SUMMARY OF THE WORK 'REDUCIBLE SYSTEMS'

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200120024-1
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 5
• Document Creation Date: January 4, 2017
• Sequence Number: 24
• Publication Date: August 25, 1952
• Content Type: REPORT

Declassified in Part -Sanitized Copy Approved for Release 2012/04/26 :CIA-RDP82-000398000200120024-1
-STAY
1
i
SUI~iA~Y Q~' TI~I~~ W~FtK e 1~+~I)l1CxSL1 SYS~!'IJN~a ~
MNrM~M'NI,yrINNNMMrNrW W,rr~NY,N~IIygWY1lIMIMI~~MYIMIYIIMNNIIIdlllll
1
r
N. 1'~ Yerltt;'~rn
~ Source: Trudy Ma~Lc?inat~.chesko~o :Cns~itut~ iineni V. A~ S~ekl,ovae
XxIx (Privodimyye Sis~temy)arks oi' 1;he Ma~tk>.ematical Insti~;u~e
1
imen~. V. A. S~~sk~.ov. X~I ~ Reducible Systems ~
~ ~ ~
a e a er~bound mana raph).
b
6
na.x~ rad , 194 9 p ~ p p
Mop caw/L ~ ~ ~
STAY
I ~
j ~ ~ ,
r'
~ R;
;!y
r,: a
~~1
/y~
r~,
I
,I `f
P
t
Y
i
:i
_
1: r;
I: ~
.I,
E.' - _ 1
i
,keu
- :
J 1 i
c~
r
u

y '
Declassified in Part -Sanitized Co A roved for Release 2012J04/26 : CIA-RDP82-0003
PY pP _ 98000200120024-1

