SCIENTIFIC ABSTRACT KRASOVSKIY, N. N. - KRASOVSKIY, V. I.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000826210017-4
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: July 31, 2000
• Sequence Number: 17
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

ACC NRs AP6017846
inequalities
f [Z U,2 di -4
dt ~'O and the pursuer knows the
values of Yi(T), Zi('r + 0), JJ(T), V(T + 0), and v(-c + 0), but the next motion of the
opponents is not known to either one, the control
-C + QV. V (V +-0)),.:
M"), V(1
is sought which ensures an encgunter of the two objects no matter what control
V(T) - n[y(T), Z(T + 0) , U(T) V(T + 0) , V(T + 0)
is chosen. It in considered that the control u together with the.control
v - n* constitute a pair of optimal controls if: 1) at Y and v an en-
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ACC NR, AP6017846. 0
~:ounter of y(t) and z(t) will certainly take place at a certain instant t w 0.0;
2) in case ev deviates from n*, the encounter takes place not later than at t - 0*.
The possibilities of solving the defined problem are discussed on the baifis of
dynamic programming methods and on the concept of attainability domains for pro-
cesses (1) and (2) introduced in the author's article (Tekhnicheskaya kibernetika,
no. 4, 1965]. The difficulties appearing here are,-indicated and the means for
overcoming them are analyzed. The presented approach to the solution of the'purs"it
problem is illustrated by a,&~,j!ple example. Orig. art. has: 26 equations. ELY]
SUB CODE:201 12/SUBM DATE: 02Feb66/ ORIG REF., 003/ OTH REF: 001/ ATD PRESS;

AP6014166 SOURCE CODE.- UR/0376/65/001/012/1551/1656
AUT11011i Krasovskly, N. N.
ORO: Sverdlovsk Section, Mathematical Institute Im. V. A. Steklov (Sverfflovokoye
otdelenlye, Matematicheskly institut) A -
TITLEt The observation of a linear dynamic system and equations with delayed arganxents
SOURCE: Differentaiallnyye uravuenlya, Y. 1, no. 12, 1965, 1551-1556
TOPIC TAGSt linear differential equation, dynamic system, linear automatic control system
ABSTRACT: The author discusses the connection b-!,'-ween the problem of observation of a
linear dynamical system and the problem of canonical representation of the motion describ-
able by differential eqr4ations with time delay. The vector differential equation
x (i) - Ax (1),
(with x an n-dimenstoma vector, and A-a(n x n) a constant matrix) describes a linear dynamic
system and the quantity ri(t) which Is observed to be connected with the vector x(t) through
(t) - P'-t W.
(2)
Card-1/2

NRI AP6014166 0
where p Is a constant n-vector. The reindts show that for any vector ftmetion. x(t) represent-
ing a solution of Eq. (1)8 the function n (t) Is the solution of equation
a, q (j)
.~.J.
J-1
where ai are constants defined by ai - Y* Tj + 0) - Y* (- Tj - 0), yo (0) Is a sectionally
constant functiong and TIis the time delay. The inverse to generally not true. Orig. art.
has: 33 formulas.
SUB CODE: 12,20/ SUBM DATEz 16Jun65/ ORIG REF: 005/ OTH REF: 001
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ACC NR, Ap6oi 535 CODE i
AUTHORSt Kragovskiy, N. N KurzhanekiX. A. B.
ORGt--~frnl ~~ ~-veraity (Urallakiy gaaudaratvonnyy univornitot)
TITLE: Toward the question of the observability of systems with time delay
SOURCE: Differentaiallriyye uravneniya, v. 2, no. 3, 1966, 299-308
TOPIC TAGS: control system, control theory, differential equation, system analysis,
time, time optimal control, dynamic system,, c oc, 12 o/ k-1,j -r,-- Si's 7F/-n
I
AbSTRACT: A study is made of operations separating the unstable coordinates of a
Zlinear second-order system w th delay according to admissible observation of a linear
c9pbination of phase coordinates. The work is related to the class of problems on thq
*trollability and obeervability of dynamic systems. The observation problem is
~fmulated for the system
dx, =ajlxl(t)+aitxl(t)+bjlxj(t-h)+blt.,cs(i-h),
di
alix, (1) 4- attxs (t) + b21X1 (t h) + bztxt (t
dl
?rith constant coefficients and a constant delay h > 0. The solution Lrxl(t)f X2(t)7 Of
UDC: 517.949.22

