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[Following isa translation of an article byT. P. Kirillova entitled "K Problems Subhchest-vo',arai opti l;n.yl• trayektorl"i )elineyrykhSlateI. (Zgllah version &bove), nIsVyse t.lk Ue .byih Zardei fhr~tr mt*tk1S---P~th*-

u4a_1 NeWS or g.Sber 3ducational Intitutions),W-3 2, Kazan, Par-Apr 61, Pr 41-53.i

Questions af the existence or optimal controls (ut erortain rest•ietions) ror the cas* of a trsaslent processescribed by a l1ear system cf differential equations have*eon Oxamned in .'&.sIl in works 11-5).

In the present artitlo wider the uMppodltion that theýontroi system Is givou 1'. non-21nolr differential equs-•t•nt, a theorem on the ezxistence of a solution of the op-'timal ccnt•ol problem will be proved. This problem wasýroposed by N. N. Kresovsoky and completed under his direo-&Lcn,

il. Let a dIfforential equation be given

+. It

?here x a (zl(t), --- , x1v(t)) Image vector, B(t) -- a matri,o1e elements o0j(t), I 1#, ... , J * 1, ... , r are con-inuous In te to, .. (L) - (u 1(t), ... ur(t) -- controllingector. As In [1-51 ue conaider that the vector u(t) islewooiie continuous and satlsftes the Inequality

..._. p akin ,(.we r.quire th-e inequality ,,a I .() I---at on thyp entire nt t*<j4ar tje relation n3Ijjv,')I>NV

S-20SWI~k~tMM~ kel 121OU ns amill3gM

a :1'-

N-- con~sltant.SThe rction f(a, t) -(f,(xI, .... nti 00 i as((x,

. 0nt)) is continuous In I° and possess continuous,o ndod p4rtial derivatives with respect to xz, ... , •,

'--I'ijL (L -- constant) arA f(O. t) -0.ex, L

* The optimal control problem consists In the followi*:;let at a moment of time t there be a point with doordin-ates x(to) w so which isýreveling along the traJeotor7 ofequation (1.1); it I* ncessary to sealot a controlling V06.4

Itor each that the point eaches the or41gins of the coordin-,atos in the, shortest time T. Ini this came u(t) is calledthe optimal controling voetor, and the ties T th, optimal

bcontrol time.

In 13 of this artiole It vwil be shown that if forSl.1 with some control u(t) uhieh satistls oondition(1.2# the coordinate origin to accessible, ther exists at

Least optimal control uO(t), also setiatsylf comDltiom(1.2) sO be@10 plcewlso coastant. The theomm is provediunder certain limitations. )i~osed UaPo a system of a IlaU-ear approximation for (1.i

Iore umose that there exists a sequenee or oontrol ves.tors u(k)(t) of the 0, (,.,)# the *ooVSPodn t -ejec-torles of x(xo, t 0 . u 19 (t)* t) of *lmob satiety t rel.-leat1one

x(x*0. i. 4r:). to+ TO)-.O. --I. 2....

'Vher* Tk ) T 1 and its Tk m T 0>o for k -,., sad the ssmsvoctors seti S the oonditioe ) ) Let there not *3dit acontrol u(t) which ucild "tiX oonation (1.2) and tbo i.jequae Iity

*(s' o .i (*). 4+*)-o for <rT.

enoeforth mach G sequeoe of oontrol veators u(k)(t)Will be called S 3ilnisaIing oequeie. our proble oPmntsn proving that If there eists a ;o;w u; 1)(t), for *

tbe origin of the oordinmtes is aoeSsble in sOme tie TIthen (wWdr certaln rostriotlova) there slsts a pecewseL

2

ou~~s~~ant opialvco)~ t) abuch that along a correspond-Ing trajectory x~~ t,.N1 Wot) 0 the origin of the coor-i'dtraOs Is I-eache1 In tl~aw To At first we vil! prove the