 Declassified in Part -Sanitized Copy Approved for Release 2012/04/26 :CIA-RDP82-000398000200120024-1
x
'
w w
r, II ~~yy.II A' rr II,~~ _ 1
~~~~Jrtl~~ ~iV~~l~~'14~r~~~~
I 1
atIMMA~Y Ole Ti~! ~ wQ1~K ~ ~;~UC~1'3~~~
SY~,~
S ~
j
~ N, P~ '~erug~t~
}gin a~sri~�~.an ua. e
ate; The fal~.aw:~n~ a,n~'armata.on ~.s tkis ar~.~~,na~. 2 l~a~e ~1 ~ S
summary found a.n the 9~wpa~;e R~asgian-langua~� want ent~.~l~1,ed f~'1'ra.vada.myy� Sistemy~~
~ed~7c%bla aystems~, reprosent~.ng No� ~T1;1� of the reg~u~.ar series Trud ~Matemati~cheska~c~
ek:l.ova Works of the Mathemat;tca,~. ,T.nstitute i.mctn9. V. A. Stek~.ov~,
,~nst~.tuta V. ~
ub1.~.sh~d 19~~6 a~~ Moscow/Zen~.ngrad; editors; CarrwMemb ~,cac~ Sc~. IJSSN,, T~ G. Pe~Lrovslc~.y;
P
i
and Professor S. M� Nikallska.y. `
:included below are tkae Foreword and 'f'able of Contents, g~.ven fa.rst~7
l
I
t
Foreword
i
r The invcsti~ation here is conducted on the baais of the theoz~y of funct~.ons of
i
matrices. .~1.1 the knowledge necessary f'ar this investigatioz~,bath relative to
theo of functions of mstrir..es and rel~~tive to the analytical theory of diff'eren-
the ry
ta. e nations c arl 1~ e found in Volume 11:I of V � ~ � Smirnov c s Caur. s e. ~~he s~ar~a.ng
` the definit~.on of r~educi~~le systems, w'ri ich was liven by ~,yapunov ~iapaunoff~
~ pa~.nt a.s
h~.s book General Problem of the ,~t bilk pf Mo ti n (aN~'~, :1,9~~, pale �
:tn the c4 letion of this worl.c T was given substantial ass~.stance by academician ,
V. ~ ~ Smx,T'Ilo~l'a ~~o whom l express m?Y profa'und gratitude.
I 'Fable of. Contents
i Pale
Foreword
' c Definitions and Thearer~~ ~
Chapter T r rasa.
19
Chapter Il. Sufficient Criteria of Reducibility
,

~ ~ ~ ,
' Aw;~d~ ul
n
~ ' ~ . - ~ i~li,
l
4
} ;
.f
i
11
S
~ ified in Part -Sanitized Copy Approved for Release 2012J04/26 :CIA-RDP82-00039800020
Declass
~ 0120024-1

 Declassified in Part -Sanitized Copy Approved for Release 2012/04/26 :CIA-RDP82-000398000200120024-1
- R - - - - - - - - - - -
I
.r r
~w
-i 5
1
i e
f
~a~~9
i
nvest:l at~.an o~ Cextain C~.aase~ of aY~tems o,~
Cha~tar ~
2~
J~~.f~'erQnt~.aw1, Ec~uat~.ox~~
;
Cha tar ~Vo Methad of Successive A~a~~rox~.mat~.o~as
A

* ~ f ~ eC~a~. Qrm
Chapter V~ Reducta.4n of bystemc o n
~ and ,~ddi�~~.ans 8g
Chapter VT. Various Comment.
9~
Summary (~glish~La~n~uaga )
g6
Contents
"UMMAI~Y En~~.ish~~,an~ua~e in the U~rig~.na1~
b
REDUC:CPZ~ SYS'fENIS '
N. Erou~i~n P. Yexugin~
P'Y
i
' er a s stern of homo~eneaus linear differential equations in the r
Cansa.d y
matr~i.c farm
dX
dt~~P
'i
'the matrix P are bounded continuous functions an the
where the elements. of
uce the new unkno~an matrix Y by means of the eeua~.ity
interval (U, oa ) . :Cntxad
C~)
y. x~
)
matrix ~ axe differenta.ab~.e,,,~"~ricta.ans on (0, ~ ,I
.where the elements of the .r~� `
r'~
we shai.~. have. an equat~.an of the form
dY (3) '
~ YB
dt
�
~ A
,
. ~ err ~ ~i
yq s
~ ~
J. ~1
} ~ ,~n p~,ldry}} 'Nb`i P! rn yr
N '1m
T ~
yy 1, ~
~ 1'�
~1YM77M~~V ' i Y k .
IR
~1 ,
Pil
1_
r
i
)
;
k
`Declassified in Part -Sanitized Copy Ap roved for Releas
, _ p e 2012J04/26 :CIA-RDP82-000398000200120024-1 '

 Declassified in Part -Sanitized Copy Approved for Release 2012/04/26 :CIA-RDP82-000398000200120024-1
f
i
,
,..oar w
4 ~ ~
I
I
;'I ~ apd dZ, , ~,'he system is said to be
where 'the matx~.x ~ depends ox~ P' ~
to ethex with d and bhe inverse {
~ dt
e there exists a matrix Z baun~ed, ~
reducibl.
hat al,l ~Q el,em~n~s a~' ~ in (3 } axe `
dc~term~.nant ~(Z~~')~ an (a~ such ~ ,
of the
n~,e rab~.~me genera. de staba.e.ite du
~
canstanta A~ ~,~.apaunaff ~,n he.s ~,amaus paper
07 eve the def~.n~.tion of reducib.~e systems
vement~' (Annales de 'loulousc~~ 19 ) ~
mou
a roblem of stabi~.ity of sal.utions o~ nanMlinear
and revealed thee.r rose for the p of
' ns. ;fin the present paper the general theory
systems of differential equate.o .
' ~ e reducibil~.ty a~' some systems of
stems is built, an the 'base of whe.ch
reduc~.bl,e sy
eneral. form is investigated. ~1e gave two
dif~'erential equations of considerably g
and make clear to what extent the
"'dent criteria of reducibility
necessary and su~~a.
sformatian matrix ~ is arbitrary.
Ch07.Ce 0~ the Iran the 50~.utian5 df
. ~ 'bilit characteri2es completely
The f irs~t cr~,terian of ,reduce. y ucibility.
rather simple necessary cand~.tion of red
~ reducible systems We also give a t of
stems from the viewpoe.n
methods for ini~estigation of the sy
We gave a number of
' to two groups the methods of successive
their reducibility, which care be dive.ded a.n
' n of the given systems.ta the systems of
a raximations and the methods of reducta.o
pp
' it of which can be easily establa,sheda
special. farm, the reduea.be.l y ~ it values at infinity
~I the s stem.5 with the matrices P hava.ng 1~.m
The reducibility of y
a and sufficient conditions of
' namely, same simple necess ry
d
e
udi
st
r
el
e.s complet y
lotions of such systems are re~~resented
'bilit of such systems are ge.veno Sol
reduce. Y
e of the independent variable t�
' ,
arias uni~'o~mly converging on the whole rang
as s
~ here the elements of the matrix.P .are
s takes places for instaxlce, e.n the case, w
Thi stem.
' rre u].ar singular point of the sy
x.
g
is an
analytic. functions and t : ~ w
,..,..4
M ~ ~ 1,
~I

i ~ ~
I~~
/~y~y y ~~y ,I
~ !~P4 1N 4b~, I~~~Y Y~'~ ;f~l Mel ~q', ~ Y~
/ tle ~
~ ~
~
' ~ {~w4R
k ;I
i! ~
V'
- ti
s';
- ~ +
r.
,
Declassified in Part -Sanitized Copy Approved for Release 2012J04/26 :CIA-RDP82-000398000200120024-1

 Declassified in Part -Sanitized Copy Approved for Release 2012/04/26 :CIA-RDP82-000398000200120024-1
- ;,~~i
' ~
;t
;
4
R
S.
y P ~~1~~~~ t~
'a~ 11
Carta~.n ~ystemsy whose coef~'icisnts have no ~.~,m~,t va~.ues at in~ina.ty, are a~.sv
investi~ated~
,r
1~e have aa.sa made c~.oar undex what variata,ons a~' the coa~~'ic~,+ants at ~nf~,n~,ty
the reducib~,~.~.ty off' the system and the xeduced system rest invariant.
We have succeeded in generali~in~ the nation of reducibi~.~.ty to the case, where
the coe~:~icients off' the system are not bounded. This has enab~,ed us to establish
the existence of bounded solutions of such systems and to repres~;nt the so~ut~.ans as
the whoJ.e a.ni'inzte ran e of the rode endent vax~.able~
ser~.es unifaarm:l.y converging on ~ ~
;~i
'I
r p
. ,r,
iti'
~
y~,
i
_ ~ 1
al~r"~ .
44 4th
~Mr�~
~ ~
(f~~,~
'

I.

7 t,
~ ~ ~ i V '~i.
1 i
,l
~ Declassified in Part -Sanitized Co A roved for Release 2012J04/26 :CIA-RDP82-000398000200120024-1
PY pP