On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)     

Stability of a growing viscoelastic rod subjected to ageing

The stability of a compressed growing rod of viscoelastic material that possesses the property of ageing /1/ is investigated. In conformity with the Chatayev definition of the stability of dynamic systems and the Lyapunov method described in /2/, stability conditions are obtained for a rod growing during a finite time interval, and in finite and semi-infinite time intervals. Some results of a numerical analysis of the behaviour of such a rod are presented in /3/.     

Stochasic stability of viscoewtic bars

This paper is devoted to the investigation of stability of viscoelsstic bars compressed by stochastic forces at infinite time interval, The problem of the bar buckling is considered in dynamic statement. Some sufficient conditions of mean square stability of viscoelastic bars are derived for arbitrary relaxation measure and different types of the end fixing     

Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

On the stability of rods for stochastic excitations

The stability of motion of an elastic rod in a viscous medium compressed by a randomly acting force is studied. The conditions of stability of the rod acted upon by a stationary process with bilinear spectral density are obtained. The dependence of the statistical moments of the amplitude of the finite flexure of the rod under stationary-motion conditions on the parameters of the compressing force and the amplitude of the initial deformation is analysed. A number of problems concerning the stability of longitudinal flexure of viscoelastic constructions acted upon by random loads were discussed in /1–3/.     

The solution of contact problems of creep theory for combined ageing foundations

Solutions are presented of certain plane and axisymmetric contact problems on the frictionless impression of a rigid stamp into a two-layered ageing viscoelastic foundation. It is assumed that the upper layer is thin relative to the contact domain, and inhomogeneously ageing. The rheological properties of the lower layer are described by the equations of linear creep theory for ageing materials. The layers are mutually rigidly adherent. The contact domain does not change with time. Depending on the relationships between the moduli of the instantaneous elastic strains of the layers, the mixed problems reduce to integral equations of the first or second kinds containing Fredholm and Volterra operators. An analytic method is proposed for solving such equations which enables an expansion to be obtained for the fundamental characteristics of the contact interaction for a force varying with time in an arbitrary manner and acting on the stamp. Cases are investigaged for the artificial and natural ageing of a two-layer foundation.     

A Synthesis of the Control of Dynamical Systems on the Basis of the Method of Lyapunov Functions

We consider the problem of the synthesis of a bounded control reducing a dynamical system to the given terminal state in a finite time. Two approaches to solve the problem, based on methods of the theory of stability of motion, are provided. One of them is applicable to nonlinear Lagrange mechanical systems with undetermined parameters, while another is applicable to linear systems. The characteristic property is that the Lyapunov functions are defined implicitly in both cases. We make a comparison between these approaches.     

Stability analysis of asset flow differential equations

I study the stability analysis of the solutions for the dynamical system of nonlinear asset flow differential equations (AFDEs) in three versions. I show that the previous two versions are not structurally stable mathematically because there are infinitely many critical points. It is important to reformulate a problem in order to eliminate any hypersensitivity in the mathematical model. I find that there is no critical point in the new version unless the chronic discount over the past finite time interval is zero.     

On approximate methods of analysing certain singularly-perturbed systems

A certainclass of sigularly-perturbed systems which have a variety of m-dimensional stationary positions is considered. When a small parameter disappears, the system also has an m-dimensional manifold of stationary positions and, therefore, the corresponding characteristic equation has m zero roots. The conditions under which the solution of a stability problem reduces to the same problem for a degenerate system are defined. As an application in practice gyroscopic stabilizing systems (the critical case corresponds to such systems) with elastic elements of high stiffness are discussed. The conditions under which the solution of the problem of the stability of steady motion follows from the solution of this problem for an ideal system (with absolutely rigid elements) are obtained. The problem of the closeness of the corresponding solutions of the complete and a simplified system of differential equations over an infinite time interval is discussed.     

