General Stability Theory Research Articles

Robust Composite H∞ Synchronization of Markov Jump Reaction-Diffusion Neural Networks via a Disturbance Observer-Based Method.

This article focuses on the composite H∞ synchronization problem for jumping reaction-diffusion neural networks (NNs) with multiple kinds of disturbances. Due to the existence of disturbance effects, the performance of the aforementioned system would be degraded; therefore, improving the control performance of closed-loop NNs is the main goal of this article. Notably, for these disturbances, one of them can be described as a norm-bounded, and the other is generated by an exogenous model. In order to reject the above one kind of disturbance, a disturbance observer is developed. Furthermore, combining the disturbance observer approach and conventional state-feedback control scheme, a composite disturbance rejection controller is specifically designed to compensate for the influences of the disturbances. Then, some criteria are established based on the general Lyapunov stability theory, which can ensure that the synchronization error system is stochastically stable and satisfies a fixed H∞ performance level. A simulation example is finally presented to verify the availability of our developed method.

Вплив конструктивних і експлуатаційних факторів на стійкість руху автопоїзда з причепом категорії О1

The issues of stability and controllability of cars and road trains (ATZ) in connection with their practical importance have been studied by many researchers. If the issues of transportation of bulky goods are studied well enough, then for the transportation of light cargo, which uses low-capacity road trains with simple single- or two-axle trailers, the issues of stability are not fully studied. Recommendations for changing the design parameters to improve a particular type of stability are different and sometimes contradict the requirements for other performance of the car, in particular, its smoothness and passability. Qualitative assessment of stability is performed on the basis of the general theory of stability of motion AM Lyapunov. Steady motion according to Lyapunov is realized in a previously unknown region of initial perturbations, which are called the region of attraction of undisturbed motion. There is a problem of defining the boundaries of this area. To solve it, the system of equations of plane-parallel motion of a road train with a trailer of category O1 was improved, from which the critical speed of rectilinear motion and the rate of oscillation instability were determined, and the factors influencing its numerical value were analyzed. The analysis of calculations showed that in the event of perturbation at speeds up to 31 m / s (30.8 m / s) the nature of changes in the lateral and angular velocities of the traction vehicle during the transient process is extinguished. Under the same conditions, when the velocity increases above 31 m / s (31.5 m / s), the nature of the change in lateral and angular velocities also fades, but there are fluctuations. At increase in speed to 32 m / s fluctuations of parameters of movement will be divergent that will lead to loss of stability and impossibility of the further movement. This movement should be considered unstable, which should be avoided. The stability of motion can be judged to a greater extent by the magnitude of the lateral accelerations acting in the center of mass of the individual links. The stability of the road train under consideration at nominal parameters of loading, air pressure in tires of all wheels, symmetrical loading of the trailer, loading on the traction coupling device within 500 N is provided, in particular critical speed of rectilinear movement made 33,97 m / s. oscillating instability 31.5 m / s), which is much higher than the normalized value of the maximum speed for road trains with trailers of category O1 (25 m / s), the maximum lateral acceleration when performing maneuvers such as circular motion, steering wheel jerk and "shift" is not exceeded the maximum allowable 0.45g.

Dynamic Stability Enhancement Through the Application of Stabilizers of Electromechanical Oscillations

Power system dynamic stability is one of key issues system engineers face. Oscillations that regularly occur in the system, limit the transmission capability of the network. The need to study the stability of power systems has been increasingly growing along with the development of power systems and their grouping into large interconnections. The focus of this paper is determining the dynamic stability of a synchronous generator, and thus the power system, by applying the general theory of stability of dynamic systems. Furthermore, the procedure for the initial adjustment of the parameters of a conventional (IEEE3 type PSS1A) stabilizer of electromechanical oscillations is briefly described based on the frequency response analysis of a linear generator model also known as the Heffron-Phillips generator model.

On the stability of equilibrium states of the dynamical systems in critical cases

The stability of equilibrium states described by an autonomous system of linear differential equations with constant real coefficients is studied. The cases when, among the eigenvalues of the matrix of coefficients of the system, there are simple or multiple zero eigenvalues or complex eigenvalues with zero real part (critical cases of the Lyapunov’s general theory of stability) are discussed in detail. Stability criteria for the equilibrium states in critical cases are formulated using only information about the order of the matrix of coefficients, rank of the corresponding λ-matrix, and the multiplicity of the corresponding eigenvalues without reducing to the Jordan canonical form. Particular examples of the use of the proposed stability criteria are given. The dynamics of phase transitions due to thermal fluctuations for systems described by the Landau-type kinetic potential is discussed separately. An identity is given for matrices with a zero determinant and a given rank, which can be used to analyze the stability of solutions of linear dynamical systems.

