SCIENTIFIC ABSTRACT YAKUBOVICH, V. A. - YAKUBOVICH, V. S.

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• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

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YAKMOVICHO V. A.: WC P~V-G-YAtll Sci (diss) -- "A sYstem of lirwir differential
equations in canonic form with periodic coofficients". Leningral, 1959. 21 pp
(Leningrad order of Lanin State U im A. A. Zhdanov), 150 copies (yj,, yo it., 1959,
117)

YAMMOVICH, V.A. (Leningrad)
Small-parameter method for canonical systems with periodic coef-
ficients. Prikl. mat* i mekh, 23 no,D15-34 Ja-Y 159. (MrRA 1212)
(Differenital equations)

16(1)
.AUTHORs SOV/20-124-3-10/67
TITLEs Oscillation PropertJes of the Solutiona of Linear Canonical
SysUms of Differential Equations (Ostsillyati3ionnyye svoystva
resheniy liney-nykh kanonicheskikh sistem differentsiallnykh
uravneniy)
PERIODICALs Doklady Akademii nauk SSSR,1959,Vol 124,Nr 3,
PP 533-536 (USSR)
ABSTRACT: The author considers the system
dx ~ JH(t)xl where x is a
2k-dimensional vector,
H(t) - (13*
0 rk) unit matrix, H(t) . H(t)* a real symmetric
Ik 0 Ik
matrix, The author uses results of I.M. Gellfand, V.B. Lidskiy,
I.M. Glazman, L.D. Nikolenko etc, in order to formulate seven
theorems after having introduced many definitions; these
Card 1/2 theorems overlap with well-known results of Sternberg ~Ref 5.7

Oscillation Properties of the Solutions of Linear SOV20-124-3-lo/67
Canonical System of Differential Equations
and others or, however, already implicitly ozeur in other in-
vestigations.
There are 10 references, 7 of which are Sovietp 2 A-ierican,
and I Portuguese.
ASSOCIATIONt Nauchno-issledovatellskiyinstitut matematiki i mekbaniki
Leningradskogo gosudarstvennogo universitets. imeni A.A.Zhdanova
(scientific Research Institute for Mathematics and Mechanics
of the Leningrad State University imeni A.A. Zhdanov)
PRESENTEDs September 22, 1958, by V.I. Smirnov, Academ--3cian
SUBMITTEN September 15, 1958
Card 2/2

16(l)
AUTHOR: YakuboTioh; V.A. SOVI,!O,- 1 24-51-9/r6i:
TITLEs Candl--"ons for t1iLe Osc-~'Ilation a-ad Nonoscilla-;ion for Linear
Canonical Systeme of DLf-fererstial Equations (Tlaloviya kol.eba-
telgno3ti I nakilebatsl,nostl d1ya linsynykh kanonicheskikh
sistem diffefentsialln.ykn uravnerily)
PERIODICALs DoKlhay A'Aad.?mi! nauk 53SR,19599Voll 12491ir 5YPP 994-997 (USSR)
ABSTRAM The -aul;ho-r considers- the a7stew
,,4
-nF.1 voitor., H(t)-(
where x 'a a 2-k-dimene
0 T
H(i,)-H(C') real symme~-rie 2kx2lc matrix. Let x
0
0.!f.k be A.inpark.7 independent solutions of (,.)~ where it is
(jX,;Lh)-O. L,.t (U) be a kv2k matr-~x, the c,)Iumne of vhIch
J4 V
are Phe vectors k- Let X(t) be the matricant of M~
see5ef 1/ ; A.rg X(t).Arg det [-U(t),IV(t The equation (1)
Card 1/2 is called7oa~.,',Ilatlng, if &rg XN for t--~Aco is not bounded;

12
Conditions for the 0,9cillation and Nonoi-lillation for SOV/20-124-5-
Linear Canonical Systems of Differential Equations
(1) is called nonosji'llating, 1-f Arg X/M is bounded. In
eight theorems the author gi7es several conditions for the
oscillation and nono-3cillation in the above sense, e.g.s
'd0,,0 it is almost everywhere
Theoremg If starting from t ~
C4 0 and define the matrices H > 0,
0
M( 64) and the vectors a and m( 6) by
0 a - a , M(6)
K H0 + 9 0 X 0 0 H0 exp [K dT
0
m ( 6") 2 K M(6')& 0
S, LP (6)
Assume that for an 6 > 0 and all 6' it holds
(3) (G;I m( 6),b)- V(G; 1 m(6),m(6))-(G; 1 b, b) >,- 6 0> 0
For the stability of (1) in the large it is necessary and sufficient that
Card 2/ 5

W55
StabIlity-Conditione in the Larp for 3/02 60/135/001/006/030
Some Non-Linear Differential i,quations of 0111 222
Automatic Control
for all co e- 6'< + co it holds
(4) 2 O-)ao)dO,~ >. (HO-1 M( 6')aof M( er)ao)
Let X-1 b
(Ha u where H is the real solution of the inequation
2 -1 b b
(KH + EUC 4"N - /14,0- K. (Ha..- IS-)(Ha -W (la$H + Hab
with
- 1 2 I)b$ > 0
/,2 _ max kf 6,) 2
0 oo < 6< +
Theorem 2: Assume that the conditional equations (cf. (Ref.6))
Card 3/ 5'

8h655
Stability Conditions in the Large for S/020/60/135/001/006/030
S. o~e Non-Linear Differential Equations of CIII/C222
Automatic Control
(9) (KU + UK*) W /AV2 UU + I (Xub Ir + bu' Kt
0 S
Ua +.Ku b 0
2 2T
where /1111-0 is given by (8), have real solutions u, U U* for all vectors b'
being sufficiently neighboring to b. For the stability of (1) in the large
it is.':aufficient,and necessary-that
(10) U>O + 2(u,a)] ~'y(OdV>, (U-1 u'u)tp(6-)2
is satisfied.
Theorem~3-' contains stability conditions for the case where the-ooeffioients
of (1) depend on a parameter.
Theorem 4 is a rougher modification of theorem 3 which is more convenient
for stability control,
The aesumptions of the theorems 1 - 4 are always satinfied if ~ is
sufficiently large.
Card 4/5

84655
Stability-Cond-itions in the Large for S/020/60/135/001/006/030
Some Non-Lindar Differential Equations of C111/C222
Automatic Contr6l
The author mentions -Ayzerman, Ye.A. Barbashin, N.N. Krasovskiy and
i.Y&p1Mo-T,----T1Yere -:are 6 Soviet- refere7rues.
ASSOCIATION- Leningradskiv irnancin styennyy universitet imeni A.A. Zhdanova
raity imeni A.A. Zhdanov)
PRESENTE-Di June 6, -1.960, by V.I. Smirnov, Academician
SUBMITTED: May 23, 1960
Card 5/5