Modes with switchings of increasing frequency in the problem of controlling a robot

Trajectories that are optimal with respect to high-speed response are constructed for a system for controlling a two-component manipulator (a robot). It is shown that when the initial conditions lie within a certain open region of the phase space, all optimal trajectories will have a segment of switchings of increasing frequency (SIF), i.e. a segment in which the control will undergo an infinite number of switchings in a finite time interval.

The synthesis of the optimal control in the R² plane containing the mode of SIF was first constructed by Fuller /1/. It was shown in /2/ that the synthesis is structurally stable in the sense that adding terms of higher order of smallness to the integrand and to the right-hand sides of the system of differential constraints does not affect the qualitative pattern of the optimal synthesis in the neighbourhood of the origin of coordinates.

The present paper explains that the synthesis in the problem of optimal control (relative to the high speed response) of the motion of the robot appears, in a certain sense, a direct product of the synthesis appearing in the Fuller problem and of the synthesis in the simplest problem of high-speed response (/3/, pp.38–47). The special aspect of the present paper consists of the proof of the proposition that switching surface is a piecewise-smooth manifold. The presence of the SIF mode is connected only with the fact that every trajectory intersects this surface an infinite number of times. In existing papers, the piecewise smoothness of the switching curve was proved for the two-dimensional problems using the SIF mode only for problems admitting of a one-parameter group of symmetries /1, 4–6/. A proof of the presence of SIF was given in /7, 8/.     

Global finite-time synchronization of different dimensional chaotic systems

This paper addresses the problem of global finite-time synchronization of two different dimensional chaotic systems. Firstly, the definition of global finite-time synchronization of different dimensional chaotic systems are introduced. Based on the finite-time stability methods, the controller is designed such that the chaotic systems are globally synchronized in a finite time. Then, some uncertain parameters are adopted in the chaotic systems, new control law and dynamical parameter estimation are proposed to guarantee that the global finite-time synchronization can be obtained. By considering a dynamical parameter designed in the controller, the adaptive updated controller is also designed to achieve the desired results. At last, the results of two different dimensional chaotic systems are also extended to two different dimensional networked chaotic systems. Finally, three numerical examples are given to verify the validity of the proposed methods.     

A general formalism for synchronization in finite dimensional dynamical systems

In this paper, an attempt is made to propose a general definition of synchronization for finite dimensional dynamical systems. The synchronization is defined here for two coupled dynamical systems with control inputs. Output functions of such systems are introduced to describe the systems’ properties on which the synchronization problem focus. Exact synchronization, asymptotic synchronization, and approximate synchronization are, respectively, defined by comparing the output functions in the corresponding ways. The definition here can also include chaos control and anti-control. The definition here covers various synchronization investigated in the references.     

On the construction and investigation of solutions of boundary value problems of thermoviscoelasticity

The uncoupled mixed boundary value problem of thermoviscoelasticity is considered in a quasistatic formulation. The temperature distribution is assumed nonstationary and inhomogeneous. The influence of the temperature on the viscoelastic properties of the material is taken into account by the introduction of a reduced time. The equations of state of the material are written in differential form as a system of kinetic equations in some tensor-type strain parameters. The system mentioned is equivalent to a Volterra integral equation with kernel in the form of a sum of exponents. The differential approach used is apparently more convenient for numerical realization /1/ (especially in nonuniform problems) and results in a substantially different mathematical formulation as compared with that based on the integral form of writing the equations of state investigated in /2,3/. Precisely for going over to the boundary value problem are the kinetic differential equations converted into an operator differential equation in Hubert space. The existence, uniqueness, and stability of the solution of the problem formulated are established, and conditions for the convergence of the Galerkin approximations and the stability of the difference approximations in time are formulated.     

The dynamic stability of an elastic column

Lyapunov's second method is used to investigate the stability of the rectilinear equilibrium modes of a non-linearly elastic thin rod (column) compressed at its end. Stability here is implied relative to certain integral characteristics, of the type of norms in Sobolev spaces; the analysis is carried out for all values of the problem parameter except the bifurcation values.