Finite-time stabilization of stochastic low-order nonlinear systems with time-varying orders and FT-SISS inverse dynamics

This paper solves finite-time stabilization problem of stochastic low-order nonlinear systems with time-varying orders and stochastic inverse dynamics. Due to stochastic low-order systems being merely continuous but not smooth, and the appearance of time-varying orders and stochastic inverse dynamics, all of control methods of stochastic nonlinear systems are inapplicable. By characterizing unmeasured stochastic inverse dynamics with finite-time stochastic input-to-state stability (FT-SISS) and applying more general stochastic finite-time stability theory, a new design and analysis method is proposed to guarantee almost sure boundedness of all the closed-loop signals and stochastic finite-time stability of stochastic low-order nonlinear systems with time-varying orders and FT-SISS inverse dynamics.

ОБЩАЯ ТЕОРИЯ УСТОЙЧИВОСТИ РАЗВИТИЯ СОЦИАЛЬНО-ЭКОНОМИЧЕСКИХ СИСТЕМ: ОСНОВНЫЕ ПОЛОЖЕНИЯ, ПРИНЦИПЫ И ПРЕДПОСЫЛКИ

At a turn of the 20th and 21st centuries, with updating of a problem of improvement of theoretical bases of safety of Russia of the Post-Soviet period in the conditions of transition to market economic model of managing and new forms of government there was a need for formation and formation of the new direction for the general science about management in social and economic systems which received the name "general theory of stability of development of social and economic systems". The specified direction is further development of traditional theories of national and economic security of social and economic systems which also those which have the state sovereignty (country) that actually opens the new scientific direction of researches of such systems are among.

Innovation and Investment Activities in Ensuring the Sustainability of the Functioning and Development of Agricultural Organizations

The study of the process of investment of innovative activities aimed at the internal development of organizations of the agro-industrial complex (agro-industrial complex) in order to obtain a sustainable profit in the medium and long term or other useful effect is relevant in modern economic conditions. The purpose of the work is to determine the value of innovation and investment activities in ensuring the sustainability of the functioning and development of agricultural organizations. The main results of the study were obtained and justified using General scientific methods: ascent from abstract to concrete, comparative analysis, the method of terminological analysis. The qualitative theory of differential equations is used as special tools. Research of stability of innovation and investment systems is based on the classical provisions of the General theory of stability of movement. The essence of functioning and development of innovative and investment activity of the organizations of agrarian and industrial complex as open system is analyzed. It is confirmed that the functioning internally generates development, determined the content of sustainable development for innovation and investment system of agricultural organizations, which reflects the economic, social and environmental results of development. Scientific and methodological provisions concerning the issues of functioning and development of agroindustrial complex organizations, taking into account their specificity, are expanded, which leads to an integrated approach to ensuring.

NONLINEAR BEHAVIOR CALCULATION ALGORITHM FOR THIN-WALLED SYSTEMS

Objectives The emergence of modern high-strength materials leads to the creation of thin-walled structures in various fields of technology. To obtain the necessary information about their behavior under load, one should analyze all the characteristic features encountered at all stages of their loading - at the initial (initial) stage of their operation, taking into account one or more types of nonlinearities, find possible critical states and, depending on the type of stability loss, study the nature of the initial stage of postcritical deformation. Based on an algorithm combining approximate analytical and numerical methods, the article solves the model problem — studying the behavior of a thin-walled spherical shell under load.Method. The study is based on solving the nonlinear problem of determining the stress-strain state at the initial - axisymmetric stage of work; critical (bifurcation) load values; analysis of the nature of post-bifurcation behavior. The work uses a variant of the general theory of stability and postcritical behavior of structures previously developed by V.T. Coiter.Result. The solution of such a general problem associated with discontinuous phenomena is carried out on the basis of mathematical ideas formulated in the theory of branching solutions of nonlinear equations. The values of the coefficients characterizing the initial stage of the post-bifurcation behavior of the shells and, from a practical point of view, the relations between the critical and limiting values of the loads are obtained. It is shown that depending on the area of the shell surface part loaded by the distributed load, the nature of the initial stage of postcritical deformation changes not only quantitatively, but also qualitatively.Conclusion. The most effective in solving problems associated with discontinuous phenomena are combinations of approximate analytical ones - catastrophe theory and numerical methods that do not require complex, timeconsuming and significant amounts of computation. Analysis of the initial stage of the postbifurcation behavior of structures allows us to assess the degree of danger of reaching a critical state, which is achieved by taking into account the values of the corresponding reliability coefficients in the calculations.

Incomplete Iterative Implicit Schemes

Abstract In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

Alternating Triangular Schemes for Second-Order Evolution Equations

Schemes of the Samarskii alternating triangular method are based on splitting the problem operator into two operators that are conjugate to each other. When the Cauchy problem for a first-order evolution equation is solved approximately, this makes it possible to construct unconditionally stable two-component factorized splitting schemes. Explicit schemes are constructed for parabolic problems based on the alternating triangular method. The approximation properties can be improved by using three-level schemes. The main possibilities are indicated for constructing alternating triangular schemes for second-order evolution equations. New schemes are constructed based on the regularization of the standard alternating triangular schemes. The features of constructing alternating triangular schemes are pointed out for problems with many operator terms and for second-order evolution equations involving operator terms for the first time derivative. The study is based on the general stability (well-posedness) theory for operator-difference schemes.