"Ifemiltonian systems Of linear differential equaticna with periodic
coefficients."
37ePort submitted for the Intl. SPTOsilun on Nonlinear Vibrations, IUPAMI
Kiev 12-18 Sep 1961
Acad. Sci. UYx SSR

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0 ill/ C 333
ATJTHORs Yakubovich, V. A.
Systems .- of --
TITLE: ii-neir--dEfferential equations of canonicel
form with periodic coefficients (Autoreview of the
dissertation)
pERIODICALs Uspekhi matematicheskikh nauk, Y. 16, no, it 1961,
223-234
TEXTi The dissertation has been maintained on June lit 1959 at the
session of the uchenyy soyet matematiko-mekhanicheskogo fakul'teta
Leningraaakogo universiteta (Scientific council of the Mathematical-
Mechanical Faculty of the Leningrad University), Opponentst V. J.
!3mirnov, Academician; M. G. Kreyn, Corresponding Member of the
Academy of :iciences, UkrSSR; G. Yu. Dzhanelid24 Doctor of Physico-
mathematical Sciences; '.I!. A. Krasnosel'Bkiy, Doctor of Physicomathe-
matical Sciences.
The resultshave been published by the author in (Ref. 17s Otsenka
kharakteristicheskikh pokazateley sistemy lineyn kh differentsiall
nykh uravneniy s periodicheskimi koeffitsientami Estimates of
characteristic exponents of a system of linear di(ferential equations
Card Va

22W
S/042/6i/ol6/001/007/007
Systems of linear differential ... C Ill/ C 333
with periodic coefficientsi , PMM .16, vYp, 4 (1954)); (Hof, 18:
Rasproutraneniye metodu Lyapunov op~-odeleniya organichenno3ti.
vesheniy uraviteniya Y" I- PkIY ~ 0, p(t+w) = p(t)na sluchay
znakoporemennoy fun) Extension of Lyapunov's method of
determining boundedness of solions of the equation Y01+p(t) Y - 0'
p(t+w) - p(t) to the case of a'function p(t) of variable signj
PMM 18, vyp. 6 (1954)); (Ref. 19: 0 sistemakh differentsialonykh
uravneniy kanonicbeekogo vida s periodicheskimi koeffitsientami
poryadka bol'she dvukh Eon systems of differential equations of
canonical fcrm with periodic coefficients of order > 2 ]DAN 1U3,
No-o' (1955)); kRef. 201 0 zavisimosti sobstvannykh znacheniy
samosopryazhennykh krayevykh zadach d1ya sistemy dvukh differentsial,
nykh uravneniy ot krayevykh usloviy L O~rl the dependence of the
eigenvalues of the boundary problem for the system of two -4,"feren-
tial equations on boundary conditions] Vestn. LGU, ser. matem.
mekhQ i aotrol No* 11 vyp. 1 (1957)); (Ref. 211 Rasprostraneniye
nekotorykh rezulltatav Lyapunova na lineynyye kayionichaskiye
sistemy s periodicheskimi koeffitBientami L Extension of some
results of Lyapunov to linear canonical systems with periodic coeffi-
cients PMR 21, vyp. 4 (1957)); (Ref. 22t Zamechaniye k nekotory-m
Card 2A

Systems of linear differential
22W
S/042/61/016/001/007/007
C 111/ C 333
rabotam po sistemam lineynykh differentsialinykh uravneniy a periodi-
cheskimi koeffitsientami CRemarks on some papers on linear systems
of differential equations with periodic coefficientai P Pxx 21, vyp. C
(1957)); (Ref. 23: Stroyeniye gruppy simplekticheskikh matri-ts i
i struktura mnozhestya neustoychivykh kanonicheskikh sistem differen-
tsialInykh uravneniy s periodicheskfzt kae-ffitsientami [Structure
of the group of symplectic matrices and of the set of unstable
canonical systems with periodic coefficients..] , Matem. ob..4A (86)t
3(195S));-(-Ref-.-24t Kriticheskiye chastoty kvazikanonicheskikh sistem
[Pritical frequencies of quasicanonical systems] , Vestn. LGU jj,
vyp, 3 (1958 )); (Ref. 25: Ostsillyatsionnyye avoystya resheniy
lineynykh kanonicheskikh uravnenly r- Oscillation properties of
solutions of linear canonical equatione3, DAN 123, No- 3 (1959));
(Ref. 261 Method malogo parametra dlya kanonicheskikh sistem s
periodicheakimi koeffitsientami. [Xethod. of the small parameter for
canonical systems with periodic coefficients3p PJW 2 1 vyp. 1 (1959));
(Ref. 27s Voprosy ustoychivosti. resheniy sistemy dvukh lineynykh
differentsialfnykh uravneniy kanonicheakogo vida a per-Lodicheskimi,
koeffitsientami 0
uestions of the stability of solutions of a
Card 3/3

22W,
S)1042/61/016/001/007/007
Systems of linear differential C 111/ C 333
system of two linear differential equations of canonical form with
periodic c oefficients-79 Hatem. ab . U (79), YYP. 1 (1955)).
A report on the results wits given among others in the seminaries of
V. J. Smir nov, Academician (Lening rad); N. G. Chetayev, Corresponding
Member of the Academy of Sciences USSR, Institut mekhaniki AN SSSR
(Institute of Mechanics AS USSR); L. S. Pontryagint Academician
(MJAN); V. V. Nemytskiy, Professor (MGU).
The author considers the systems
dx ~ JH (t) x (2)

where x is a column vector with the components p I'*"9Pkl q19 ... pqk
Otjh lj~hjjj 0 1k
H(t) j,h k)9 J - I
Card 4/ 1113 jh Yjh 11 1k 0

22413
C
S.;/042/61/016/001/007/007
Systems of linear differential C 111/ C 333
Ik -- unit matrix. The c.Lq (~, Y are real functions of t and integrable
according to Lebeegue; ck4h(t) ' ~"h~(t)' I jh(t) hj (t) almost
everywhere.
Chapter I considers the structure of the space L n ~H(t)" of all
H(t) - H(t+,C), to which there correspond solutions of (21 which
have certain properties. E. g. let ~jZ be the set of the H(t) -ror
which (2) possesses a certain number of bounded solutions for
t -+w , while the other solutions are estimated by
~X(t)~'e C ecle-t ; these properties are said to be stable with respc-ct
to small variations of H(t). It is stated that the sets V.* thus de-
fined are either domains or are decomposed into at most denumerably
many domains. The author gives clear models of the space L for k
and - 2.
In chapter II the author considers the convex properties of the
stability domains in Ll here~~Y, is a stability domain, if the
solutions of (2), the H(t) of which belongs to VY., are stable.
is called convex in the direction of growth, if from
C iLr d 5/9