The realm of problems connected with the Lagrange-Dirichlet equilibrium stability theorem and its converse involves specific difficulties when considered in the infinite-dimensional case: stability in infinite-dimensional systems is investigated relative to certain integral characteristics such as norms /1/, and as the latter may be chosen with a certain degree of arbitrariness, different choices may result in different stability results. On the other hand, there is no relaxation of any of the difficulties encountered in the case of a finite number of degrees of freedom.

We shall consider a certain natural mechanical system with a finite number of degrees of freedom. If the first non-trivial form of the potential energy expansion is positive-definite, the equilibrium position is stable. A similar statement has been proved for infinitely many dimensions as well /1–3/, using Lyapunov's direct method, and the total energy may play the role of the Lyapunov function.

The situation with respect to instability is more complex. In the finite-dimensional case, if the first non-trivial form of the potential energy expansion may take negative values, instability may be demonstrated in many cases by means of a function proposed by Chetayev in /4/. A general theorem has been proved /1/ for instability in infinitely many dimensions, relying on an analogue of Chetayev's function. Such functions have also been used /5, 6/ to prove the instability of equilibrium in specific linear systems with an infinite number of degrees of freedom.

However, Chetayev's functions /4/ are not suitable tools to prove the instability of equilibrium in most non-linear systems. Another “Chetayev function”, which is actually a perturbed form of Chetayev's original function from /4/, has been proposed /7/, and it has been used to prove instability when the equilibrium position is an isolated critical point of the first non-trivial form of the potential energy expansion.

The majority of problems concerning the onset of instability of equilibrium configurations of elastic systems have been considered from a quasistatic point of view (see, e.g., /8, 9/). Problems of elastic stability and instability were considered in a dynamical setting in /2, 5/, where stability was investigated by Lyapunov's direct method. However, most of the results obtained in this branch of the field concern linear systems, and there are extremely few publications dealing with the onset of instability in non-linear elastic systems using Lyapunov's direct method. This is because in an unstable elastic system the quadratic part of the potential energy may change sign, and therefore the analogues of Chetayev's function from /4/ are not usually suitable for solving these problems. Dynamic instability has been studied or a specific non-linearly elastic system /10/, with the fact of instability established by using an analogue of the Chetayev function from /7/.

This paper presents one more example of a study of dynamic instability crried out for a non-linearly elastic system by Lyapunov's direct method.     

Inverse problems of reconstructing parameters of the Navier-Stokes system

An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem consisting of reconstruction of the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the results of approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For solution of the problem by this method, the concepts and constructions of open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the results of approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For solution of the problem by the dynamical method, the concepts and constructions of the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications and regularizations of the methods for solution of problems proposed that are based on one piece of a priori information or another about the desired solution and solvability conditions of the problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Stabilization of the motion of a dynamic system under conditions of uncertainty

The problem of the stabilization of the unperturbed motion of a dynamic system when there is incomplete information about the system parameters is considered. The solution is sought by Lyapunov's second method in the class of dynamic controllers and generalizes the result obtained in /1/ to controlled dynamic systems. Similar control problems were considered, in particular, in /2/.

The solution is used to stabilize the permanent rotation of a rigid body by a controlling moment with zero x-component /3/.     

Equations of the motion of linear viscoelastic fluids and the equations of the filtration of fluids with delay

A new variant of the equations of the motion of linear viscoelastic fluids, namely Maxwell, Oldroyd, and Kelvin-Voight fluids of arbitrary order, is indicated. This variant is especially convenient for the investigation of dynamical systems, generated by initial-boundary-value problems for these equations, and for the investigation of the hydrodynamic stability of the flow of these fluids.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 132–137, 1987.     

Finite‐time control of impulsive hybrid dynamical systems in pest management

This paper deals with the dynamics of a class of hybrid dynamical systems, which are subject to time‐dependent impulsive perturbations within a finite‐time interval and describe control strategies for integrated pest management. By using suitably defined Lyapunov functionals, sufficient conditions for the finite‐time contractive stability of the null solution are found by means of monotonicity arguments. Finally, a numerical simulation illustrates the theoretical analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

A Perturbation Result for Dynamical Contact Problems

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.