New Mixed Elements for Maxwell Equations

New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal-continuous Lagrange elements of any order on simplexes in two- and three-dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations. Essentially optimal error bounds (only up to an arbitrarily small constant) are obtained for eigenvalues and for both singular and smooth solutions. Numerical experiments are performed to illustrate the theoretical results.

Two-level schemes for the advection equation

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax–Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank–Nicolson scheme) and fourth order. The conditionally stable implicit Lax–Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

Effect of vehicle motion stability after impact/crash on traffic safety

The article presents the application of general stability theory to the study of road traffic stability immediately after an impact (crash, collision). It turns out that when modelling a collision, vehicles can be treated as colliding masses and dynamical systems can be assigned to this phenomenon.

Splitting schemes for unsteady problems involving the grad-div operator

In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the solution vector. In this paper we discuss various possibilities to decouple the equations for the different components that result in unconditionally stable schemes. If the spatial discretization uses Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The stability analysis of these schemes is based on the general stability theory of additive operator-difference schemes developed by Samarskii and his collaborators. The results of the theoretical analysis are illustrated on a 2D numerical example with a smooth manufactured solution.

Splitting schemes with respect to physical processes for double-porosity poroelasticity problems

Abstract We consider unsteady poroelasticity problem in fractured porous medium within the classical Barenblatt double-porosity model. For numerical solution of double-porosity poroelasticity problems we construct splitting schemes with respect to physical processes, where transition to a new time level is associated with solving separate problem for the displacements and fluid pressures in pores and fractures. The stability of schemes is achieved by switching to three-level explicit-implicit difference scheme with some of the terms in the system of equations taken from the lower time level and by choosing a weight parameter used as a regularization parameter. The computational algorithm is based on the finite element approximation in space. The investigation of stability of splitting schemes is based on the general stability (well-posedness) theory of operator-difference schemes. A priori estimates for proposed splitting schemes and the standard two-level scheme are provided. The accuracy and stability of considered schemes are demonstrated by numerical experiments.

A theoretical model for the porosity–permeability relationship

An analytical permeability model is proposed for incompressible flow in porous media. Based on a general non-equilibrium thermodynamic stability theory, the model is derived from maximizing the entropy production rate to achieve a stable system. The resulting formula is in a form similar to previous empirical equations, but with calculable coefficients. Both three-dimensional lid-driven cavity and cubicle natural convection problems have been computed, via a modern computational fluid dynamics (CFD) solver, for numerical investigations of its performance.

Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain

The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985---1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finite-dimensional case. The current setting leads to simplified, more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our analysis makes use of standard tools from convex and variational analysis.

Postbuckling behavior of stringer versus ring reinforced geometrically imperfect cylindrical shells under pure torsion

This paper deals with the buckling and initial-postbuckling behavior and imperfection-sensitivity of stringer or ring stiffened cylindrical shells under pure torsion. In particular, the effectiveness of stringer versus ring reinforced shells is examined for various types of stiffener and shell parameters under different boundary condition. The initial-postbuckling single-mode analysis is performed using Koiter’s general theory of elastic stability. By changing the parameters of stringer or ring, we are able to find out how does the stringer or ring affecting the buckling and post-buckling behavior of cylindrical shells.

Explicit schemes for parabolic and hyperbolic equations

Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit–implicit splitting of the problem operator (Saul’yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes belong to the class of unconditionally stable schemes, but they demonstrate bad approximation properties. These explicit schemes are treated as schemes of the alternating triangle method and can be considered as factorized schemes where the problem operator is splitted into the sum of two operators that are adjoint to each other. Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for hyperbolic equations of second order. The study is based on the general theory of stability (well-posedness) for operator-difference schemes.

Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting

Via a general construction, we are able to establish a quite general and comprehensive stability theory for Feynman's operational calculus in the time independent setting. In particular, we are able to establish stability of the operational calculus with respect to general types of the time-ordering measures. While the domain of the operational calculus is somewhat restricted as compared to the "standard" version of the operational calculus (established by Jefferies and Johnson (Russ. J. Math. Phys. 8:153---171, 2001; Mat. Zametki 70:815---838, 2001; Adv. Appl. Clifford Algebras 11:239---264, 2001; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5:171---199, 2002), the advantages of this relatively minor domain restriction are significant in that the stability theory (with respect to the time-ordering measures) as it stands to this time is contained, essentially in its entirety, in the principle result of this paper, Theorem 2. Moreover, Theorem 2 allows immediate, and rather far-reaching extensions of the stability theory that, until now, have not been possible.