On stability in one critical case : PMM vol. 43, no. 6, 1979, pp. 963–969

The problem on the stability of the trivial solution of an autonomous system of ordinary differential equations is solved in the critical case of one zero root, m pairs of pure imaginary roots, and q roots with negative real parts. It is proved that the presence of the zero root, as a rule, leads to instability, which can be detected already from the form of the second-order series expansion of the right hand sides of the equations. In the degenerate case necessary and sufficient stability conditions have been indicated for a model (simplified)system; it is shown that the absence of additional degeneracy the instability of the original system follows from that of the model. Sufficient conditions for the asymptotic stability and instability of the original system have been obtained under the fulfilment of the necessary stability conditions for the model system.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

On the spectrum for the artificial compressible system

Stability of stationary solutions of the incompressible Navier–Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number ? which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small ?. The result is applied to the Taylor problem.     

A problem in the theory of mixing layers

The classical problem of the free steady mixing layer which is formed as the result of the interaction between two parallel homogeneous flows which move with different velocities and come into contact in a certain section is considered. Subject to the additional condition that the first derivative of the solution in a class of self-similar functions is positive, a boundary-value problem is studied, for values of the self-similarity index m > 0, which describes the mixing of two viscous streams of the same fluid for m = 1 [1] and for m = 2 [2]. The method of investigation used [3–5] enables the third-order non-linear equation to be reduced to a first-order equation and enables the corresponding solutions (Gz) to be constructed in a parametric form as a function of the values of m. A knowledge of the behaviour of the velocity profile of the main stream can be used to investigate the flow stability. The results obtained form the basis of the subsequent construction of the solution of Lock's problem [6] and the investigation of the uniqueness of the solutions obtained.     

On the stability of the steady-state motions of systems with quasicyclic coordinates

The stability of the steady-state motions of a system with quasicyclic coordinates under the action of potential and dissipative forces and also forces which depend on the quasicyclic velocities is investigated. The results are applied to the problem of the stability of the steadystate plane-parallel motions of a rotor on a shaft which is set up in elasticated bearings with a non-linear reaction /1/.

The stability of the stationary motions and relative equilibria of systems with a single cyclic (quasicyclic) coordinate has previously been investigated /2/ from a common point of view. The question of the stability of the stationary motions of systems with quasicyclic coordinates under the action of constant and dissipative forces has been considered in /3/. The results obtained in /2/ have been generalized /4/ to systems with several cyclic (quasicyclic) coordinates and, additionally, a third regime of uniform motions, which includes the regime considered in /3/, has also been investigated.     

Stability of nonlinear viscoelastic systems under stochastic excitation

The dynamic behaviour of viscoelastic system with due account of finite deflections but under condition of small strains is described by the system of nonlinear integro-differential equations. On an example of a thin plate subjected to loads, which are assumed as random wide-band stationary noises and applied in the plate plane, the stability of nonlinear systems is considered. The stability in a case of finite deflections of the plate is considered as stability with respect to statistical moments of perturbations and almost sure stability. For the solution of the problem, a numerical method is offered, which is based on the statistical simulation of input stochastic stationary processes, which are assumed in the form of Gaussian ”colored” noises, and on the numerical solution of integro-differential or differential equations. The conclusion about the stability of the considered system is made on the basis of Lyapunov exponents. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stability of a class of reversible systems

The problem of the stability of the point of rest of an autonomous system of ordinary differential equations from a class of reversible systems [1] characterized by the critical case of m zero roots and n pairs of pure imaginary roots is considered. When there are no internal resonances [2, 3], the point of rest always has Birkhoff complete stability [2]. Internal resonances may lead to Lyapunov instability. The conditions of stability and instability of the model system when there are third-order resonances may be obtained from a criterion previously developed [4] for the case of pure imaginary roots. The results are used to analyse the stability of the translational-rotational motion of an active artificial satellite in a non-Keplerian circular orbit, including a geostationary satellite in any latitude [4, 5]. The region of stability of relative equilibria and regular precession of the satellite is constructed assuming a central gravitational field and the resonance modes are analysed.     

On the stability of projected dynamical systems

A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by Dupuis and Nagurney (Ref. 1). This paper initiates the study of the stability of such PDS around their stationary points and thus gives rise to the study of the dynamical stability of VIP solutions. Examples are constructed showing that such a study can be quite distinct from the classical stability study for dynamical systems (DS). We give the definition of a regular solution to a VIP and introduce the concept of a minimal face flow induced by a PDS, which is a standard DS of a lower dimension. We then show that, at the regular solutions of the VIP, the local stability of the PDS is essentially the same as that of its minimal face flow. Hence, we reduce the problem, in this case, to one of the classical stability study of DS, a more developed discipline. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.This research was supported by the National Science Foundation under Grant DMS-9024071 under the Faculty Awards for Women Program. This support is gratefully acknowledged.     