0
2
2 433.
S/042/61/016/001/007/007
Systems of linear differential ... C 111/ C 333
H1 G IU 1 ' H2 C W(, ' H1 IS H2it follows H C* z1Z , if H1 :~-* H A:~ H2*
It is stated that all stability domains are convex in this sense for
k = 1,2. Under certain assumptions the stability domains can be
subdivided into convex subdomains for k > 2. These properties are
used for obtaining sufficient stability conditions for (2).
In nhapter III the author extends the Lyapunov method for the in-
ve3tigation. of the Hill equation
zj + P(t) Y - 0, P(t +V) - P(t) (4)
dt2
to more general classes of canonical equations.
In chapter IV the author considers systems (2), the coefficients of
which are not periodic. He investigates oscillation properties of the
solution,and degenerated self-adjoint boundary value problems. A
geometric definition of the oscillation character of the equation (2)
is given, where results of J. M. Gellfand and V. B. Lidakly
Card 6/9

22423
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Systems of linear differential ... C 111/ C 333
(Ref. 10: 0 strukture oblastey ustoychievoski kanonicheskikh line-
ynykh sistem differentsiallitykh uravneniy s periodicheskimi koeffitsien-
tami Fon the structure of the regions of stability of liaear canoni-
cal systems of differential equations with periodic coefficients] I
UMN 10, vyp. 1 (63)s (1955)9 3-40) are essentially used. Degenerated
self-_`Wdjoint boundary value pro~blems are investigated for the
equation
dx , J(H (t) + )~H (t)) X (6)
W 0 1
The author gives necessary and sufficient conditions for the
existence of at least one and of an infinite number of eigenvalues
tending to infinity.
Chapter V and VI have applied character. The author investigates
the equation
dx J [Ho + EH( 1Dt, E) ] x
it- ~ (8)
where is the exciting frequency and H(e, F_ H 1(8)+ eH 2(8)+...
Card

22413
S/042/61/016/001/007/007
Systens,of linear differential ... C 111/ C 333
especially the case of renonance is considered, where JR 0 possesses
eieen values uhich are congruent mod iG. The author describes
parameter resonance and the construction of the dynamical instability
domains of (8).
In a longer footnote the author rejects the objections of N. P.
Yerugin who criticizes in (Ref.38: Metody iseledovaniya.sistem
lineynykh differentsiallnykh urayneniy s periodicheskimi koeffitsien-
tami fMethods of investigation of linear systems of differential
equations with periodic coefficients'
,), Inzh.-fiz. zhurn. AN BSSR
No. 2 (ig6o) that the author uses results of other authors without
referen,7e in chapter 5 of his dissertation.
A, M, Lyapunov, L. D. Nikolenkov J. K. Glazmanq N..N. Bogolynhov,
Yu, A, Mitropol'skiy, J~ Z, Shtokalo, J. G. Malkin and K. A~ Breus
are mentioned.
There are 44 Soviet-bloc and 2 non-Soviot-bloc references. The referen-
ce to English-language publication reads as follows:
Card 8/9

22Wj
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Systems of linear differential ... C ill/ C 333
R, Sternberg, Variational methods and non-oscillation theorems.for sy-
stems of differential equations, Duke Math. Journ. jj, No. 2 (1952)9
511-312.
Card 9/9

29024
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D274/D302
AUTHORa Yakubovich, V*A.
TITLE: Unbounded-stability conditions for a second-order
differential equation
PERIODICAL: Leningrad. Universitet. Vestnik. Seriya matematiki,
mekhaniki i astronomiip no. 4, 1961t 83 - 91
TEXT.- Several theorems on unbounded stability of dynamical systems
are proved. The equation
d2x + ,dx + CF(et)x = 0 (1.1)
dt2 dt
is consideredp where V _--09 e :;- Op T(s + 21r) = cp(s)g the function
cp being either continuous -on -the interval [09 23YJ or having a fini-
te number of singularities and
Card 1/6 /cr(s)/de -< oD .

29
0,
8/0 61/000/004/002/008
j
Unbounded-stability conditions ... D274 D302
~D
In the followingp such functions are called 2V-periodicp p'ibcewise
linear and integrable. According to M.A. Ayzerman (Ref. 1: Dostato-
chnoye usloviye u8toychivosti odnogn klassa dinamicheskikh sistem
* peremennymi parametramie YMMp vo 15, noo 3P 1951)9 a dynamic sys-
tem described by Eq. (1.1) has unbounded stability if for any posi-
tive Op the condition gx
x(t) --), 0, - --+ 0 for t --0 + oo (1.2)
dt
is satisfied. Another author considered Eq, (1.1) with piecewise-
constant functions
M2 for 0 t < Vt
M2 for V t 2$r;
'he arviVeFd~ at the- 6onditions
2
V 2> ~E (1 T2 d (1-3)
2 M 2
Card 2/6

- --------- --- ---
29024
S/043/61/000/004/002/008
Unbounded-stability conditions D274/D302
where Cf 2 = 1(M2 + M2
7
A more general result can be derived from the author (Ref. 3: DAS
SSSRv 879 3t 1952). This result is formulated as Theorem 1: Denote
by QM,or the set of 2ff-periodic piecewise-linear, Integrable func-
tions cf(s) for which
2Tr
max cf (s) = M2 1 (p(s)ds = a2'~"D 0
23Y

are preassigned numbers. For unbounded stability of Eq'*- (1-1) with
any function q(s)E I'M
cr, it is necessary and sufficient that in-
,
equality (1-3) hold. Belowp a very simple direct proof of theorem
1 is given. An analogous result is formulated in a more elaborate
way, when min q(s) is pre-assigned instead of max cpks)p viz. Theo-
rem 2: Denote by Vm'o' the set of functions cf(s) for which
Uard 3/6
L EMWA
h 5,
014 M'T.12 A
N
'
Q`

29024
8/04Y361/000/004/002/008
Unbounded-stability conditions D274 D302
D
win cp(s) = M2 1,
2S(
are pre-assigned. In order that
for any function T(8)(F-'Pm,d' it
1) for V-e-mf inequality
ch V t + coed
V27 7 2
2fr
rf(E3)ds
Eq.
is
2
m 2
=cf2
(1.1) have unbounded stability
necessary and sufficient that
sin 0 (1 - 4)
_ V2
should hold for any 0 - 0 (1.6)
-P - 2 2 _ 2)
m2 T m
holds for any t ---,,-0. Purther, the analogous problem of evaluating
the characteristic exponents of equation (1.1) which belong to the
classes of functions 4Vm,(,, and Vm,cft is considered. As this problem
is equivalent to evaluating the characteristic ecponents for Hill's
equation
2
dt 2 + P(et)y = Of
the latter equation is considered, whereby the function p can also
take negative values. Two theorems are formulated. Theorem 3: In
order that for any positive e and p(e)p the approximation
y O(ePt)9 t ---.* +00 (108)
to*the solution o1L Eq. (1-7) should hold being a pre-aB81-
Card$/6
-,~fl V 11VA'.
Ni~

