Временное исчисление и его применение в контрольных задачах тема автореферата и диссертации по математике, 01.01.02 ВАК РФ

REPUBLIC ' OF GEORBIfi Tbilisi I. JavakhisiWi1i Stats University (TSU)

With the right of a Manuscript

' GELflSHVILI KOEfl

CHRONDLOGICRL CRLCULRTIO.N fiND ITS RPpLlCflTION . IN CONTROL TRSKS

Ol.Sl.OE - differential equation

THESES DULLETIM Submittanse of a thesis for a candidate's rfegr«;^

Tbilisi

- 139;3

The work has bean done at the Department of Cybernetics and Applied Mathematics, Chair of Control Theory, Tbilisi state University

Research suparvisarsi Doctor of Science (Physics & Mathematics), Academician

G. Kharatishvili Candidate of Science (Physics & Mathematics) I. Gogodze

Official opponentsi Doctor of Science (Physics I Mathenatics), Professor .

. ' • 6. Kharibegashvili . Candidate of Science (Physics & Mathematics) D. jgarkava

The defence of the thesis nil le take place on

-23» ' VI 1993

at /¿j PM in the Large Lecture-Halle of Physics (Tbilisi State University, second building) at the open meeting of the Academic Certifying Council (FM 01.01.CN1-3).

The thesis may be acquainted in the Scientific Library of TSU (2, University Street, Tbilisi 382043).

The theses Bulletin w«*s distributed on "22- V 1S93.

Academic Secretary of the Academic: Certifying Council, Candidate of Science (Physics & Mathematics),

Assistant Professor 0. /Vft^je0. Napetvaridze

Many problems for the equations with partial derivatives car» be transferred in to problems for common differential equations in Banakh space and further studied by methods well known for common differential equations, for the stationary case such an approach was developed by Hille, Iosida and Phillips and" forth» nonstationary case --by T.Kato. In the honstationary' case the most full results permit conditions which are difficult to prove and interpret. The last and also the freshness of this results partially explain the fact, that in the theory of the optimal control this approach did not find due application till now.' At the same time, in the case of similar approach to many problems of the optimal control, in the case of concentrated or distributed parameters, it is very important to gain . the necessary conditions for the optimality of high orders.

Independently important is a version of chronological calculus developed in PM-groups CChronological calculus is proofed, by academician R, Gamkrelidze and developed in works of his pupils}, used for the investigation of differential equations With unbounded operators. Here different versions of multiplicative intogralB. aro generalized. Derivatives in groups are defined for the'first time, an«)' analogs of well known correspondence of operation» of integration and differentiation are set'.

Scientific novelty and theoretical value of the thesis.

In. the thesis PM-spaces and PM-grdups are defined and studied. PH-spaces of regular one-paramoter sub-groups of some concrete PM-groups are calculated. . >

Chronological calculus developed in PM-groups is a new tool of research of differential equations with vnboundwd operators.

Besides this it has an independent scientific interest: first of all it generalizes different versions of multiplicative integrals; secondly in PM-groups differentiation is defined and connection is settled with the operations of differentiation and integration.

With the help of chronological calculation a theorem of existence and uniqueness of solution is obtained for quasi linear differential equation with an unbounded linear part.

For a quasi linear controlled system with an unbounded linear part necessary conditions for the second order optimality are obtained. Obtained results are applied to a concrete hyperbolic, symmetric system. .1

Practical value of the work. PM-groups appear naturally during the analysis of linear differential equations wi^h unbounded operators. Examples analyzed in the thesis show that the testing of the axioms of PM-space and PM-groups does not comprise any difficulty.

For the application it is important to calculate PM-spaces of. regular one-parametric sub groups of concrete PM-groups what also does not comprise great difficulties. It must be noted that this process is analogical with the construction of algebra Li for the group Li.

Tho obtained theorem of existence and uniqueness of solution of differential equations is free from some traditional hardly examined conditions C Y-admissibility, stability 3.

The scheme of obtaining of necessary second order conditions . developed in the thesis can be applibd to the< controlled systems with concentrated parameters as well ' as to many problems with distributed parameters and is illustrated in particular on the example of hyperbolical symmetric system.