A control scheme based on ER-materials for vibration attenuation of dynamical systems

Based on a bang-bang control scheme acting on so called “electrorheological” fluids (ER-fluids), a vibration suppression method is proposed for a class of n-dimensional systems subjected to unknown perturbations. The proposed controller relates to robustness vis-a-vis unknown but bounded disturbances. Two approaches for designing the control scheme are presented and compared. On the one hand we employ Lyapunov stability theory; on the other hand there is an obvious reason for minimizing rate of energy change due to the spring/damper elements by varying the ER-fluid properties appropriately. The system under investigation is an n-degree of freedom one consisting of masses and spring/damper elements. The spring/damper elements contain an ER-fluid; their stiffness and damping properties are changed by varying an imposed electrical field. The changes in spring and damping properties can be effected in microseconds since the control does not involve the separate dynamics (inertia) of usual actuators. Detailed derivations are presented for a two-dimensional case and simulations are carried out for examples of smooth periodic and discontinuous periodic excitation forces.     

Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.     

THE STABILITY OF STATIONARY SOLUTION FOR OUTFLOW PROBLEM ON THE NAVIER-STOKES-POISSON SYSTEM

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.     

Feasibility of time-limited control of a competition system with impulsive harvest

A kind of time-limited control model on a competition system with impulsive harvest, described by impulsive differential equation with the initial and boundary value problem, is presented. The existence of solution of the model, corresponding to the feasibility of the short-term control, is discussed. By the comparison principle, the conditions under which the model has a solution are found by a series of the upper solutions, and the conditions under which the model has no solution are also given by a series of the lower solutions. Finally, the practical meanings of those conditions are explained. As an example, if other parameters are given, the times of the impulsive control is estimated and the theoretical results are verified by numerical simulations.     

Stabilization of a quasi-conservative system subjected to high frequency excitation

The existence and stability conditions for the steady motions and equilibrium positions of non-linear quasi-conservative systems with fast external perturbations having quasi-periodic and random components are investigated. A change of variables is proposed which reduces Lagrange's equations of the system to standard form. It is shown the averaged system of the first approximation has a canonical form and the effect of fast perturbations (not necessarily potential) is equivalent to a change in the system's potential. This leads to stabilization of unstable equilibrium positions and to the appearance of additional stationary points different from the equilibrium positions of the unperturbed system. The approach used is illustrated by examples; the stability of a pendulum on an elastic suspension when there is suspension point, and the steady motion of a sphere subjected to a high-frequency load. The critical loading of a double pendulum loaded by a pulsating tracking force is estimated. A form of wide-band random perturbations capable of stabilizing the system is considered.     

Optimal stationary control for dynamic systems with Markov perturbations

Affine continuous and discrete-time dynamic systems with homogeneous jump Markov perturbations are considered and the existence of an optimal stationary control under a quadratic cost is discussed. In order to solve this problem some new stability results for linear systems with Markov perturbations are given.     

Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).     

An algorithm for routing messages between processing elements in a multiprocessor system which tolerates a maximal number of faulty links

An algorithm for routing messages between processing elements in a multiprocessor system is proposed. The algorithm can always be performed when the system is a connected graph. Comparisons are made with some known algorithms. The algorithm has been illustrated on an n-dimensional hypercube but can be applied on an arbitrary connected multiprocessor system.     

On stability of autonomous systems with internal resonance : PMM vol. 39, n 6, 1975, pp. 974–984

When investigating the stability of the trivial solution of an autonomous system of ordinary differential equations in the critical case of n pairs of pure imaginary roots an essential role can be played by the presence of integral linear dependences between the system's frequencies or, in other words, by the internal resonance. Various special cases of this problem were examined in [1–6]. Our aims are: to obtain a special (normal) form of the differential equation system with internal resonance of most general form in it; to ascertain the conditions under which the presence of internal resonance does not permit the application stability investigation methods developed for resonance-free systems; to solve the stability problem in one of the most important cases of odd-order internal resonance, generalizing the preceding investigations. In the solution of the last problem the necessary and sufficient conditions are given for the stability of the model (simplified) system. Using Chetaev's theorem we show that as a rule the instability of the original system follows from the Instability of the model system. Cases of structurally-unstable instability (^*) for which the model system does not resolve the problem of stability are outlined. The results obtained are extended, in particular, to Hamiltonian systems.     

Duality and existence theory for nondifferentiable programming

Necessary and sufficient conditions of optimality are given for a nonlinear nondifferentiable program, where the constraints are defined via closed convex cones and their polars. These results are then used to obtain an existence theorem for the corresponding stationary point problem, under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the programming problem. Furthermore, a dual problem is defined, and a strong duality theorem is obtained under the assumption that the constraint sets of the primal and dual problems are nonempty.     

Stability of the nonconstant stationary solution to the Poisson–Nernst–Planck–Navier–Stokes equations

We study the three-dimensional Cauchy problem of the Poisson–Nernst–Planck–Navier–Stokes equations. We first show that the corresponding stationary system has a unique semi-trivial solution under a general doping profile. Under initial small perturbations around such the semi-trivial stationary solution and small doping profile, we obtain the unique global-in-time solution to the non-stationary system. Moreover, we prove the asymptotic convergence of the solution toward the semi-trivial stationary solution as time tends to infinity.