SCIENTIFIC ABSTRACT GAMKRELIDZE, R.V. - GAMOV, A.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000614210015-1
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: September 17, 2001
• Sequence Number: 15
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

Doklady Akad.Nauk 110, 7-10 (1956) CAU 3/3 PG - 707
If the 2n-disonsional vector (X, is a solution of the system
i i
f (X, U)
(2) 9 f ,
9 Zi Y-* ,Z)xi
whamthe(piecewine continuous vector u(t) always satisfies the condition
H~x~ -W t),u(t)) - M(x(t),-Y(t))> 0, then u(t) is the optimal control and
x t is the corresponding locally optimal path.
Starting from a fixed initial condition x(t 0) - 10 and changing the condition
ik(t 0) -10 , then (2) with these conditions and the condition H(x(t),tP(t),uM).
M N(x(t),'Y(t))?-O determines the set of all locally optimal paths through
the point 10 - X(t0) and the corresponding optimal control mechanisms u t).
INSTITUTION: Math.Inst.Acad.Sci.

': ! ..~ -_I; -~r , . .I . , I ; I , I : I
': I : IW L___ I -Ii v
GINKRILINS. R.V.
. - - -
L.S.Pontriagin's seminar on mathematical problems of the oscillation
theory and automatic control. Usp.siat.nauk 12 no.3'267-272 NY-JQ '57.
(MIRA 10:10)
(Automatic control--Study and teaching) (Oacillations--Study and teaching)

AU THOR: GANKRELIDZE R.V. 20-1-1/44
TITLZz On the Theory of Optimal Processes in Linear Systems (K teorii
eptimal!nykh.proteessoy v lineynykh.sistonakh)
PERIODICIL: Doklady Akad. Nauk SSM, 1957,Vol-116,Nr.I,pp.9-11 (USSR)
ABSTRACT: Given the linear differential equation
(1) i - Ax + bu,
-where x and b. are. nwdimension&l.. vectors, A in- &.linear Irasm-
.formation matrix and u a u(t)..Js & pi*cewise, contiwmme, function
with infinitely many. points of -discontinuity, where lul .4, 1. Lot
0 And El be two points of the a-dimensional ~pk&** space. The
authar se*ke a Itontrolling" u ;. u(t) such- that the Image point
x(t) xhic:h-.mov*s on the trajec cry. of (1),. in ..whartest. -.time
comes from the point To into the point ~1. The corresponding
.u(Q is -Annoted.. as _ optimal. contralli ag and, tho.-zorresponding
Imaj"tary is. denoLted. as. optinal-.trajectory.
.-Thenrems..All optimal u(t) and all.optinal A(t) cer"opending to
theup which for t - 0 leave the point I , &" contained In the
controllings and trajectories which result by the solution of
Card 1/2 the system
I -:` l-,~-'~7'a-~~l-z'~31,~-7~'l;~f';~-l~~I~.-',T

On the Theory of Optimal Processes in Linear Systems 20-1-1/44
(2) i - Ax + bu, x(O) - K, -~- - Alf I u - signf-b.
The initial values %f(0) of the solution tp(t) satisfy the
single condi-tion
kf(O) [Ax (0) + bu (0) 0.
The.systen (2.) i&Fr*v*s.th* maxima principle formulatedty
Bol*yanskiy.,.G&akrwli-ds* and Pontryagin but.not-proyod in the
general case fRef. 1_7
ASSOCIATIONt Mathenatical Institute 1m.V.A.St&3avAc#dSc.USSR (Matematicheskiy institut
in. V.A.Steklom AN SM)
SUBMITTEDi April 4, 1957
PRESENTM BY: P. S. Aleksandrov, Acadmleian, April 5, 1957
AVAILABLE: Library of Congress
CARD 2/2

AUTHOR: Gamkrelidze, R,.V. 30V/38-22-4-1116
TITLE: The Theory of Optimum Processes "With Regard to the 7elocity
in Linear Systems (Teoriya optimallnykh po bystrodeystviyu
protsessov v lineynykh sistemakh)
PERIODICALt Izvestiya Akademii nauk SSSR, Seriya matematicheskaya,1958',
Vol, 22, Ur 4, PP 449-474 (USSR)
ABSTRACTs About three years ago Pontryagin took the lead of the Moscow
seminary on the theory of oscillations and automatic control.
One of the first visible results of this occupation was a pre-
cise definition of the notion "optimum" LRef 1). The given ri-
gorous definition soon allowed the formulation of new results
([Ref 1] and the report at the International Congress 1958
in Edinburgh). The author - a coworker of Pontryagin - con-
siders in detail the linear special case which is described
by the equation
j . OC AC + b1u1+ b 2u2 + .* + b rur
where Ae is an n-dimensional vector, Ot. a given transform-
Card ation matrix, b i constant vectors and ul(t) so-called re-,

The Theory of Optimum Processes With Regard to the SOV/38-22-4--1/6
Velocity in Linear Systems
gulating functions which are subject to the conditionlu
The existence of an optimum control is proved, i.e.t If it
is possiblo to go from g 0 to 1iI by means of an ad-
missible control, then it is possible 41so by means of an op-
timum control. The optimum functions UJ prove to be relay
functions, i.e. they assume the values + 1 aad perfomi flnitelyy
many jumps. Of special interest is the proof of the fact that
there may be a sequence of control functions uJ , so that the
1 k
transitions from to take place in periods t k Mo-
notonely decreasing to T : t 1>t 2 > .... > tk> .... > T , but
that there is not necessarily an admissible limit control uj
for which the transition exactly lasts the time T. The opti-
mum control can only be aimed at by a stronger "tremor" of
the regulations u~ . For the case r - 1 (one final control
J
element) the author provaA two uniqueness theorems.
There are 3 references, 2 of which are Sovi,t, and 1 American.
ASSOCIATIONs Matematicheskiy institut imeni V.A. Steklova 1kademii nauk
SSSR (Mathematical Institute imeni V.A.Stek] v of the Academy
Card 2/p of Sciences of the USSR)