"ihe contents № the work

In the introduction is given the motivation, aim, subject and the contents of the work.

In 1.1. is introduced the.notion of PM-space Cpseudometrical space? are discussed standard questions, connected with, the definition of the notions of convergence and continuity., sub-space and multiplication' of spaces etc. in PM-spaces.

According to the definition, the three (X.ClfO) Cor simply XD, consisting of set-carrier X, d;X*X— >R+ and 8;X— >R+, is called pseudometrical, if:

CPMj^ CX.dJ is'a metrical space.

cpm2) S^«Cx)J= S^cm(i)>, VleB. VBcX. c

where E is the closure of the set E in the metrical space CX.cD. Where the pair (d>0) is called pseudometrics.

When necessary, metrical spaces can be viewed as PM-spaces.

• Csupplement 1,2.}. Example 1.9 shows that PM-space is 'not necessarily topologlcable; theorem 1.1 shows that PM-spaces are in the class of pseudo topological spaces.

In 1.2 are defined and studied PM-groups Cpseudometrical groups^ and PM-spaces of regular one-parametric sub-groups of PM-groups. According to the definition the three (G,d,8) is called PM-group, if following axioms are fulfilled Cfor each geG • its inverse is marked by g) : CPMG^

(pmg2) (P^Gg)

tF::o.3)

G is a group.

(d»0) is a pseudometric on G. dtgi.ga)^^.^).

d^eSa.^^)^^ )fl(g,h)0(rc2).

(PHGe) «(g)=«(g). : (PWSy) ^^m^WBe)' ■

It 1< kM«ti ihut topological groups arise in connection with the consideration of groups of continuous transformation Cfor •xupl* th* {roup OT automorphisms of the space X, example 1.33. The -question of topologibility Cand more over metricabilityD of the PH-group OLClUX^i Ceutomorphisros of X, the restriction of which Is autoKarphlsws of . defined in the example l.S 'Is rather

Lcat«d reveals non accidental connection with nonbounded operators, Actually in 1.3.4 is proved that regular

orteparaMtrlcal subgroups of the group GLCX.XQ3 are ' exponential functions oT nonbounded operators of. a definite type.

In 1.8.1 the definition of' PM-group, ' subgroups and multiplication is introduced», the continuity of group operation« Is proved and are also considered other questions connected , with PH-groupa.

In 1.2.S is researched a set of regular . oneparametrical subgroups. It is done In the following way. On the set of oneparametrical subgroups on the group O the ecjuivalence relation i» defined: iatt))^ «.tb(t»t<aR . if exists CUP«®«. «nd 0a(t), Op(t) -- are infinitesimal functions of t , such that

a(a(t),b(t))ft|t|-K^(t),' |d(a(t)H(b(t))sp|t|+Op(t).

The class of equivalency, to which belongs the trivial subgroup 6CO~e . Ce - unit, in S> is cabled • **t of regular oneparametrical subgroups of the group G and 14 marked through Rq.

Is naturally endowed with the structure of PM-space and . it. is proved thaV completeness is a qudlity inherited from G.

-In 1.3 for some PM-groups that are often met in the supplements are constructed and characterized PM-spaces of regular ooeparametrical subgroups. In spite of the infinite' dimension and

coordlnitlM thas» constructions are analogous in the contents with the construction of algebra Li for the group Li. they are natural and conipact.

In 1.4 is developed differential and Integral calculus in the PM-group. The functions of real variable receiving the magnitudes in the PM-group are integrated and differentiated.

1.4.1 1« given to X-integrals. The main difficulty consists in the definition of the integrand as in the group O the aulUpUntion of the elements of the group by scalar Is not defined. Taking into account that, in . Banakh space X refection Ct-XDC, where tjb^R , and X is art arbitrarily element of X is a continuous'oneparaaetrical subgroup, the X- integral is defined in U» following wayi as integrands appear families ÍO^ ( * ) t t } >

belonging to the space of regular oneparamatrical subgroups Cthe only continuity Is not enough). Right and left I-inUgral« are defined naturally. So the'Unit, of integral products

