization of water, generally in excess, is essential and alcoholic solvents such as ethanol are often required to homogenize the solution when organic compounds coexist. As the common sol-gel method using water allows for limited uses of organic substrates due to their low solubility and stability in aqueous solution, modified variations of sol-gel method are reauired.     

Principal Resonance of Parametrically Excited Moving Viscoelastic Belts with Geometrical Nonlinearity

Nonlinear dynamic analysis is performed on moving belts subjected to geometric nonlinearity and initial tension fluctuation. To incorporate more accurately the damping mechanism of belt material,linear viscoelastic models are adopted in a unified form of differential operators. To circumvent high-order differential vibration equation of time-varying coefficients and with gyroscopic and nonlinear terms, where analytical solution is almost impossible, a systematic approach is presented by reforming the motion equation and directly using the method of multiple scales. To exemplify the procedure, the solutions at principal resonance are obtained and their stability conditions are derived for employing a Kelvin-Voigt model to reflect the property of the belt material. The solutions and stability conditions successfully reduce to those for using Kelvin model and elastic model, which validate the present approaches. Numerical simulations highlight the effects of tension fluctuations and translating speeds on the stability of the belt vibration.     

The stability safety factor calibration based on the reliability index

When the bridge structure stability safety factor of the first type is 4, the research that whether the structure reliability index will reach target reliability index under the more-likely-to-happen collapse situation of the second type is necessary. The stability calculations of the first and the second type are made respectively for single layer and single span rigid frame bridge. Based on the critical load obtained from the stability calculation of the first type, the stability safety factor of the first type is taken as 4, and the first order reliability method is used to program and calculate the reliability index. Then, the load effect under the stability reliability index cal- culation of the first type and the critical load of the second type are employed to calculate the reliability index of the second type. The evaluation of structure stability safety factor is discussed according to reliability index. Based on the discussion above, parameter analysis of the stable critical loads of two types is made, and the in- fluence of critical load change on reliability index is researched. The result shows that stability analysis should identify collapse state; when the stability safety factor of the first type is 4, but the structure has the collapse of the second type, the reliability index cannot be ensured to reach the target reliability index under certain condi- tions.     

Molecular recognition study on a supramolecular system (Ⅷ) --The fluoroscopic study on inclusion complexation of adamantanol with cyclodextrin derivatives

The fluorescence spectral titrations have been performed in phosphate buffer solution (pH 7 2) at room temperature to give the stability constants for inclusion complexation of a series of chemically modified β cyclodextrins bearing aromatic groups with adamantanol. The molecular recognition mechanism is discussed from the viewpoint of the structural factors, electron effect of the substitutent groups and the size and chain length of the fluorophore. The results indicate that the complex stability of the cyclodextrin derivatives bearing a positively charged pyridinio moiety can be enhanced by the electrostatic interaction between the host cyclodextrins and the adamantanol guest and the microenvironment change plays a crucial role in the inclusion complexation.     

Foam drainage wave coalescing and its energy evolution

Liquid foam is a dense packing of gas bubbles in a small amount of surfactant solution. Liquid drains out of foams until equilibrium is reached due to the compromise between gravity and capillarity, which greatly affects the stability of foam. Based on a series of work on foam structure and drainage we conducted previously, this paper reports the results on coalescence of an original forced drainage wave at a low flow rate with subsequent drainage waves with higher flow rates. The evolutions of viscous energy and surface energy during the process of coalescence are theoretically analyzed.     

Three-Dimensional Modes of Fiber Suspensions in the Taylor-Couette Flow

The linear stability analysis of the fiber suspension Taylor-Couette flow against axisymmetric and non-axisymmetric disturbances is investigated. A generalized complex eigenvalue problem generated from the linearized set of the three-dimensional governing system equations around the basic Couette azimuthal solution are solved numerically with the Chebyshev spectral method. In a wide range of radius ratios and the magnitudes of counter rotating, critical bifurcation thresholds from the axisymmetric Couette flow to the flow with different azimuthal wave numbers are obtained. The complex dispersion relations of the linearized stability equation system for vortex patterns with different azimuthal wave number are calculated for real axial wave numbers for axially extended vortex structures.     

Evaluation of crustal stability in the lower Yellow River region

The evaluation of crustal stability in the lower Yellow River region （LYR） is vital to the engineering project of the Yellow River embankment and construction of the Huang-Huai-Hai plain. From three aspects of tectonic stability, surface stability and medium stability, this paper analyzes in detail the factors such as tectonic activity, tectonic stress, exogenic geologic disasters and ground material composition which affect crustal stability in the study area and the spatial distributions of subordinate degree of the evaluation factors. It then constructs the crustal stability evaluation model of fuzzy mathematics based on GIS by using AHP to among the various factors find out the interrelations comprehensive evaluation of fuzzy mathematics and the functions of spatial overlay and spatial database of ArcGIS. Based upon this the crustal stability evaluation is carried out, the results show that the areas of stable and instable regions are basically the same, being 48.72% and 51.28% respectively. The area of moderate stable region takes a dominant position, being about 70.58%. The crust instable region presents zonal distribution and under the control of tectonic faulted belt. But the region where the LYR runs across is mostly instable with a total length of the river being 568.345 km, or 78.93% of all. The river reach where the crustal stability is relatively poor along the LYR is concentrated in the Lankao-Dongping section.     

Research on the stability of control systems described by fractional-order transfer functions

The stability of control systems described by fractional-order transier functlon torm is mainly investigated. The stability analysis of integer-order linear systems was extended to the fractional-order control systems. The stability definition of fractional-order linear control systems is presented in temas of the Lyapunov‘ s stability theory. Using the theorems of the Mittag-Eeffler function in two parameters directly derives the stability conclusion. The illustrative examples are also given by simulation results.     

Optimization of block-floating-point realizations for digital controllers with finite-word-length considerations

The closed-loop stability issue of finite-precision realizations was investigated for digital control-lers implemented in block-floating-point format. The controller coefficient perturbation was analyzed resultingfrom using finite word length (FWL) block-floating-point representation scheme. A block-floating-point FWL closed-loop stability measure was derived which considers both the dynamic range and precision. To facilitate the design of optimal finite-precision controller realizations, a computationally tractable block-floating-point FWL closed-loop stability measure was then introduced and the method of computing the value of this measure for a given controller realization was developed. The optimal controller realization is defined as the solution that maximizes the corresponding measure, and a numerical optimization approach was adopted to solve the resulting optimal realization problem. A numerical example was used to illustrate the design procedure and to compare the optimal controller realization with the initial realization.     

Predictive PI control and its robust stability analysis

It is difficult to analyze robust BIBO stability of predictive PI (PPI) control system due to the time delay of process. A novel technique for stability analysis of this kind of system is proposed in this paper. The state space form of PPI control system is given, and its characteristic polynomial is derived by a simple method, so the robust BIBO stability analysis of this kind of system is also the analysis of corresponding characteristic polynomial. Applying Kharitonov theorem and edge theorem, the BIBO stability can be judged for arbitrarily given process parameter intervals. The system robust stability with variable process parameters is described respectively, and some beneficial conclusions for the design of PPI controller are obtained.