AUTHOR% Gamkrelidze R V. SOV/2C-123-2-3/50
TITLEi ~nthe ~Gene~ral Theory of Optimal ProcesseB (K obafichey t,
optimallnykh protsessov)
PERIODICALs Doklady Akademii nauk 33SR9 1958, Vol 123,Nr 2,pp 223-226 (USSR)
ABSTRACT: The present paper originated in the seminar on the theory of
oscillation and on the theory of automatic control under the
diractknof L.S.Pontryagin. The paper Joins [Ref 11 and contains
Lne pruof of a general maximum principle (formulated at first in
[Ref 11), if in an optimal system a functi(nial of the form
T2
r fo(x(t),u(t))dt reaches a minimum. The proof baeas on
T,
I
variation processes and convex sets, where arbitrary bounded
measurable functions with values in a topological Hausdorff
space are admitted as control functions. The author's problem
generalizes the question given in [Ref 11.
There are 3 references, 2 of which are Soviet, and I Engliah.
ASSOCIATIONildatematicheskiy institut imeni V.A.Steklova AN SSSR
(Mathematical Institute imeni V.A.SteklovAS USSR)
Card A

160)
AUTHOR: Gamkrelidze, R.V. ')'OV/20-125-3-2/63
TITLE: Optimum - Rate Processes With Bounded Phase Coordinates
(Optimallnyye po bystrodeystviyu protsessy pri og-ranichannykh
fazovykh koordinatakh)
PERIODICAL: Doklady Akademii nauk SSSR,lgRg,Vol 125,Nr 3,PP 475-478 (USSR)
ABSTRACT: During the last years the author and others investigated
optimum control processes with the aid of the maximilm prin-
ciple set up by L.S. Pontryagin. (see lRef 5,62), The principle
supposes that the image point can attain all poasible
positions in the phase plane. In the preeent paper the author
investigates the exceptional case i In the phase space Xn let
the points and _~2 be given, they lie in a. closed domain
G. It is asked for a control function u(t) with the following
properties 1.) The phase point ' -1 on a
151 is to move 'o ~2
trajectory of ;i = fi(x,u) entirely lying in G 2.) This
motion is to take place in a minimum time. The author states
Card 1/2

3
Optimum - Rate Processes With Bounded Phase Coordinates ~J70V120-125-3-2163
that in this case the maximum principle is only piecewisely
applicable and he says in-which way the optimum trajectory
of the problem can be pieced together.
There are 6 Soviet references.
ASSOCIATION: Matematicheskiy institut imeni V.A. Steklova AN SSSR
(Mathematical Institute imeni V.A. Steklov AS USSR)
SUBMITTED: December 179 1958
PRESENTED: December 25, 1958, by L. S. Pontryagin., Academician
Card 2/2

GA.,IIKRI~,II I)Z E, R. V._, Doc Phys-Nuth 6ci-- (uiss) "Theury of optimal pro-
cesses." Moscow, ZAcademy of Sciences iSSR Pub1i:jhj..rq.--. House-
/I I ;Go.
15 PP; (Academy of ~Scicnces Mathematics Inst im V. A. Steklov.);
200 copies; free; bibliography on pp 14-15 (15 entries); (KL, 2~j-60, 121)

Re
:31
a
H pt
jail
t ef

16(1) S/038/60/024/01/001/006
AUTHORS: Boltyanskiyj.G., Gamkrclidze R.V.
and PontryaTJn-
, L
TITLE: Theory of Optimal Processes. I Maximum Principle
PERIODICALs Izvestiya Akademii nauk SSSR. Seriya matematicheakaya, 1960,
Vol 24, Nr 1, PP 3-42 (USSR)
ABSTRACTs The paper contains a detailed. representation of the results
published by the authors in fRef 1-6, 102. At the
Mathematical Congress in Edinburgh L.S.Pontryagin has
reportea about the most essen results.
There are 10 references, 7 of which are Soviet, 1 Gorman,
and 2 American.
SUBMITTED: May 14, 1959
Card 1/1

8o862
i6,el- 50 0
,'t.IJTIIOR% Gamkrelidze, R.V.
TITLEs Optimal Regulating Processes
3/036/60/024/03/02/008
for Bounded Phaae Coordinates
PERIODICALt Izvestiya, Akademii nauk 333H, Seriya matematicheskaya, 1960,
Jol. 24, No. 3, PP. 315-356
TE'XTs k function u(t) -'(u it.S.Ou r) callajd admissible if it is piece-
wise continuous an 'd piecewise smooth and if it has only discontinuities of
first Irind. Let B be a i6l.oaed domain of the phase spacen n)
let the boundary of B be a regular hypersurface of the Xn with a con-
tinuously variable curvature. Let the real scalar functions L(,X, U),
f'(3E,u) be continuoue'and continuouply differentiable with respect to all
coordinates of the vectors. I and u ." The motion of the image point
x - (xi ...,,n) is described-by
(1.1) 7(71 U)
wherp
(I*2) (f 1 (j,u), fn(-T,U))
Card 1/3

80862
Optimal Regulating Processes for Bounded S/038/60/024/03/02/008
Phase Coordinates
For given initial values, to every regulating u(t) there corresponds a certain
traje6tory T(t) of (1.1). Let U(5,1 ~2) be the set of all u(t) with the
property that the corresponding trajectories i(t) connect the points'~,, B
with eachother and lie entirely in B. Problem i Determine in U (~11 ~2) that
regulating u(t)9 ti :Et