< limit is taken at jflj-« , «here 0={ t^t^^St^'"i^St^tf}, jBl^naxí^-t^^ Jl=1,.,.,n) >, i»-called right Cleft) X-integral on Ctg.tj] from ÍBtf'J^ett >t J • Right il«ru X-integral

1« marked by ^ O^df) J ^ a^fdOj. Formula is received td

'. ■ ts,tf ... ...

connect right and'lef\ X-integrals. Por th<> continuous integrand

the «.xistance of , X-inte^rels, continuous and evolutionary

dependence from the limits of integration.are proofed:

In 1.4.a and 1.4.3 ' directional • X-derivatives • and X-

deri-vatives Cleft and right};in PM-groups are defined and studied.

According to our information this is the ' first case of defining!

- iS —

derlvitlves in the group. In spite of this definitions and results have got a natural form. We shall note particularly the formulas of X-differentiation of X-integrals ,at the limits of integration:

It ( A^IK ' . it ( ^ '

i t»Q * t» O t » tj

£ ( X a<H]=%- • Lit ( X vH]=\ • ->

were D I(t) denotes the right X-derivative of the function f in t-t0

thi. point tn • which according to the definition denotes

following: a= D i(t)eIL. and following two conditions are

t"t0

fulfilled:

(D^) d(I(t0)I(t0+AtJ,a(At)) is infinitely small unit from At . (Dg) There exist -(£>0 and infinitely small unit 0(At) such, that

Q<0 (TTE^Ti (t0+A t))-I<a I At I+o (A t).

If in this conditions are taken the left perturbation of the function f in the point tQ that is f(tQ+At)f(tQ) in stead of I(tQ)f(t0+At) the definition of the left X-derivative is got.

In chapter 2 the task of ■ optimal control with a terminal functional for the quasilinear system with an unbounded linear part is studied. Banakh, spaces considered here can be real or complex. In order to reach Jnaximal generality the derivatives are considered to be linear depictions of real Cor may be turned into realD Banakh spaces. Such a convention does not lead to any ambiguity because further there appears no need to consider the derivative in - other

In 2.1 the necessary conditions of optimality of the second order are adaittcd. Controlled object is described by the equation

and starting condition:

01 (1)

x(t0)=30,

U(') is called control and gets its meanings in sub-set V of

some Banakh space. Each control having only a finite number of

discontinuity Call of the first order3 is called admissible.

{A(t)>. .. , is a family of linear but not necessarily bounded td Vq 1J

operators on the Banakh space X and 1; [tg,^ ]»V»X— >X. •

To the corresponding to Cli homogeneous system is applied the following result, which is proved with the help of chronological calculation.

T h e'o rem 8.1 . Let X, Xq be such Banakh spaces, that Xq densely and continuously embedded into, X and unit globe of space Xq is closed in X . Let tA(t)>t6[t t j be the family of

generators of Cq -groups on X and let

CU Set D(A(t)), Vtclto.tJ, "coincide with a set

{xeX | Sup in"1!«'*0!-:)!.^^ <x|t|ii

C1U Xq continuously embedded into D(A(t)),- viewed with the norm:

Wdcacoj-IXIX+^J

CUD . 3a>0 .is such that for any . OeiR, tettQ,^] takes place saco, a|a| saco , sacts, a|a|

I® |BCX3Se W |® |BCX03-®- •

A A

(lv) For any te[t0,tj ] at t-»t uniformly along, s when 0<|8|£f takes place

sA£t3 sACtJ,

« * )|bcx0,x>*P-

0'

In this conditions there exists a unique family of operators

<S(8,t)) . _r, . - on X with the following properties: * O* 1

• CaJ. For any B.r,tett0, tj 1

Tit.t)^, KB,t)=(T(t,s))"1„ T(t„r)T(8.t)=iE(a,r).

CfcO. TCs.O Is strongly continuous in X along s,t and for any BitElt^ttj] takes place

o|t-s| al*--*!

|X(s.t)|K)D£e , T(B,t)X0=X0, |T(s,t)|BCXo^e .

Cci. For any XgXq and B.teltQ.tj] are fulfilled both the opposite as well as direct differential equationsi

(d/at)T(a,t)x=A(t)T(a,t)x. 0/as)T(s0t)z=-T(B.t)A(B)x.