1COrrectn*8s Of lermas lagortant for t)k.e aibsoqpunt develop-

ILsas 1.1. If 404iI(4- _~() *(~ - control vectors4 ~he equAt~on (l.l)(,1ren for er spornding solutioge,t$~ x(xo. t, Uc W1(t, t) W¶k at * (X0, to,, U t

t)an accurate estimate is

fwhoro M'ma (1ad) Itsi

Proof. Soeing that

A'21 (1) A# 4 ' if)+BUS) Ism (*)) ds*

tren, tu~b?.rsettS %he second eqution from th~e first, we obita in

i.u ccaridln4 to the conditi'ul, K{S.it rollowst that

Thu~s, ut have

- ~3 -

veil) jr: (S) All" (

I• .•"'t• x'•'It) < LA Ill,"s)- jO (Olds +m MS 4 NMI

S gntrytin max Itb,,t) I 4%;t5. by N and applying leiii [7g ottatn the destred lneqAhalty

From sUCe %n Inequality, it esiily follows that if

rues E a •' i) -- a'(I)l. * ) --Q'"

(or *1ch3 - & 2, .... r, a ) 0 and k-.. then Xzkl(t)_.X Ofror k

LOOM 1.2. If U -- siniaisin equenc or theý,?uatlon (1.1)v thena the sequenc, of trejectorles x(Zop to#kj k)(t), t) contains at least one waiforly convergent sub-sequence . l - , o. "" (I. t). X. X Ai i.. + T.

I;,By the symbol is(•)#- W*('•)2 esint the 8"Buo'of tne s@elt In hich the Inequa !•l ~ )-•/ (1) 1;s to Sat-

= ~ Is le 0 . Sal 22 1 1 i 1.........

2y 6&4@ 1:* f olow. that tb* sequenao x )()(t) Is boun4,d Unif OiM in I 4 t4+. 31n~c* the~ f.Mntiofls f (X. t

kha oneu z)t are bound~ed In t (unifors In k), tLeln from

1Ih' In~alt

CPAlows *.he equ -4egpeo contim~uty' of the runetions xc t

Tfl..s, V tanwe g nt tons 4 (1), k - . 2. ... . 1# < j + T are,u~nifornuly bounded sz4 equi-degroo continuous. COn~q'SCt1Y

!1611 Lnelixeztst &L 21-aif Cue uzi4fpoSy Conlvergent sub se-QUeTICe x Mt of %rhe sequence * x (t), 14< <7o .[Subsequently, *e will coohslder OtAt limits the generality

ofrwasoningi If u(k)(t) .- SinolsIing sequence, of the con-

Itrol vectors, Owbn the correspond¶sUeOIts$ Ct equation (1.1) z(XO0 , to.0 t), t) converse unl-irorvuiy to @one *oen.L1-.houe function XOt) tI~ ~g-T

W* will examine thoe quaation of perturb" motion for

vItht~ he veri3tion of tths zontrol Ou(t). As is kno"n

(7 T1, p-296) , it has trse form

jI ).

*her thefunction 0(.3x# t) stis-fie the r*lationships

r feqain(.) Thas asomabvLet u~(k)(t) -- mininsaIng sequence of the *outr Too-

... xu~,If ~ ~~.T. It the matrizi function

Is computed along tho curves xC(t), x (k(t)v k w 10 2#_

*,then the equation of perturbed motion cOAn subsequienti?beWritten &sB

týtelinear approximation eqlation aorrhSpofl4IDS to *4U5

6 () .MVk)(),then for the corrospoMinS S0lutILOfSýNY~ý) a~k)t) of eaaations, (1.3) sand (1.4) the follow-

114 to tU;

LOMS 1.3. The sczlutions ckf the systems, (1*3) almd(1-4) satisfy the inequaslity

IJA (1 - I* 1)k in1. 2....