ACC NR: AP6010535
such a system for t,;- t - 0 in determined by means of initial vector functions
0
y --((Pl (t), T2 (I)),- given on the initial time interval Lr-h, 07. A linear operation is
defined iri general terms, and it io shown that this type 0? operation can be useful in.
determining system observability along a coordinate or a linear combination of
coordinates. The necessary condition is stated in the theoremi The system (1.1) is
observable along the coordinate yi(t)q (i - 1, ...j k) in that case and only in that
case where 61~11'1 + bdj" ~/- V
The numbers d(i) and d(i ) are compone'nt"o of.vectors d (i) , which are solutions of the
1
systems
-B e'x p *( -'"I- I'h) A- I E)'d0(i -1, k)p
and y,(t) is a coordinate'sepairited through applicdtion .of.the investigated operation.
The necessity condition is proved and sufficiency follows from the proof given. Orig.
art. has: 31 equations.
M CODE: 12/ SM DATEI 31.Tul65/ ORIG MWt 005
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-- A-PTOi2540 SOUICE CODE: U
ACC NRi k/0666/66766/6 i10209102
AUTHOR: Krasovskiy, N. N. (Sverdlovsk)
ORG-. none
TITLE: Problem of pursuit in the case of linear single type objects
SOURCE: Prikladnaya matematika i meichanika, v. 30, no. 2, 1966, 209-225
TOPIC TAGS: control theory, control systom, control optimization, minimax strategy
ABSTRACT: The problem on the mininvix of time T until the meeting of the pursuing
motion y(t) and the pursued motion x(t) is studied. The two motions involved obey
the relationships
dy / dt = Ay + Bit, dz / di Az' BV
where yz are n-dimonsional vectors of the phase coordinates of controlled objects;
u, v are the r-voctors of control forces; A, B are constant matrices for the
respective measures. This problem was stated by IN. N. Krasovskiy, Yu. M. Repin, and
V. Ye. Tretlyakoy (0 nekotorykh igrovykii situatsiyaldi v teorii upravlyayertykh sistem.
Izve All SSM, Takhnicheskaya kibernetika, 1965, No. Q. A simplo rale is established
for the case on firxiing u and v such that minu maxv T = maxv minu T with'the
condition y(r -T) - z(r-T) where 'r is the current moment in time. This rule is of

`41
A:~~- : lis L"
AC&SSION NR: AP4043291 S/0040/64/023/004/07WO724~;~
AUTHOR: Kr
rasovskt , H#
TITLE: On the approximation of one problem of the analytte design
of regulatocs it%a slatem vith -it time tag
ISOUR CE IPrikladnava matematiks L mekhanLka, v. 28 no.' 4 1964,-
716-724
TOPIC TAGSt regulaeur design, timelag S Y s t 0 MO.P..~An -AILS; En- tn"_~.Lt
optimal control approxLmdtL*n,,Be1ltman equation
ABSTRACTi The problem of optim-fring control by mLuLmizing the stand-
ard error Le stu Led In contr6l aystems ascribed b3r-the equation
dx
=x'A x+ D,- (c + 6tA
Where x is an n-dimani.ional-vector of phasw coordLn-ati3s" ts t s -a --I ------
acaler control action, Y :-:0 is a constant delay and A, B, and h are
constant matrices. Thel.dptimlat contrat n" in system~(I) uts deter-:
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ACCESSION M AP4043Z51:.
~mined earlier by the autho--~ La the form of a linear functional can-
tatni"g functions et and 6i:*(Prikladmaya mntemmtika i mekhanika,
1962, v. 26, no. 1). Since the Vrocess of deternining ai and 6t is
complicated, the author.~proposIas to''Approximate this problem by thd
,analogous problem for the controL.aystam described by the ordinary
differential aqdationt
%'U'Z + MY01 MYO-0
AV + H~Cul) + W(i I.%
ate er, T' -M are a-dimensLonal vectarst~
:where n to a positiv-aL 9 1 and'y
and A, 3, dnd b ire the some matrLce-a at it equation (1). Tito optl-
vial cqntrol ~* for the central aystem (2) was derived in the form at
a linear functional containing functions ai~T-) and Wvl) by A. It.
Lotov (Avtomatika L 19610 v. 22, no. 4) and R. E.,
~Kalnan (Proceedings Sympostun of Dedfoark Viffer-atial EqU4ttorts
Ctir(12/3

ACC NR: AP6033203 SOURCE CODE: Ult/0040/66/030/005/093;-/0938
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AU-MOR: Krasovskiy, N. N. (Sverdlovsk)
VKU: none
;TITLE: Controlling a plant which is subject to aftereffect
~SOURCE: Prikladnaya matematika i mckhanika, v. 30, no. 5, 1966, 936
!TOPIC TAGS: linear control system, automatic control parameter, feedback automatic
!control
IiUSTRACT: This paper examines the probleri of damping a linear system with the after-
ef f ect (lag) fi',(') = Ax (1) + Cz (J - v) + bu , which "is required to be transf erred by control
u = u(t) from 'a given initial state x(t) = xo(t) (-T ~ t