In definite conditions for f Cin particular when the conditions of the theorem 3.3 are fulfilled} for each allowed control there exists the only solution of task C13 which gets the value at Xq and is differentiated at X .

On the trajectory of task C13 is defined functional JCu5,\

JtuHWti )-Hnln, (2)

which must be minimized. As usualcontrol giving minimum to the

functional, is called optimal.

Methods of research of- task C13-C83 is bayed on the

calculation of. the increments of the functional and the trajectory.

For each allowed c.ontrol the increment .of the functional i?

expanded in the Tailor series by 6, ££0» .

i *

JiUg ^^Wtuj^te.v.u/'SMfeW.v.uX^/Zj+o^). (3) 7

where U^ g „ is • needlolike variation of the control u:

C. tel8,&*e), (t), tcie.&fO,

and , tig do not depend upon 6 . In the case of the optimality cS u troa C30, taking into account the kind of coefficients £Sj , IL,, the e&nciitlons of tho I and IX order optioality are got. T>w» win result tuts got. the foil owl t-.g form.

' t> arguments continually are bounded accordingly in B( X) and

B(RX,B(RX)) Cher® RX denotes turning of the space X into real^.

-H9--

Theorem 2.3. Let the data of the task, C13-C23 satisfy following limitations.

CO Banakh spaces X, Xq and the family of the operators iA(t)}, f4 , satisfy the conditions of theorem 2.1.

te[tO'S1

Clll f continually reflects ItQ.tj, ]»V*X->X .

Ciiii SqEXq. Hltg.tj I.V.XqJcXq and for any compact subset VQcV we have:

Sup(|r(t„V0fi(to,t)i^)|x^ | tGltQ,^]. veV0}«o.

civD ar/ax b ^f/te2 are defined everywhere, depend upon Ls continually are t ?X)) Chero Rx der

.aiaVt^.v^) .

CV3 -—--- is the set bounded in B^X-) .

dx *

CVI3 <J>:X—>tR and- (jl* „ <j|" are defined everywhere.

Then, in task c15-c25 for the optimality of admissible control - u, particular on the region OsItQ.t^], EieG0>O, it is necessary: ■

13. Fulfillment of the condition of maximum on the set [tQ.tjlXO

0<t)i(t,u(t),z(t,u))=^W(t)r<t,v,a:(t.i}))K

vsv

S3. Fulfillment of inequality

(s(t)A?-i(t,u(t).s(t.u).))Ayf(t,u(t),3(t,u))+

a4vf(t,u(t>,s<i,u)) •

-HSKt)-—--JLi(t,U(t).2(t.U»<0

• dx v ,

for all. VQj(t), telnto , where 0(t), tclt^,^) is "a piece-constant multiple value map. ' >

Herei •

{<|)(t)>te[t ^ j is «-weakly continuous in (RX)* family

which satisfies the conjugated equation: •

0$(t)x Tf 0i(t,u<t),x(t,u)k* -,

h — [£Att>H-M-) №\x> ■

«JJt^ )=-$* (xc^ ,u)),

VXsXq. t - any point of continuity of u and

iJ)(t)eD((A(t)+aft>u/ax)*)

The family iS(t)>t£[t >L jCB(RX, (RX)*) is such that for any given Xj^ »^eX

(0(t)z1)x2=(§(t)x2)x1, Vtslto.t/i, '

t->(i(t)X^ continuously maps [t^ntj] in to Kp - ,

a , 9%,u . r r ait.u .

-(• (tjx, )za=-<p (t) (-^-x^-f® (t) (Ait).-^-]^]^

3it.u

dx

U(t1)=-<|)"(x(t1.u)),

t - is any point of the continuity of control u and

■ «(D^eDtiAttJ+flit^/te)*); Avf(t,w,x)=f(tiv„x)-I(t.w,x), ftu=i(t,u(t),x(t,u))e

Cas usual, u Is called special on OCit^tt^] if for each t€C? there is '...■••

u(t)=-jR£v| Iji(t,u)r(t)w,x(t,u))=ma3:^(t„u)f (t,7,x(t,u))^iu(t)}3.