?"of. Oni the Vounds of legi 1.1 we have

fo

".Irsa th~at the mtrix fl4WtiOfl to o@utS.Rm*OU aeoGOdlf to

condition, uma uaO fumction O(i2 (k, t) satisfi~es the

apschltz 0oftations,

whore I __ constant, Spafs w.1 Ole Ong wazto for suffi~ciently

'inM recalling the abov&1 so have

is. C it s.) ofor

'If. we subtract equation (1.4) from (1.3) and Integrate'h rough,, cosdrn the correct inequality

Ala)~.efor -O

"~We f Ifally obtain

~whre 0 for 4.- 0,, unllle the value Is awiif ore11

C

J2. We will *exaine t'ne equation

I her'e the elements iet(~ or the matrix P~"l(t) are equal

to the frwotIons f& alaulat*4 al~ong Uw curves z (k(t),

Ve will Intro4uat notation& If it is meeossary to ex-pul.ne the J-calum er the matri.x Bart) then~ we viii write it,13 13,(t), ?or a ftalar Grivatlve of the vectors t andF )(t) B3(t), *her* j)t mtrix,, laver"e to the twa-Jdav!ental matrix of equation (2.11 with b(t) a 0,, we will A

ye suppov* that the relatIons

- ~7 -

•l~r,•'(~ajOl-,, •,2• j l .. ,r k-0. is .... (2.2)

hold only at distinct isolated points. As Is known [•],lf a control vector w(t) is such that a peter moving frmzo along the trajectory of equation 2,1) encounterg the or

11$n eor the coordinate* at moment of tbwe o, then the vec-tor w(t) is a solution of the systes

4, ,(9) a (4) W (t) Il. (2.3)

Mere none of the numbers z0 are conneted with the original

problem.

Now we will examine the m•inlsing sequence of the con

trols u(lk)(t). The control time for the trajectories whichcorrespond to them equals Tk and T k -T for k-- , Tk -- decreasing sequence. Consequently, there exists a Sequence• <, hilch satlsles the Inequalities Tk - -k < T and the relationship l -- 0 for k - We. We suppose

1 I.f)a (t),,",(I) dt--nr.,. *no1. 2.... (2.4)

It Is clear that of the points ) 1, 2, ... ,along the trajectory of the equatlon421) the origin oft•e coordinatesto accessible (even with the controlUk(t)•)). Ifr by the synbol o('r7 - ,%) Is •,4.r•too4 theset of points of phase space a. for which the origin of thecoordinates ts accessible In time t< rs-- along the tra-Jectory f, the equation (2.1) then, It to apparent that the1points z k, k w 1& 2&, 000P belong to the corrospondlng re-i

Froa the results of the worie (8) it follows that eachof the regions O(Tr - ik), k m 1, 2, *,., is convex andclosed. We construct lines frou the origin of the coordin-ate* to the intersection with the boundaries of the region&

(T - 'ic), I• k 1, 2, ... , and we designate the greatestdd ANe te 0 off.^ bo__22daaofn of

8

Ithe regions G(k- Irk) one ~ l-, k ;; t 2,#..lo'otained by the rersulto of such a construction by dk. Then!it can be proved, considering the resul•t or eAticle (9],,tat d, -. 0 for k-..

Zubsequently we will speak about the fact that the Bo-',, •df.enceB of regions G(Tk - TO) converge to the region 0(T).

S;l the present case, the Limiti region G(T) rcnai&ts of.p:intb, for which the origin o,1 the coordinates in acces-sBile in time t• T ar.orn the trajentory of equation (2.1)

,for k - O, I.e., when the functions Ar_ are caloulated al-

jong the curve x0 (r), and is, as can be ihown, considering10o supposition (2.2), a non-empty set. We will designate'the boundaries of the region& 0 ( T k - Ok) by rk re3pectively.IThe following assertion is proved;

Larns 2.1. All limit points of the sequence s(k) be-Ilong to the bouniary i- of the •igon G(O), 0

Proof. In vte4 of tha above mentioned, the sequencelot regions O(Tk - •, ig uniforinly bounded in k. Consequenýt'7 the set of Points zIkj hAs at least one limit point zO.'• s:,4ll prove that z 0 telong to r.