29024
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Unbounded-stability conditions ... D274/D302
gned number). it is necessary and sufficient that
PmU_L!_CP_ for p + P > 0
2 vimax max cp
V >_;'11 .9)
-V- Pmax for pmax + pep < 0
hold. Theorem 4 is analogous to theorem 2. In the following, the
4 theorems a::e proved. There are 10 references: 9 8oviet-bloc and
I non-6oviet.-bloo.
Card 616

LI! T'::
LU'll?) yc.~,N.; T.1.3,~]~,, VIC,')
Abrarovich Tartt-:ovz':,, cn hic !'-M-
naW,. 16 nc.,.r,:225-2lO S-o
(,.rl c:
A:,

YAKUBOVICH,, IEA-
Cr,~ ,onD oe inVmited stability for a certain differential
Nuatlon -~-f the second order. Vest,WU 16. no.19:83-92 161.
(Differential equations)

31906
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0 D299%304
AUTHOR: Lak (Leningrad)
TITLE: Arguments on a group of simplectic matrices
PERIODICAL: Matematicheskiy sbornikq V. 55, no. 3; 1961, 255-280
TEXT% The group of simplectic matrices is related to the linear
system of canonical differential equations
8H BH
p = F- , - (j = 19 ... 9 k) (0.1)
q qj op
i i j
where H is a quadratic form of pj, qj, with real coefficients. The
concept of argument on a group of simplectic matrices was intro-
duced by I.M. Gellfand and V.B. Lidskiy in connection with the
structure of -tability regions of Eq. (0.1) with periodic coeffi-
cients (Ref. 1: 0 strukture oblastey ustoychivosti iineynykh kano-
nicheskikh sistem differentsiallnykh uravneniy s periodicheskimi
koeffitsiyentamip Uspekhi matem. naux, v. '10, no. 1 (6j), 1955, 3
Card 1/6

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Arguments on a group of D299/D304
- 40). Various definitions of the argument ari? given, e-clh of the
definitions !eading to some new interpretation of the number of
stability- and instability regions of Eq. (0,,I) with periodic co-
efficients. The main application of the arguments consists -in stu-
dying the oselliaiory properties of the solutions to Eq,, (0-1);
this howeverg is the subject of a later article by the author. The
prenent article is devoted to defining the various arguments, their
properties and their equivalence. Let ArgX be a real function (of
even sign) of the matrix X C-G, satisfying the conditions: 1) The
function ArgX is defined for any matrix XCGO 2) It ~ArgX)o denotes
one of the values of ArgX9 -then the other values are
(ArgX.). = (ArgX)0 #- 2m,7f ( m - +_ I , � 2 9 ~ . - ) ;
3) Each of the values (ArgX) M is a continuous function of X4-G-1
4) There exists a closed curve U(t)C~G with index unity, so that
6 ArgU(t) ~ 2--r. Any function which satisfies the above 4 conditims
is called argument on the group of simplectJo matrices. This defi-
Card 216

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5/03 61/055/003/00,'-'/004
Arguments on a group of D299YD304
nition can be rephrased: Assume a continuous mappin is given of
the group G on a circle, whereby a closed curve UM6 G, 0 ,,;; t -,-::: 1,
of index unity exists, so that if t varies frcm 0 to 1, the corres-
ponding point traverses the circle once in the positive sense; by
definition, ArgX is the argument (in the ordinary sense of the
word) of the point on the circlet onto which the matrix X is map-
ped. A theorem is proved which shows that the various arguments
defined above are equivalent in a topological sense. Assume, fur-
ther
k
Arg*X Arg pj (1.7)
J=1
the matrix X is divided into 4 k x k -matrices:
X = ('1 '2); (1.8)
V1 V2
the notations
Card 3/6

Arguments on a group of ...
Arg,X = Arg det
Arg2X = Arg det
Ar93X = Arg det
Arg4X = Arg det
are introduced. It is also
zione 9
31906
S/03 61/055/003/OC2/004
D299YD304
(Ul - iV,),'
(U2 - iV2)0
(Ul + 'U2)P
(V1 + 'V2)?J
possible to take as arguments
Of - iv,),
(U" + iV11).1
ArgAB X = Arg det
Arg H X = Arg det
AB
(1-9)
the fixnc-
0"11)
Theorem 3. Each of the functions defined by formuias (1.7), (1-9)
and (1.11)p satisfies the conditijns of the definition and can,
thereforep be considered as an argument on the group of simplectic
matrices. The equivalence of various definitions is then examined.
bet Arg'X and Arg"X be two different argumentsp i.e. two functions
on G which satisfy the 4 conditions of the definition. The argu-
ments Arg'X and Arg"X are called equivalent if a positive constant
Card 4/6

319o6
3103 61/055/003/()02/004
Arguments on a group of ... D299YD304
C exists9 so that for any continuous curve X(t)C G, the inequality
/j!!~Arg'X(t) - AArg"X(t)/---C
holds., Further, a model is constructed which shows that varlous
arguments, both equivalent and non-equivalent can be introduced
which satisfy the 4 conditions of the definition. Theorem 4. The
above-introduced arguments
ArgjX (j = 09 19 2, 3, 4)p Arg!BXP Argil X9 Arg*X,
AB
are equivalent. This theorem is important for studying the oscilla-
tory properties of the solution to Eq. (0.1). Its proof is compli-
cated and involves several lemmas. There are 3 figures and 20 re-
ferences: 14 Soviet-bloc and 6 non-Soviet-bloc, (including 3 trans
lations). The references to the English-language publications
read as follows: R.L. Sternberg, Variational methods and non-os-
cillation theorems for systems of differential equationsg Duke
Math. Journ. 199 no. 29 19529 311-322; W.T. Reid, The theory of
the second variation for non-parametric problem of Bolza, Amer.
Card 5/6

Arguments on a group of ...
31906
S/039/61/055/003/CC2/004
D299/D304
Journ. Math., 57, 19359 573-586; W.T. Reidp A matrix differential
equation of Riccati type, Amer. Journo Math., 689 1946, 257-246.
SUBMITTEDt January 7, 1960
~y
Card 6/6

YAKUBOVICII, V.A.
Structure of the functional space of complex canonical equations
with periodical coefficients. Dokl. AN SSSR 139 no.1:54-57 Jl
161. (RIRA 14:7)
1. Laningradakiy gosudarstvannyy universitet im. A.A Zhdanova.
Predstavleno akademikom V.I. Smirnovym.
(Differential equations) (Groups, Theory of)

YAKUBOVICH, V. A.
Convex properties of the stability domains of l1rear Haziltonid-n
systems of differential equations with periodic coefficients.
Vest. LGU 17 no.13:61-86 162. (MA 15M
(Differential equations, Linear)

GELIFC14D
A.O.; LINNIK, Yu.V.; YAYUBOVICII, V.A.-, L114141K,
CHUDAKOV, N.G.;
,
IU.V.; CRDAKOV, N.G.; _
IAKUBOVICH, V.A.
An incorrect work of N.I.Gavrilov. Usp.mat.nauk 17 no.1:265-267
Ja-F 162. 04IRA 15:3)
(Functions, Zpta)
(Gavrilov, N.I.)