I veV

Jt oust b® noted that in the finite-dimensional case our conditions.

of opticiality coinside with the results of G&basov and Kirilova.

In £.2 hyperbolic symmetrical system is considered. For any

5s{1 ,..„,n), XeiR11, te[0,l)] and Ve£kiRm it is assumed, that

EjtZ.t) is Ermit RiN matrix. b(X„t,V) is a IWJ real matrix,

N

yCx.O and cCx,t,v5 belong to C and is analysed the controlled

syit-tjci:

ay n ay

—e £ a. (x.t)-^(x.t,u)y=c(r.t,u)0

:t j-i J (4)

Ly(x.0.u)=y°(r), XcR11. tel0.1l].

Th» admitted control u gets its meanings in QdR® and has only a final number of points of' discontinuity Call of the first kind}.

Concerning the data in C43 we assume that: 013. t->aj('.t) continuously maps tO,T)l in (^(R11), J=1,..,n. CIX3. (t,V)->b(',i,V) continuously maps [0,1)]»Q in C(Rn), is bounded, and (№/8Xj

CIJI3. (t„V)->C(%t,V)< continuously maps iO,t)]»fl in L3(Rn), y°eH1 (Rn) and •¡0(»,t,7) J te[6,T)], Yeity is bounded in H1 (Rn).

Here CCE3 denotes set of all such N»N matrix-meaning

functions which are continual and bounded on E, C* (R^) is set of all N'N matrix-meaning functions a such that a belong 'to

They are viewed as Banakh corresponding supremum norms.

Now C43 is interpreted on Hilbert space X ■

e'=A(t)y+i(t,u,y),. <0)=y°, Vtero.l)], VyeY.

where

n ay

' A(t)=- £ a, (Z,t)-,

J-1 J Ox,

i(t,v,y)=c(',t,v)-b('t,v)y(«),

X=L2(Rn), Y=H1 (Kn).

and for each admissible control u the Koshl task C33 has got the only solution, differentiate Cln X3 in points, of continuity u and receiving meanings in Xq .

On the solutions ■• C33 a functional J(U)*$(y(* ,t}.U)) is defined:

J<u)=$(y(« ,T|,u))=J" r0(y(x,t),u))<lx-ffiln, (6)

• . Rn where satisfies limitations:

CIV3. f^iC^R is continual and for some d>0 and |l>0

|f0(z)|<a+p|z|2, . V^eC*1.

and (fla/fllj) spaces with

(5)

CV3. Is defined evgryvfhoro, continuously depends upon £

and for sore«? (^>0« there is

l^(s>|<at+p1 |ej„ vzec". CVI3. Is defined everywhere, is continuous by z and Iq' (E). is bounded.

CVII3. (ф')р(У) is continuous by у Cthat is <j>'' is defined everywhere^, .

In these conditions the task of optimal' oontrol is set correctly

fi)y(" ,t,U)

-=A(t)y(-,teu)+f{t„u(t).y(%t.u)).

8t .

y(X,0sU)=y°(l), Xe£Rn, te[0,T|], (7)

J(u)=<S>(y(',T).u))=J Г0(уй,1ьи))йм1п,

Kn

the theorem 3.3 can bty applied to the task С7} -and fo.r th® formulation of the'confirmations of this result an operator . is calculated conjugated to ACt3. For any yeD(A(t))p te[0,t)]:

[A(t)]*y=-iA(t)- £--]y. D(lA(t)]*)=D(A(t)).

' L dZj J

Th«? conditions of th© optimality of II order, that are results of

the application of the thso.-еш 2.3 to the task СТО, are

forlr.ulsttsd in tho thesorem 2. 4 . It must be noted that using tho

cpecificl ty of tho-task, the received conjuncted equation has got a i

usual strong sichtr '

r—ij(r.t)- A(t)- £-—ft)' (x,t,u(t))K>(i,t),

at L i!ij J

(8)

■ • »

t - is tho point of continuity of contra! u , e t )еВ(Л( t) } ,

ccniinual fatdly <ф('Л) >t6t 0>rJ satisfies In КЬг(КП) thj

cc-ivjvt.vcied «-»¿¡ii-alLon CB}.