Assume tha contrary. Then, in lit of the fact that•ne sequelice or rogicnrs 0(Tk - T•k0 overges to O(T) for

-, tere exista it subsequence zý , every point of whichýcnrl u surrounded by a srtere of radius pk,(pk, -. pO for t--- ) ) P0 0 Q, whereupon, all points of the spheres be.'10ng tO the corrtspoudlg regions of the coordinate lines t,he intersection with the boun4ary 1'k, of the region F(Tk,

Such an a&Uxliary ocnstruction ifr each point S-(kj) ofitne ofa 1 sci of the sphere of radius Pk, %itn center at the!Spoint z0, puts it in corre4 ordence with somn point (k1 )of the boundary Lr•" Ii .(z-3 .. the "atio of the dim-

tance from the origin of th kOaOrdinSt*s to some point ofthe surface of the sphere s (radius of the sphere Ph 1 ,

, .. . - -

center at the point to the distance from th originof the coordinates to the corresponding point z of the'surface Fkj, then it is not difficult to see that the funo-

,t~on ((ki)) sati.ri a the inequality 0 < m( () 1 ad:s continuous in it "Tl

As is known [3-4], ;or each point Z(kl) there exists aunique optimal control V ,wich is a piecewise conatantfun tion, 'rom equation (2.3) it follows that tiaeveotorfQ(!Rkj)w;k)is a controlling one for the point- kZThus, for each point of the surface of he elhere z acontrolling vector taken a function cp(iz--(. kl) From theproperty of continuity of the function w' k I mtial values and from the coin

obtain ~ ~ ~ ~ 12 that0 th0ucin1~i~ ~ o(obtain that the function e(x(jk (k )(t) also is continu-ous in • 01)

We will examine the function

V1,) (t) - U`11) () + A is (Z-=111) P(" al)l" (2.5)

where A 1tJosfies the inequality 0< < Seeing that thevector u (kjSt) is a controlling one for the initial condl.-ditions zO , then for the substitution of A into the men-tioned limits, formula (2.5) gives a control for each inter-

ior point of the sphere of radius pk, with center at zo 'rkdjor X = 1 we obtain controlling vectors for the points

zk of the surface of the sphere.

It is clear that the function %(kl)(t) is continuousIn te Initial values and batlujies the inequality

SWe rwmber that we call w 1(z, t) continuous, accordingto the Initial values z, if for each t > 0 there existssome 8) 0, that the inequality

Ms P ( 1, 019 (S,. t) - VO* (.I'. t) I :;.,) < a. a > o.Is fulfilled only if

- 10

< IV

aa Ale, (z) I[ u.e I I,•t) - utt (t(2.6)

¾ k,,:, exami.ne the linear approximatirng equation (1.4) with

Let 6x,(,%tO) - 0. Then the variations 6x (t) are

I. "t. 2. 7 1kF

ve'rtalic,3u t, fornulas (2.t and (;.5' agre

.1la Ir ýSpaice z itt the momen~t 01, time t M t0 + Tj Tc h

~n c the ccr~iietqlesi reached from~ each oint of therax-dlus p, N tn. itf"osram (Z-.7) that the

f'- "he rnjectorl. s xt of the s•c

rx rt L!e xrimant t . tO+Ik - 'Tkformn an ellipst by a non(k' a r Tk of the sphere

• ~~~~( +O T k i h

~F k'At ~ t + Tkt - kr) ,F.7(T) for 1...a

Iie tre seopenc cf the entlpsoi45, defined by I2.7)0 has,.,!* l an ellipsoid, which can be fcun from the o re

,~'of radius po by a diependent transformation 70 (T). But

,.... , i.• ' T1. 0 for L - a. we conclude: there ex-

,jSt somJe.rumn~~ber 11, beginning wi;zi which all the ellipsoldO

11) c-jntann the origit ofo the coordinates.