S/ 31907-'62/056/001/001/003
B1 12/B1 38
AUTHOR: Yakubovicho V. A. (Leningrad)
TITLE: Oscillation properties of the solutions of canonical
equations
PERIODICAL: Matematicheskiy abornik, v. 56(98), no. 1, 1962, 3-42
TEXT: The author studies the oscillation properties of the solutions of
the canonical system Jdx/dt - H(t)x, where
j
'0
0
=( k
and H(t) - WA(t). The method applied is a geometric rather than an
analytic one. It is based on the fact that each fundamental volution
matrix X(t) is a symplectic matrix: XitjX - J. The symplectic group
G - has the topological structure of a multi-dimensional torus. This
resuftlis due to I. M. Gellfand and V. B. Lidskiy (Uspekhi matem. nauk,
t. X9 vyp. 101) (1955)t 3- 40-). The canonical'system considered ic said
Ca~d_ 1/3 : -

31909
S10391621056100110011003
Oscillation propqrties of the ... B112/B138
to be of the oscillation type if the fundamental matrix X(t) "rotat-)s"
without limitation on the torus G for t-->+cv . The equivalence of
different definitions of oscillation is shown. A series of criteria are
derived for whether a given canonical system is of the oscillation*type
or not. G. A. Bliss (Lektsii po variatsionnomu inchisleniyu, Moscow, IIL ,
1950.), 1. M. Glazman (DAII SS,3R, t. 116, ITO. (1958), 423 - 426., t..L12,
No. ) (1958), 421 - 424.), V. A. Kondratlyev ~Uspokhi matem. nauk, t. Xii,
VYP-2 (-L) (1957), 159 - 160., Trudy Mosk. matem. o-va, t. 8 (1959),
259 - 282.), L. D. Nikolenko (Avtoreferat dissertataiig Kiyev,1956.-,
DAN SSSR, t, 110, No. 6 1956), 929 - 931.), IA. G. Kreyn (DAN SSSH, t. 13,
No, 5- (1950), 445 - 4478 V. B. Lidskiy (DAN t. 102, ITO. ~ (1955),
87 - 880.) are referred to. There are 4 figures and 34 references: 26
Soviet and 8 non-Soviet. The four most recent references to En-lish-
language publications read as follows: R. L. Sternberg, Variational
methods and non-oscillation theorems for systems of differential equa-
tions, Duke Math. Journ.9 19, No. 2 (1952), 311 - 322; J. H. Barrett,
A PrUfer transformation for matri~_differentia_! equations, Proc. Amer.
Math. Soc., 8t No. 3 (1957), 510 - 518; W. T. Reid, The theory of the
second varia7tion fo-r non-parametric problem of Bolza, Amer. Journ. Math.,
.a (1935)2 573 - 5a6i ',.7. T. Reid, A inatrix differential equation of
Card 2/3

'0
371
S103 62'/056/001/001/003
Oscillation properties of the B112 B138
B,
Riccati type, Amer. Journ. Math., 68 (1946), 237 - 246.
SUBMITTED: January 7, 1960
Card 3/3
fn

3?3 713
S/020/62/143/006/007/024
B125/B112
AUTHOR: Yakubovioh.
TITLE: Solution of a few matrix inequalities in the automatic
control theory
PERIODICAL: Akademiya nauk SSSR. Doklady, v. 143, no. 6, 1962, 1304-1307
TEXT:- The author derives a necessary and sufficient condition under which
a Hermitean matrix H - H* exists for a given matrix A and for given
vectors a and b, so that the matrix G - gg*-, where G - -(AI'H + HA), and
g -(Ha+b), is positive definite. This condition reads
1 2Re ((A-i&o) -1 a,b) > 0 for -co 0 L > 0 exist, that for any t >., to and any solution x (t), this'
inequality holds true:
Further, forced oscillations are investigated assun-dng that the solution x. (t) is
known. Among other things, it is found that, in the absence-of external influences'#
the condition of absolute stability with a specified decrement is similar to Ve Mq~!'
Popov',3 condition. Orig. a~rt. has: 50 formulas.
ASSOCIATION: none
SUBMITTED: 04Nov63 A ENCL: 00
SUB CODE:' DP IE NO REF SOV: 025 OTHER: 008

STARWINSKY, V.1-1. (Moscow); (Laningrad)
IfContribution to the Lyapunov nothcd o." determining r,--riodic solutions"
Report pro3ented at the 2nd All-Union Congrens on Theoretical and Applied
Mechanics) Moscow 29 Jan - 5 Feb 64-

GAIrIMAKIM , F.R. (Mlorcow); YAKUBOVICH, V.A. (Lcriirwrad):
17Absolu-'%-.e stability of non-linear controls."
report presented at the 2nd All-Union Conryess on Theoretical and Applied
Mechanics, Noscm,, 29 Jan - 5 Feb 6h.

