Asymptotic stability

It has been possible to study the asymptotic stability of the system, i.e., the manner in which the fluctuations of motion of the lead car are propagated down the line of cars. The steady state behavior is easily derived. Because of velocity control between the following cars, a steady state is eventually reaohed in which each car moves with speed u, and, hence, A = 0 i.e., the study of the system involves no time lag. 

If C is orbitally stable and, in addition, the distance between B and C tends to zero as t - oo, this form of stability is called asymptotic orbital stability. 

For a systematic account, see A. M. Liapounov, General Problem of Stability of Motion, Charkov, 1892 W. Hahn, Theorie und Anwendungen der directen Methode von Liapounov, Springer, Berlin, 1959 L. Cesari, Asymptotic Behavior and Stability Problems, Springer, Berlin, 1959. 

Since xi0 is the known solution and xt is the perturbed solution, an important case arises when <( ) - 0 for t - oo. In such a case the stability is asymptotic. 

A slight modification of the preceding theorem permits establishment of the condition of asymptotic stability ... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Asymptotic stability. Let us describe rather mild conditions under which... 

This provides support for the view that the solution is completely distorted. From such reasoning it seems clear that asymptotic stability of a given scheme is intimately connected with its accuracy. When asymptotic stability is disturbed, accuracy losses may occur for large values of time. On the other hand, the forward difference scheme with cr = 1 is asymptotically stable for any r and its accuracy becomes worse with increasing tj, because its order in t is equal to 1. In practical implementations the further retention of a prescribed accuracy is possible to the same value for which the explicit scheme is applicable. Hence, it is not expedient to use the forward difference scheme for solving problem (1) on the large time intervals. 

The scheme of second-order accuracy (unconditionally stable in the asymptotic sense). Before taking up the general case, our starting point is the existing scheme of order 2 for the heat conduction equation possessing the unconditional asymptotic stability and having the form... 

Asymptotic stability of the three-layer scheme. The object of investigation here is the three-layer scheme... 

The shape of the magnitude plot resembles that of a PI controller, but with an upper limit on the low frequency asymptote. We can infer that the phase-lag compensator could be more stabilizing than a PI controller with very slow systems.1 The notch-shaped phase angle plot of the phase-lag compensator is quite different from that of a PI controller. The phase lag starts at 0° versus -90°... 

For the pseudo-binary mixture (a = 0.5) of sulfonate and nonylphenol with 30 E.O., figure 2 shows how the concentration of each of their monomer calculated by the RST theory (1), varies as a function of the overall surfactant concentration. It can be expected that the asymptotic regime in which monomer concentrations are stabilized will correspond to a plateau of the adsorption isotherm for the surfactant mixtures considered. 

Plume stability Plume is stable asymptotic conditions reached... 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

Aiming at a more formal analysis, the asymptotic stability of a steady-state value S° of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value... 

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