.,We return to equation (2.1). Prom the continuity of

L,-ie , ng vector w,( (t) (soee (2.5)) the Initial

ka-,lu z eki and from fo + 1ja (2, '4) it follows that if wo.ie..re•se the contro 6u, W +, r - '2,e

5e=tK .•t (•)(O %P L (k/)- o( fo (L -.- ,

Q.r",, 0t•. i a 1,then the controls uh 1rt) r h

Correrpond to the u iatlni conditions z of

!• cn~lLn vctrw 11)t -se(.) h nta

Wonsequent17, for the deorease of the controlS of 6u i'(t)fn time e <• Is the rali. of ttso sph.sres decrease In time 49.

We designate a raftus-vector coming from the center(to + - Tic,) of an ellipsoid 4efind b equation

05.7) by j then, obviously, for a decrease In the radius

f a sphere pk, in time a the magnitude of %be rsd•ua-vectoz

e0omes equal to g%,"

But all ellipsoids for I > 11 contain the origin of tho0oordinates. 0P9lYIns laa 1.3 we obtain mat ItA4 U) -

4A4 (t)I ab1 - UheI *k 15 equal to the < N-wiheIs of the ellipses and A is equal to the distance from

he origin to tMe po in "t)D > s A )

13.Thus, for the vaiation x'"(').' w•en the oontrolling

sotors ar" transformd according to the formula .1i (-z0)(1h)() - u'9 (t)h the points6 of the trajectories £(&i() +

+ 9 (tJ) in a moment of time t - to + Tki '- kl comprisea continuous set 04l, agreeing with lmia 1.1. In view ofthe above mentione and lmas 1.1, we arrive at the follow-

I The ellipsoids of the radius-vootor e•f, for a < 'I Mi > 93s which contain the origin,, art continuously mpe

Into continuous sets Qk1 ,!t'•so contain the origin.

Conseque•-tly, we find la the conditions of applicabil.11ty of Ae te.soroe on th. existence of a root ((10], p 573)

that in our case it corresponds to the existence of a tra-jectory for equation (1.1). along whieb the origin Is reacfrom the point xO in time t m - 1 kt < T. But this toImpossible according to the condition of the problem.

It means -., 0 for I .. I n other words, the point

12

Sbelongs to the bound-try L of the region G(T).- The lem-m is proved.

Insofar as the point z belongs to the boundary of theregion G(T), then the time T is the optimal control time foi1the Initial values zo, and the optimal controlling vectorýIs found from the relation

==A/(t • stg r -, ;; (:) 8.i(1)1). (10.=0 = - I - o.. r. (2.8)

where the vector 10 ti obtained by the condition

I 14. r m

§3. Now we shall prove on the basis of the results of§1 and 12 the following theorem.

Theorem. If 'or equation (1.1) for some control P)t the origin of the coordinates is accessible in time T1 ,

lar•d th'e linear approxinating equation satisfies the condi-10onS ("1 .2'2 in the region lxg(t)I<nrMIV T er,'(M=-mSxlb,,Q )I,

.to<4 1#-+ 71,). then there axlate at least one optimal control*wfticii Is a plecewise constant funitIon and is defined by therormulas (2.8).

Proof.') i1 the controlling vector u(l)(t) is not op-;Llmal, tihen there exists a control u(2)(t), for which the!orig.in is accessible frcn point x0 in t.me 12 < Tl.

By reasoning similarly, we w•ll convince ourselves ofthe extstence of a minimizing sequence of controlling vec-tors u k)(t), k - 1, 2, .. , for which the control time is

Irespectively equal to T , Tk+1 < Tk, 11M Tk = T > 0 for k-.

The inequality T > 0 follows from the fact, that the,right parts of equations 11.1) are bounded. It can occurthat the number of controlling vectors in a minimizing se-quence is finite; this correspords to the case, when for

TV The theorem can also be proved by application of L. S.)fntryagin's princi.ple of the maximum.

