Astability thorem for systems with discrete parameters and arbitrary circulatory forces

The stability of linear, time-independent dynamic systems with finite dimensions is studied by the direct Liapunov method. Sufficient and necessary conditions for stability in the sense of Liapunov are given for systems with arbitrary non-conservative circulatory forces.

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Zusammenfassung Die Stabilität endlich-dimensionaler, linearer, zeit-invarianter dynamischer Systeme wird mit Hilfe der direkten Liapunov'schen Methode untersucht. Notwendige und hinreichende Bedingungen für die Stabilität im Sinne von Liapunov werden für Systeme mit beliebigen zirkulatorischen Kräften angegeben.

     

On the stability of dissipative systems with circulatory forces

The stability of dissipative systems with circulatory forces is revisited. It is found that a general statement recently made by J. Teichmann [6] is erroneous and the validity of a Thompson-Tait-like statement can be derived for less severe conditions (semi-autonomous systems) than those of P. C. Müller [5].

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Zusammenfassung Die Stabilität eines dissipativen Systems mit zirkulatorischen Kräften wurde unterscht. Es stellt sich heraus, dass eine allgemeine Aussage von J. Teichmann [6] falsch ist, und dass die Gültigkeit einer Stabilitätsaussage in der Art von Thompson-Tait für eine weniger streng (halbautonome Systeme) Bedingung als in P. C. Müller [5] abgeleitet werden kann.

     

Lyapunov direct method for semidynamical systems

The stability of closed invariant sets of semidynamical systems defined on an arbitrary metric space is analyzed. The main theorems of Lyapunov’s second method for the uniform stability and uniform asymptotic stability (local and global) are stated. Illustrative examples are given.     

Second method of Lyapunov and comparison principle for systems with impulse effect

In the present paper questions of stability and boundedness of the solutions of systems with impulse effect at fixed moments with respect to a manifold are considered. The investigations are carried out by means of piecewise continuous vector-valued functions which are analogues of Lyapunov's functions. By means of a vector comparison equation and differential inequalities for piecewise continuous functions, theorems of stability and boundedness of the solutions of systems with impulses with respect to a manifold have been obtained.     

Second form of the generalized Hamilton-Jacobi method for nonholonomic dynamical systems

Summary New light is cast on the generalized Hamilton-Jacobi method for nonholonomic dynamical systems of Chetaev's type. As opposed to the method characterized by the introduction of unknown multipliers, the method presented here proceeds using independent variations and is illustrated by Appell's example.

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Résumé Des nouveaux résultats sont obtenus sur la méthode généralisée de Hamilton-Jacobi pour des systèmes dynamiques non-holonomes du type de Chetaev. Contrairement à la méthode caracterisée par l'introduction de multiplicateurs inconnus, la méthode présentée utilise des variations indépendantes et est illustrée par l'exemple d'Appell.

     

Direct Lyapunov method for hyperbolic systems on the plane with time-periodic coefficients

In the framework of the direct Lyapunov method, we establish tests for the exponential stability of solutions of boundary value problems (the mixed problem and the Cauchy problem for hyperbolic systems of the class in question) in the L ₂-norm. In these tests, the conditions on the derivative of the Lyapunov functional along the trajectories of the system are weakened compared with the known results for the case of any smooth coefficients. The main result is illustrated by the example of the mixed problem for the telegraph system with small friction periodically switched on.     

On the stochastic stability of systems with discrete parameters and arbitrary circulatory forces

The stochastic stability of linear dynamic systems with finite dimensions and arbitrary non-conservative circulatory forces is studied by the direct Liapunov method. Sufficient conditions for stability in probability are given for three stochastic models.

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Zusammenfassung Die stochastische Stabilität endlich-dimensionaler, linearer, dynamischer Systeme mit beliebigen nicht konservativen zirkulatorischen Kräften wird mit Hilfe der direkten Liapunovschen Methode untersucht. Hinreichende Bedingungen für Stabilität in Wahrscheinlichkeit werden für drei stochastische Modelle gegeben.

     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

The method of parametric Lyapunov functions for Markov dynamic systems

The notion of parametric Lyapunov function is introduced for Markov dynamic systems. The existence of a function of this kind is shown to be a necessary and sufficient condition for the strong stochastic stability of an equilibrium. In terms of parametric Lyapunov functions, a sufficient criterion is proved for asymptotic strong stochastic stability in the case of Feller Markov chains. Some examples are given showing the efficiency of the method proposed.     

Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors

This paper proposes a new method for computing the Lyapunov characteristic exponents for continuous dynamical systems. Algorithmic development is discussed and implementation details are outlined. Numerical examples illustrating the effectiveness and accuracy of the method are presented.     

Lyapunov function method in input reconstruction problems of systems with aftereffect

For certain classes of systems described by differential-functional equations, the author gives a review of algorithms for dynamical input reconstruction. The proposed algorithms are stable with respect to informational disturbances and computational errors and are based on the Lyapunov function methods and also on modification of the N. N. Krasovskii extremal aiming principle known in guaranteed control theory. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Optimizing Lyapunov functions for systems with structured uncertainties

The robust stability problem of a nominally linear system with nonlinear, time-varying structured perturbationsp _j ,j=1,...,q, is considered. The system is of the form $$\dot x = A_N x + \sum\limits_{j = 1}^q {p_j A_j x} .$$ When the Lyapunov direct method is utilized to solve the problem, the most frequently chosen Lyapunov function is some quadratic form. The paper presents a procedure of optimization of Lyapunov functions. Under some simple conditions, the weak convergence of the procedure is ensured, making the procedure effective in solving the robust stability problem. The procedure is simple, requiring only numerical routines such as inverting positive-definite symmetric matrices and determining the eigenvalues and eigenvectors of symmetric matrices. It is expected that the optimal Lyapunov function may be used in a robust linear feedback controller design. The examples demonstrate the effectiveness of the method. As shown when considering a system of dimension 24, the method is effective for large-scale systems.     

A direct method for calculating Lyapunov quantities of two-dimensional dynamical systems

We discuss cosmological models involving homogeneous and isotropic Yang-Mills (YM) fields. Such models were proposed recently as an alternative to scalar models of cosmic acceleration. There exists a unique SU(2) YM configuration (generalizable to larger gauge groups) whose energy-momentum tensor is homogeneous and isotropic in space. It is parameterized by a single scalar field with a quartic potential. In the case of the closed universe the coupled YM-doublet Higgs system admits homogeneous and isotropic configurations as well. While pure Einstein-Yang-Mills (EYM) cosmology with the standard conformally invariant YM action gives rise to the hot universe, Einstein-Yang-Mills-Higgs (EYMH) cosmology has a variety of regimes which include inflationary stages and bounces, and exhibits global cycling behavior reminiscent of the multiverse developed in time. We also discuss other mechanisms of conformal symmetry breaking such as the string-inspired Born-Infeld (BI) modification of the YM action or field-theoretical quantum corrections.