Stability and oscillations of phase systems

Siberian Mathematical Journal -     

微分系统的稳定性

This article deals with the reflecting function of the differential systems.The results are applied to discuss the stability of these differential systems.     

Stability of indefinite systems

Absrtract  The paper considers a continuous system

Stability in Lyapunov systems

The stability of the stationary point of a Lyapunov system [Malkin IG, Some Problems in the Theory of Non-linear Oscillations. Moscow: Gostekhizd; 1956.], which describes the perturbed motion of a dynamical system with two degrees of freedom, is investigated. It is assumed that the characteristic equation of the first approximation of the system has two pairs of pure imaginary roots and that the quadratic part of the integral is not sign-definite. Both the non-resonance case, as well as cases of lower order (second-, third- and fourth-order) resonances are considered. The necessary and sufficient conditions for stability are given in cases when the problem is solved by a combination of the first non-linear terms of normal form.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Stability analysis of queueing systems

This paper contains a survey of some results on the stability of queueing systems obtained by the authors by means of the method of trial functions (developed from Lyapunov's direct method). In addition, the paper contains a series of new results on the stability of regenerative processes. All the qualitative statements are accompanied by the construction of quantitative estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 41–61, 1979.     

Stability of inner planetary systems

The model of the planar restricted problem of three bodies is used to evaluate the stability of the inner planets of planetary systems with arbitrary mass ratios. A quantitative measure of stability is introduced by finding the difference between the critical value of the Jacobian constant (at which bifurcation may occur) and the value of the Jacobian constant that corresponds to planetary type orbits. Hill's definition of stability is used according to which inner planetary orbits are stable if they are bounded in a region enclosing only the larger primary. For small values of the massparameter (<10^–3) the maximum value of the dimensionless radius of the orbit for Hill-stability is given by 1–2.4 µ^1/3.

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Zusammenfassung Die Stabilität von inneren Planetensystemen mit beliebigen Massenverhältnissen wird am Modell des ebenen restringierten Dreikörperproblems untersucht. Aufgrund der Differenz zwischen dem kritischen Wert der Jacobi-Konstanten (wo Bifurkation eintreten kann) und dem einer Planetenbahn entsprechenden Wert wird ein quantitatives Stabilitätsmass eingeführt. Dabei wird die Hillsche Stabilitätsdefinition verwendet, d.h. eine innere Planetenbahn heisst stabil, wenn sie ein nur den grösseren Zentralkörper enthaltendes Gebiet nicht verlassen kann. Für klein Werte des Massenparameters (<10^–3) beträgt der maximale (dimensionslose) Radius einer Hill-stabilen Planetenbahn 1–2.4 µ^1/3.

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Dedicated to Professor Eduard Stiefel     

Stability of nonholonomic Chaplygin systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1100–1106, August, 1989.     

Stability of uncertain discrete systems

The uncertain system

$x_{n + 1} = A_n x_n , n = 0,1,2, \ldots ,$

is considered, where the coefficients a _ij (n) of the m×m matrix A _n are functionals of any nature subject to the constraints

$\begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $

Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.By using a special Lyapunov function, a bound δ ≤ δ(α₀,α_*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a _i,j (n) = 0 for j < i.It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals.     

Stability of infinite-dimensional sampled-data systems

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space and the control space are Hilbert spaces, the system is of the form , where is the generator of a strongly continuous semigroup on , and the continuous time feedback is . The answer to the above question is known to be ``yes' if and are finite-dimensional spaces. In the infinite-dimensional case, if is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is ``yes', if is a bounded operator from into . Moreover, if is unbounded, we show that the answer ``yes' remains correct, provided that the semigroup generated by is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.

     

Stability of non-Markovian polling systems

A stationary regime for polling systems with general ergodic (G/G) arrival processes at each station is constructed. Mutual independence of the arrival processes is not required. It is shown that the stationary workload so constructed is minimal in the stochastic ordering sense. In the model considered the server switches from station to station in a Markovian fashion, and a specific service policy is applied to each queue. Our hypotheses cover the purely gated, thea-limited, the binomial-gated and other policies. As a by-product we obtain sufficient conditions for the stationary regime of aG/G/1/ queue with multiple server vacations (see Doshi [11]) to be ergodic.Work presented at the INRIA/ORSA Conference on Applied Probability in Engineering, Computer and Communication Sciences, Paris, June 16–18, 1993.     

Stability of difference rational systems

For systems of difference equations with rational functions on the right-hand sides represented in a unified vector matrix form, we obtain stability conditions and calculate a value of the radius of a disk for the domain of asymptotic stability on the basis of the second Lyapunov method. Kiev University, Kiev. Translated from Ukrainskii Matermaticheskii Zhurnal, Vol. 51, No. 3, pp. 428–431, March, 1999.     

Stability of solutions of pulsed systems

We present the principal results in the theory of stability of pulse differential equations obtained by mathematicians of the Kiev scientific school of nonlinear mechanics. We also present some results of foreign authors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 98–111, January, 1997.