Showing papers on "Asymptotology published in 1967"

On the asymptotic behavior of solutions of the wave equations

Abstract: In the theory of scattering for hyperbolic equations, it is necessary to estimate the behavior of solutions to the unperturbed problem as well as the perturbed for large I t I. At present most estimates for the wave equation or the relativistic wave equation are in the sup norm. (See [l]-[5].) The purpose of this paper is to present some simple but rather interesting estimates in L2 of solutions to

On asymptotic stability in dynamical systems

Abstract: 0. Introduction. The basic theorems of the Lyapunov's Direct or Second Method as applied to stability and boundedness problems in ordinary differential equations are well known (see, e.g., [1], [2], [3], [4], [5]). Krasovskii [5], Hale [6], [7], and others have extended this method to functional differential equations, and Zubov [8], Auslander and Seibert [9], Bhatia [ 10], and some others have developed the same in the setup of a dynamical system defined on a locally compact metric space. In [9] Auslander and Seibert formalized the long suspected duality between stability and boundedness by restricting themselves to a locally compact separable metric space. In proving the converse theorems of the direct method for stability and asymptotic stability of an equilibrium point or a compact set M, one invariably proves the existence of a Lyapunov function with desired properties in a sufficiently small neighborhood of the given set. For converse theorems on global asymptotic stability and boundedness or ultimate boundedness in which an added condition is needed on the Lyapunov function a separate proof is generally given. An exception is the use of the proven duality in [9] to obtain theorems on boundedness and ultimate boundedness from those on stability and asymptotic stability. Now every asymptotically stable set has a region of attraction (the smallest invariant neighborhood of the set) which is an open neighborhood of the given set (see, e.g., [10]). One of the purposes of this note is to construct suitable Lyapunov functions on the complete region of attraction instead of in just a small neighborhood of the given set. This has the advantage that the theorems on global asymptotic stability and ultimate boundedness become immediate corollaries of the results we obtain. Our method of proof indirectly exploits the situation that if M be a

Asymptotic behavior of stieltjes transforms. i.

Abstract: Several theorems are proved concerning the asymptotic behavior of Stieltjes transforms as |z| approaches infinity, in a sector of the complex z plane which does not include the cut in the transform. The asymptotic behavior of the transform is related to the asymptotic behavior, for large values of the argument, of the function whose transform is taken.

The asymptotic representation of a class of $G$-functions for large parameter

Abstract: and for a, ß, a i, b¡ suitably restricted. (Our analysis will reveal that many of these restrictions may be dropped.) Since fi\\x) has an asymptotic representation in descending powers of \\x, (2) may be interpreted as a summation process which converts the generally divergent expansion into a convergent one. Important special cases of (2) yield expansions for the confluent hypergeometric function \\p(a, c; \\x) and Lommel functions. We will treat only the case Q — P — 1 > 0 since the case P + 1 ^ Q may be handled by an elementary analysis. In the former, n(M), as we shall see, has the unusual behavior of exponential decay as n —-> °o f in contrast to the latter case, where $>AM) behaves as inverse powers of w!, or at worst (P + 1 = Q), algebraically in n. In Section II, we first prove three lemmas; the first establishes an integral representation for $„(ö)(x); the second estimates for large n a closely related integral, and the third gives the desired asymptotic formula for $n(e)(X). Our main theorem follows when we find we can express <Ên(A/)(A) as a linear combination of the functions $„(e)[X exp (wi(Q M 2k))]. Section III is devoted to examples. There are quantities and assumptions about them which occur frequently in this paper, and they will always be as below :

A method of asymptotic expansions for singular perturbation problems with application in viscous flow

Abstract: Asymptotic expansion method for solving certain classes of singular perturbation problems with application in viscous flow

On estimating the region of asymptotic stability using Liapunov's method

Abstract: The problem of estimating the region of asymptotic stability is formulated as a minimization problem, and the Davidon search routine is applied to obtain the minimum. This approach is compared with the tangency method of Rodden for one example problem.

Experimental determination of the region of asymptotic stability by reverse-time simulation

Abstract: The experimental determination of the region of asymptotic stability of a second-order time-invariant system may be considerably simplified by taking advantage of the nature of the trajectories which form the boundary of the region. These trajectories are easily found by reverse-time simulation. Applications of the technique are presented.