- 13

ýqUatlon (1.1). besides ttae control. u(1)(t),, there exists ar lnite number of controls, for wbioh the origin is aoces-Bible in time t < T 1 . In Luch a case we Can immediatelyjapeak of the existence of an optimal vector u(t).

The vector u(t) ib computed by the formula (2.8). Thisir2act will be demonstrated below for the more general case,namely, when the members of the minimizing sequence are die-tlinot.

I nu, let (t) be a minimizing sequence of control-ling vectors of the control (1.1). As was shown in the baslIf the lemma 1.2, the sequences of *-rajectories .h:z(.is, uW'11t),) uniformly converges to x6(t) for k-.

Xj () ap*rM T, eL T],

where

A,=-M.XIl,(1) . to<d <-.t. TI,

1(see lemma 1.1), then we can construct a sequence z k)

'which is defined by f rmulae (?.4). In the basis o£ lema12.1, the sequence 4 has at least one lit)point z0,)which belongs to the boundary a(T). Let z " -. z0 for -

** =•.We obtain the difference

•r

J(FD(e B(uF;j).-p' (1) B8(1)0 u jdt -

14 u()()aQ).u~(~d

+ --Fi() F;'- (Ol) (1) a'~"(1)t) d -

We introduce the notation

- _- 1* __

-:• + 4" +f IFilt)- I'F (1)) 8 (t) I(11 dl-

t01

- (' J•j (e)8(1' b ) . (t) di=C,,

4t Is clear teat ck - 0 for e-6 . But

cki .m f Fc-' (t) B(1) dt(t) a i-- (1))ld.

'We scalar multiply the vector z0 (see (2.8)) by ck,

(t (. cat,) == I" ;P-[(Fo-'(f) Bj (i)j) [ujo (1)- ur• (i)l di.

iBut uo(t)=NIsign(PfFIo'(t) B;l(•), (%)=--l and by the condi-t.on~ of the problem I Ua,(.oj)j N< , consequently,

sign (PIFV' (1) B1 (Ql))-- sign (ut (t) -- urO (t)).

Thus, there to obtained

(Pc) =1 (4 IFiO (M)BQ JIj I()urh?(t)I dt.

)rom (&0 ckL)-- 0 for e•-. w and condition (2.2). follows that

rmesE( I uj* (I) --- Ir (tl•>a)--O fore and for each -1 23r. That means the sequence u(kj)(t) converges by uas-

re to the f)nction u0 (t). It is not diftioult to see thatthe vector u (t) is optimal for the control (1.1).

Actually, seeing that the functions x(k )(t) are solu-Ions of the equations (1.1) there are the relations

__.- 15 -

F-a

j L )$A• -. xO(t) uniformly for e-, u(kl)(t) - uO(t)r-ia,týre, the function f(x, tv is continuous A, consequen.

'y il"i, can In the limit be put under the integral sign.

0 I.

SThuA, the trajectory xO(t) corresponds to the control

( of equation (1.i). The equality x(xO, to, uO(t), to

ST T 0 immediately "ollowo from the continuity Of the

ýunctione xk(t) and xb(t) in t. We a8 the same time provedht.at the optimal controlling vector u (t) -, computed accor-

Lding to formula (2.8), i.e., it is a piecevise constant funa-

Ition. The theorem is proved.

I The restrictions (2.2) imposed upon the linear' approxi"itiou equations, can be expressed in a form, which In a!Series of -ases permits in view of (1.1) judging the exle-Xence of a solution of the problem of optimal control with0 pIecesi3e constant controlling vector of the type (2.8).Such conditions for a linear equation with constant coef-fIcients are given in work [3). Sxfficient conditions wille Introduced below for the fulfillment of (2.2) in the casef a linear equation (n,<3).