:ACCESSION NR: AP4036716 9/0020/64/156/002/Ob78/0281
jAMOR: Yakubovicho V. A.
TITLE: A solution for certain matrix inequalities'encountered in a nonlinear
control theory
SOURCEt 'AN SSSR-P Doklady*,, v. 156, no. 2. 190j', 278-281
TOPIC TAGS: matrix inequality, nonlinear control, Lyapunov function, matrix
i
spectrum, closed halfplane, absolute stabLlity, Lurie method, quadratic form
ABSTRACT: This paper deals with special systems of differential equations for
..;automatic control) 'and a bounded class of the Lyapunov function (e.g., quadratic
I I
forms). The existence of this function was converted into a purely algebraic prob-
lem involving the existence of a solution to certain specialized matrix inequalities.
~Through a series of tuathematical arguments and the construction of theorems, a
!solution was presented. In his third theorem, the author assumes that P0
Therefore, in order to have a matrix U - 11* in which F(p,H) > 0 when 0 ~-. p :r. P0, it:
V Is necessary that In the case of a certain !r k 0 vmd all w 2: 0 j, the following
]equatIon be carried out:
J*Card 1/2

A
.13 Z; a -~A
A J",',l R iTakUborich, V. A.
T
T
~ LE: Frequeney conditions for the ab,,q0jLjt,3 Stability of n-rilinear
control gptem-
CITED SOUIEE! Tr. Menhvuz. konfere sii p0
pq~l. ~GkhA~n. 1962. Kazan', '196
135-10
TOPIC TAG3: nonlinear automatic c;ontrol. system, absolute stab:Llity, stability
c0nditionj Lyapunov f -
unctlQn.
MANSLATION.- The controlled systern is studiedl X . pX 4, q
T(6)p vhere d'. r*x,
P is a constant Hurwitz matrix, q and r are constant columns, and the function
satisfies the condition (6) 6 < ;-L 6-2 1 LA -
r I r.-,Q:
-CLW".L11J1 v-L IFLle InVeSrAgated System are derived by
anuy-8~19 Of the kyripmov f=tjo- - - F- (H is a positive-defiat
Card 1/2

L 58921-65
AR5016489
H
at
matrix). It ii shown that tNq obta-Ined results arO VG!id 01P611 in the Casewhen
(6) is a piecewise-ccntir~uw-s fu=,inn :ae third matrix droblem Is and
its solu'-'.orr tr,~at ta-~ "wea.K
c) ~ cw3 frop
3 5 -7 13 X I S ' 1-1 43
jith 1~ 'rho frequency c(.ndition f(,r atsoiutu z5
case vner, T is a -,-)rTtinuou!; hy-terosis funcLion. Jk syste!-a with t.)e placcrw.so-
,q -:~ nr or, 0 iz a.-- s c c o nR id e r ed 31-Lfficieat f requency
n d J t (J 7, abo].i;1.43 sygtern Ei~au.IjJLV, Ur,;,-Laer 'jiln ' ',e Of ?Cpl:,V,
IF ot~taj-j;V -,n4 '-vapunav function, A. Kh. Cre ~i g
I U _q
SUB GUIDE 13, M& L: )c

IL 08829-67 EWT(d)/EWP(l) I J Pic) BB/(;G--
ACC NRs AT6022616 SOURCE CODEt UR/3040/65/0001004/0003/0071
Yakbbovich,--V-.-- A. 31
IORG: none
TITLE: Some general theoretical principles for the construction of learning percep- I
'tion systems. I
MURCE: Leningrad. Universitet. Kafedra vychislitellnoy matematiki i Vychislitel'nyy i
I I
tsentr. Vy-CEF-M-ef naya, ie-kh-niVa i voprosy programmirovaniya, no. 4, 1965, 3-71
FOPIC TAGS: information theory, perceptron, autcmaton
(ABSTRACT: General principles are given for the construction of automatic systems wbicIT,
!model hu;nan perception and identification processes. A large class of models of the
lparceptron type is discussed. A general introduction is given, de."cribing the general
I
Troperties of a perception system. The genera .I formula for such a system is
N
S (X) -Aai (x).
where x.7 are certain numerical coefficients, s(x) is an elementary "concept" resulting
1from the perception of image x, and are certain so-called "a-elements"-functional
ad
Card 1/2

L 06629-67
ACC-Nk1-__AT0_226l--6--
1 0
,elements which react on incoming images. The method for choosing these a-elements is
Idiscussed. It is shown that the collection of functions representing a-elements is
complete in the field of images under certain sufficiently general assumptions regard-
ing the functions given on the receptor field ("retinal'). It is shown also that for
ian optimal choice of coefficients xithe probability of error in perception will be-
come arbitrarily small providing the number M of a-elements is sufficiently large,
along with the number m of elements of the learning sequence. The requirements for an
,algorithm of learning and perception are discussed. Systems are studied which are bas.
led on the defined notions of L- and C-optimality. The L-algorithm is shown to be theo.
retically superior but extremely unwieldy and hence unsuitable for circuit form; while
algorithms satisfying C-optimal requirements, converging only with sufficiently great
Ivalues of N, are quite simple and can be circuit-programmed. These algorithms are re-
icurrence solutions of linear inequalities and equations. Opig. art. has: 225 fo=u-
,las, 9 figures, I table.
SUB CODE: 09,12/ SUBM DATM OlAu&64/ ORIG REN 017/ OTH REr: 006
1 Card 2/2 not
I

TWV -I Po-4/Pq-4/Pf-4/Pz-4/Pk-4/P1-4
ACCESSION NR: APML91j~ UR/~)103/65/G26/004/0577/9590
AUTHOR: Yakubovich, V. A.(Leningrrz)
,\,TTTLE-: The method of matrix inequalities ii, the theory of &tability of nonlinear
conLrylled eystems. 11. Absolute itabilitv in a class of nonlinqarittes with
derivatLve condftio"
SOURCE: Avtomatik2 i telamekhanika, v. 26, no. 4, 1965, 577-590
nonlinear class stability, derivative
TOPIC TAGS: control syntem stabil _y~
conditi-n, absolute nonlinear system stability, matrix inequality
ABSTRACT: The author studiad the system
dz
- = Px + qy(o), a
dt
ulterattt ) 1.0 a functLon 41ffa.reatlahle a,t every point and sattsfyiog
14(;a + 00)
(2)
and either
(3)
f

-L 48952-65
ACCESSION NR: AP14011900
or
(4)
and derived A new frequency condition for the absolute stability of the nonlLn
earirv -laqrcs gaf-iqf-ing (7) and (3). or (2) and (4) and an additional cerditiott
t t
ri~-,wte avi t r i
A TTI
r near
Marrfx inequalities discussed earlier (Doil. AN SSSR, v, 143, ac. u, iiuzj.
--Md uae-, the Lext. '~rtg. art. hag-
-appendix- -contakas the proof of four cheor,
T--
f
ASSOCIATION: None
SUMITMD: 23Sep63 ENCL: 00 SUB CODE: IC, KA
NO REF SOV: 013 OTHER: 005
2/2

AUTHOR: Yakubovich, V. A. ningrad)
TITLE: Method of matrix inequalities in the theory of stability of nonlinear
controlled systems. Part 3 -Absolute eLability of systems having hysteresis-type
nonlinearities
SOURCE: Av-tomatika i telemekhanika, v. 26, no. 5. 1965, 753-763
TOPIC TAGS: nonlinea:- automatic cont-ol, automatic control, automatic control
design, automatic control system. automatic control theory e_)
ABSTRACT: The article proves Phat the V. M. Popov frequency condition holds
tiue in the case of a hysteresis -ty-pe nonlinearity if the parameter that enters the
above condition hag a definite si-n that depends an the direction of following the
5
hysteresis loop. An autornatic-control uystern is considered which ran be
dz
descriLe,.' by th-eae equations: dt ~P1 r'.r. where P- 0 x'J ,9 a
Card I /Z

61; r*x jr, a scalar
0, k-
q I *--
constant rnatrix; q, r -x are constant vectors nonlinearPy,
ysteresi8 functi,n. For a backlash-type-
product; ~Pc) i -- .4 %,
. TEO*, j t ity io determined, Orig. art. haa: 3iigurr-sand
the condition of absolute stabil
55 formula3,
ASSOCLATION: none
SUBNUTTED: l7ju-163 ENCL: 00 SUB CODE. DP, LE
NO RF- F SOV: 0 1 1 OTHER: 000
Card 2;1?

