APPLICATION OF A.M. LYAPUNOV'S THEORY OF STABILITY TO THE THEORY OF DIFFERENTIAL EQUATIONS WITH SMALL MULTIPLIERS IN THE DERIVATIVES

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000100230011-4
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 5
• Document Creation Date: December 22, 2016
• Document Release Date: May 10, 2012
• Sequence Number: 11
• Publication Date: March 4, 1952
• Content Type: REPORT

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"Application of A. M. Lya
unov1s Theo
of Stability to the Theo
F
4 0~' Differential Equations with Sina11 multi liars in the Derivatives
i
I. S. Gradshteyn; Doklady Akadernii Nauk SSSR, Volume 81,
No. 6, pages 985-6+ Mp5C0~riwLeningrad: 21 December 1951.
STAT
STAT
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APPLICATION OF A. M. LY.APUNOV t S TFIEORY STABILITY TO THE
THEORY OF DIFFERENTIAL EQUATIONS WITH SMALL MULTIPLIERS IN THE DERSV T.i
I. S. Gradshteyn
~ote; the following article appeared in the regular Mathematics section
of the thrice-monthly Doklady Akadernii Nauk SSSR, Volume 81, No. 6 (21 December
1951), pages 98986. The contents of this article were read earlier at the
18 September 191 Session of the Moscow Mathematical Society and were printed
in the Uspekhi Matematicheskikh Nauk, Volume VI, No... 6 (46), Nov/Dec 19;1.7
In my article (2) I indicated the connection of the theory of stability
of differential equations with small multipliers (factors) in the derivatives
with the first method of A. M. Lyapunov iapounoff7 in his investigation of
the stability of motion. Further investigations have shown that this con-
nection concerns not only the first method but the entire theory of A. M.
Lyapunov (1).
1. Definitions. Let (y#, p*) represent a set of singular points of the
family of systems of equations
dYi/dt hi(Y,p) (i - 1,
(1)
where Y is a n..dimensional vector and p is az?dimensional vector-parameter.
For the points of the set R representing the part (properly or improperly)
of set # p ) let the solution of the system (1) possess the following
properties: for any e i 0 and 0 < a z e one can show O 0 such that for any system of family (1), the singular point
Declassified in Part - Sanitized Copy Approved for Release 2012/05/10 : CIA-RDP82-00039R000100230011-4
 Declassified in Part - Sanitized Copy Approved for Release 2012/05/10 : CIA-RDP82-00039R000100230011-4
of which enters R, from the inequalities
/(o,) y 1< r (i 1,2,...,n)
i i
for t ::32 0, follow the inequalities
/y(t)py/p -~.~T# jr the region
qs ,y, ) of its definition is con-
tinuous in all variables and possesses derivatives of the first order in q
continuous in all variables;
c) the function V and
are positive definite functions,
v f: h.(x*,Y + q,T#)
In such a case the family of systems of equations (1) or, what is the same,
Y=y +q,
defines motions that are uniformly asymptotically stable relative to the
compactum R.
The theorem of A. N. Tikhonov (1j,) on the existence is a partial case of
this theorem; corresponding to Lyapunov's function
11
n? w
Submitted 20 Oct 1951
Submitted. by Academician I. G. Petrovskiy
22 Oct 191.
-3-I
J
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Literature Cited
1.
A. M. Lyapunov.
General Problem Converning Stability of Motion. Moscow-
Leningrad; 1950.
2.
I. S. Gradshteyn.
Doklady Akademii Nauk SSSR 65, No. 6, 789 (191i.9).
3.
I. S. Gradshteyn.
Ibidem 66, Noo 5, 789 (19L9).
)4.
A. N. Tikhonav.
Matematickeskiy Sbornik Mathematical Symposium, 27
(69), No. 1, 1Li.7 (1950).
wpL
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