We will examine the linear approximating equation for'(1.1), with the corresponding variations 8u(t), when the

iatrix function - is calculated along some curve x(t)

!-ri = t a 1 +B() u()

t will also be proved that the assertion Is also true if

he vectors, B,(s), RIY(t). d, t) P (t), where P (j())

dBi(f) _p(t)B(t) for all J and x(t) arising from the equa-

16

lone (1.2)0 are nonool~litt ar, then the relation (2.2)1 istrue. Actually, we will exa•ine the vector Hj(t) - V- (t)

d - ( 1 tPI " '1 1, t h e nl3 (t)s seeing that -di() Q 7,te

%L dt "--t)

"We call PP (t)B,(J) by R (t), then we will obtain

jaformula for the determination of the second derivative 1"

H; () .-= F -I - P(:) R (t)

di Let the relation (d ( tli t)]) -a ) hold In a sets,diatinct from a collection of is4oated point, Then the relations (IE 3(t)) - C, (IFO(t)) - 0, and (1-Hjlt)) = 0 hold±n the same set. •t l tAhis Is Impossible, seeing that aooor]lding to condltion the veotora

Sdi

are non-collinear and the mtrix 1P-l(t) is non-singular,what was required to prove.

Let the vectors

Bi (g), R~" (t) - - PQ() BI It), - P (I) R" (t)

tbe non-collinear. We will uabsequently call such a factcondition (A). Then the theorem of existence of optimaltrajectories oip be expwressed ast if for equation l1.1) for'some control utL)(t) the origin is accessible for t -TIOand the linear approximating equations satisfy the oondi-tions (A) in the region

17

i)eoeuse constant function and 1.3 defined by formula (2.8).M shall write conditions (A) in clear form for n - 2, 3.

*or n - 2 we have

2

A bq(1, bi (1)dt ax -,t aa~l

y the s1mbol [(1)(t)]k is soent the k-component of the

otor R 1l)(t) .he entry Rk - 1, k, ...k m1, mehe veotor R(15(t).ro

Then condition (A) for n - 2 reru9es to the non-collin-arity of the vectors (bl, b 2 j)" [Rt 1 )(t)Jkg k - 1, 2. For

,k-3

3" t,: " ),, _ . b,,), k I, 29, 3.

*e defIne the vector, f (g) j5 - - P(j)R•' (). Seeing thatWitRj")()=-• p)a() M 8

then

0 'Bj (t) dP(t) Bj()-2P (i) i) + )Bdt dt I

here pu(I),,,- means,

m,,t I ", '!,

nd, consequently, we finally have

itN _ • =. . = =. . J, i I :-l-B

I'j lit -. l 1,j•1 f4 .hrdt3 S$,I XIm ) MWI

Thus, If the vectors b bj 1 b2j' b 3J)9 (P-!1)(t)]k k-S1, 2, 3, # - 1, r, are non-collinear, then oondition

.21 for the linear approximatin" equation will hold, Uhl!In thia cas" (for n w 3) for verifieation it is sufficientfir (2.2) condition and necessary to compute the derivativeski until the second order exclusively from the functionsbi(t), f(x, t) in x and t.

ýTa llskiy Polyte*nnjoa.& I.nstitute Receivedimeni S. M. Kirov 29 Jan 1959

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6. L. A. Lyusternlk, V. I. Sobolev. Elements of FunctionaAnalysis. QITTL (StaV. PublihsIA1 House of Theoreti-cal and Technical Literature) E.-L., 1951.

17. V. V. Nemytskly, V. V. Stepanov, Qualitative Theory of

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Differential Equationis, 01fI, -

8. N. I. Akhiyezer, X. 3. Kreyn. On Certain Problems ofthe Theory of Moments, Article IV, GOOTXZ WV (StateUnitad Publishing House of Scienoe and Tsehnology),1938.

ý9. F. N. Kirillova. On the Correctness of a Statement ofa Problem of Optimal Control. News vusov, Mathes. NoL •4, PP 114-125, 1958.

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