A~
ALMiOR: Yakubovich A.
ncy criteria for absolute stabll-'Ly'and disaipativity of c
TITLEt Freque
sTgtems with one d-i f ferenti able noulinearity
Va 160, nO. 2, 1965# 29&301
SOME: 0 MR. Doklady
TCFPIC" TAGS: absolute stability criterion, disaipativIty criterion, nonlinear
control aysteia, global asymptctic stability
Alk3TRACr- A study is made of the 8tability of rionlinear control system 4escribed
by the system of differential equations
dz Idt = Pr + 99(a).
Vhere a a r*x, is a H u rv itt matrix, q 13 a ver-mor, and (p(c) is a differ-en-
tiable f~mctim sWarying the following conditions:
r
-at (2)
Card 1/3

L 25383a-65
jLcCEMION 11R.- AP5004585
where u0, a, and aL2 are finite numbers. An expression w(w) containing certain
pars ters ;1, T2, 6, and the transfer Mriction of the linear part of the system
t A intrr~duced vhl -h t La usad f,., lefi-iing criteria --)f the absolu',e stability
r
of the system (1). Conditions vhich parameters T, )~ U, ir2 "k a' a and ;iNe;
U"iafy thle. the aclutLon X of i-.-+fm (11 h aay=ptatically stable in the large
are presented in the form of tvo theorems. For the control sy3ten described by
equations
dr / dt = Pz + TV (0) + / (t, z) r',c (3)
vinere f''L, 1', is a c-nntinunim fun~'.I~n -r x 3r, .1 tan I
conv,prg,-~fi trywo-rd tero up. y vl~h t, Pnd ind-r thp saz.* ansi-mptionF!
as fo-. system 1) except for conditkn w

q
a
'k2 0, 7'
---I rp I
A -,r -,Tmr,-o

L 24596-66 EWT(d).,
AGG NR: AF6oo9hl5. SOURCE CODE: UR/0020/66/166/006/1308A3
AUTHOR: Yakubovich, V. A.,
Leningrad �tate UniXers Aariov eningradskiy gosudarstvannyy
ORG ity JM, Z
univernitat)
TITLEt Recurrence finitely convergent algorithms for solving systems of
inequalities
SOURCE-- P14 SSSR. Dokladyj v* 166, no, 6j 1966j 1308-1311
TOPIC TAGS: algorithm., Euclidean space., Hilbert space,, set theory, vector,
real function
ABSTRAM The infinite sXstem-of inequalities
> 0.
is examineds where x is the unknown vector of the euclidean or real Hilbert space
Rl; aj are arbitrary vectors of some set M of a euclidean or real Hilbert space
R2; and 9.(x, a).is a real functionii The derived algorithms have a number of
common characteristics with relaxation algorithms& Besides being simple, they
Met
C d.1 -519-95

L 21696-66
ACC NR AP6009415
have high reliability, Five theorems are proved, Superposition of the aleorithms'
Xi +i - fjCxj, cj(x a (x , a )7 for x, G G will be a finitely convergert,
algorithm for solution o4 the 12qualities (P (x, aj) > 0. Simple finitely
convorgenb algorithms for solution of the inequalities + 2(hjp X) + (11j)
xP
>0, whore Hj m H* > 01 can be easily obtained by this mlethod, ' "The author
x
This'paper was presented'
thanks Vo Is Smirnov for a number of valuable comments."
1--SR[Fnov; a~adiimioian3 on 22 Jujw 1965. Orig, art, of 4 equatigns.
by V. ha
SUB CODEt' 12/ SUBM DAM 06JunO/~ ORIG REFI '0051, oTH Rat oo4
-Card-2/2 ~A

L-27783-66 EWT(d)/r/E11P(1) IJP(c) GG/BB/JXT(CZ)
ACC NRt AP6012911
SOURCE CODE: UR/0020/66/167/005/1008/1011
AUTHOR: Kozinets, B. N.; Lantsman, R. M,; Yakubovich..V.-A.
ORG: Lithuanian Scientific- Mesggrcb JnfjtltuW for Forensic Examinations Vilnius
(Litovskly'Nauchno-isaledovatel'Bkiy Institut sudebnoy ekspertizy)
TITLE: CrIminalistle examination of similar handwriting by means of electronic computers
SOURCE: ANBSSR. Doklady, Y. 167, no.'5, 1966, 1008-1011
TOPIC TAGS: computer application, adaptive pattern recognition, electronic computer)
digital computer
ABSTRACT- One of the most difficult tasks In criminallatic examination if; the Identification
of similar handwriting. The presengttthors developed a program for a learning digital
computer which bases the recop-mitt .rocess on learning according to the algorithm which
follows a training sequence. The graphical object Is first converted Into digital form by
means of characteristic features. The processing of data Is carried out by associating to the
stereotype of the handwriting of a given person a sampling of convex sets. Computer recogni-
tion.o.f.1tr"pe wd frrg~!d signatures of the personnel of the Lithuanian Scientific Research
-~xamlnattlons-(Lltovskly nauchno-looledovatellskly institut sudebuoy
With the r;euits of --Identification's grid
ic-Ktj~, 27 -enin
compare by experts of the L
Scientific Re.s~'---rch Laboratory of Forensic Examinations (Leningradskaya n-auchno---
Issledovatel'skaya ~abokatoriya sudebnoy ekspertizy), of the scientific technical department
L 11TW.- RIG OR

I-ACC-NR, -MoUi ~1~
of the UM UOOPLO (nauchno-tekhnicheskly otdol), and the scientific- technical group of the
highway department of the militia MOOP RSFSR (nauchno-teklinicheskaya gruppa dorozhnogo
otdola militsit). Results are shown In Table 1.
Table 1 Handwriting recognition
Recognition, ppreent
Signature Experts Machine
Meteyavichyus 58.3; 68.3; 70 88
Shtromas 75.4; 78.9; 80.7 91.2
Chyapas 76. 0; 80 84.2
Poshkyavichyus 1 90. 0; 92 100
A more detailed accotmt of the Investigation will appear In Symposium No. 2 of the Lithuanian
Setentific-Research Institute for Forensic Investigation which planned the study in conjunction
with the CoMputer CentAr of 1&n1ng
rrad Ufilyergity VychIslItel1nyy tsenter Unin&radskogo
universiteta). The authors express their &Tatitude to the experts of abovementioned
instittitions. The paper was presented by Academician Smirnov. V. 1. 20 Jul 65, OrIg. art.
has. 1 table.
SUB CODE:, 05p 06 WBM DATE:' 17juI64 ORIG REF.., 001~
1/2

L 04900-67
ACC NRi- AT6022670 SOU11CE CObE: Ulil6666ld7Cl6b~-1066-1662i*10-02'8
AUTHOR: Kozinets, B. N.;
Lantsman, R. M.; Sokolov, B. M.; Yakubovl94,y.,,A
ORG: none
TITLE: Handwriting Acognition and discrimination by means of electronic compUters
SOURCE: Moscow. lastitut avtomatiki I telemckhaniki. Samoobuchayashchiyesya
avtomaticheBldye sIBte (Self -instructing automatic systems). Moscow, 1zd-vo Nauka, 1966,
21-28
T, OPIC TAGS: pattern recognition, automaton, character recognition, computer application
ABSTRACT.- The general problem of machine recognition and discrimination of handwriting,
the development of the necessary algorithms, and the theoretical principles underlying the
process of teaching an automaton handwriting analysis are discussed. The discussion Is based
primarily on certain theoretical work In this area that has been carried out at the VTs LGU.
A detailed explanation Is given of the manner in which the handwriting or "graphic" material Is
converted Into a system of numbers suitable for computer processing, and several different
metrization techniques are described. The principle of the "dynamic stereotype of writing"
(a fundamental assumption of the method proposed) is Introduced as a means of neutralizing

L 04900-67
ACC R, AT60226'10
ations from an established and c[uantized standard. The
random or deliberate handwriting devi alidity of this hypothesis are statedr and it is si-,ownl
necessary and sufficiejj~ conditions for the v er than those which disre-I
re In all cases much SIMPI tain machine. experl-
n this assumPtion EL f the results of cer
that algorithms based Q, afid, an analysis IB made o arison Of the algorithm adopted
gard it - ExampleE; arg, given . ined, Including a comP ents
ments using tile general. techniques outl The theoretical considerations and exPerim
with, others founded on different approaches, oying computers for the differen Itiation
cribed substantiate the possibilitY in principle Of CmPl
des iting styles . 0rIg,, art. has: 8 figures,
of similar handwr
Sui3M DATE: 02Ma-r66/ ORIG -REF" 003
SUB CODE:09,06~1"/

- __ I -
ACC Pill, AT6024067
A/U
AWHOR: Yakubovich, V. A.
SOURCE CODL:' UR/291;4/66/000/003/0051/OG',~rg
ORG: none
TITLE: Regions of.dynamic instabilitX.in Hamiltonian systems
SOURCE: Leningrad. Universitet. Kafedra vychislitellnoy matematiki i Vychislitel'nyy
itsentr. Matody vychisleniy, no. 3, 1966, 51-59
'Topic TAGS; lincar differential equation, Hamilton equation, dynamic stability
ABSTRACT: The system studied is: C + Ell (t, it, (0.1)
dl
where x is a vector solution, J is a non-degenerate real skew-symmetric matrix, and'
matrices C and 11 are real-valued, sy7rmetric for real c, 0, and depend analytically on
the parameters :hi the region
-co < 01< 0 ,- 02 < + c0, I E< eo < 0-- (0.2)
'A oint of instability is defined as follows: a point (cu, 6u) in region (0.2) is a
~ p
point of instability if for c = cu, 0 = 6U equation (0.1) has solutions unbounded on
:(0,-). Regions on the plane (c,0) into which an open set, obtained from the set M of
I
,all instable points by discarding boundary points, falls are called regions of dynamic lie
Card 1/2

. . ........
L o9h51-67
i.1 ACC NRs -AT6024067 critical for equation (0.1) if
5 called (0,00). 7be general
iinstability of equation (0-1). 71he value Ou i e to the point lculat-
an be fownd arbitrarily ClOs is given for ca
G of instability c I is studied, and an algorithm Of Orige art*
point [3gions of in-stabilit: c instability for small values
~fom Of such 14 es of regions of dYnIi
ling the boundari 5 figures.
76 foxmulasp
j~;SUB CODE: 12/ suDH DATE: 16Apr63/ ORIG-PXF'- Oll
2/2

'l
I
M
SOURCI; CODE: UR/2944/66/000/003/0076/0104
AU,IiOR: Ftsi i nV. N.; Yakubovich, V. A.
03k": none
TITLE: Vac computation of characteristic exponents for linear systems having periodicl
coefficients
SOU"iCE: Leningrad. Universitet. Kafedra vychislitellnoy matematiki i VychislitelInyy
-tsentr. Metody vychisleniy, no. 3, 1966, 76-104
.TOPIC TAGS: linear differential equatioii, dynamic stability, approximate solution
':ABSTIRACT: An equation is derived for determining the characteristic exponents of the
:system
IC + ED (t, E)l X,
'For c 0, they reduce to a given characteristic exponent au of the unperturbed equa-
tion dx
CX.
dF
.Torm,ulas arc also derived for the calculation by successive approximations of charac- ;1
I
;teristic exponents for perturbed systems with periodic coefficients, for a vector equa-,
I
i,tion of degree m, and for systems with multi-parametric perturbations. Some conclusioni,-
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!~UB CODE: 12/. SUBM.DATE:.'.16Apr63/ ORMPtEr: 011/~ OTH REF: 006
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ACC NR1 AP7001539 SOURCE CODE.1 UR/0020/66/171/003/0533/0536:
AUTHORs Yakubovichj,__Yj_Aj___
ORG: Leningrad State University im. A. A. Zhdanov (Leningradekiy gosudarstvennyy
universitet)
TITLE: Periodic and nearly periodic limiting states of control systems with certain,
generally speaking, discontinuous nonlinearitiea
SOURCE: AN SSSR. Doklady, V. 171, no. 3, 19661 533-536
TOPIC TAGS: control system, automatic control system, system analysis, differ6Atial
equation
ABSTRACT: Consider the differential equations of a control system with :4 nonlinear
units T i - vj(a'd dx1dt=Px+q(p(a)-~1(t), a=r*x.,*
The matrices -P, q, r, x, lltpj(aj)li*'are real and have respective orders
V, v, v X Y, V X'xV"v5