Optimal Control of Parabolic Singular Perturbated Variational Inequalities

Copyright © IFAC Singular Solutions and Perturbations in Control Systems, Pereslavl-Zalessky, Russia, 1997

OPTIMAL CONTROL OF PARABOLIC SINGULAR PERTURBATED VARIATIONAL INEQUALITIES. Vladimir Y. Kapustyan Projellor, Hcad of Department, Dnepropetrov,c Tran6port Univer6itll, 2, Acad. L4zaryan Street, 320010, Dnepropetrovd:, Uiraine

Abstract. In this pa.per the a.uthor's investiga.tions results in the optimal control singula.r perturba.ted problem's solution for pa.ra.bolic varia.tional inequa.lity a.re reduced. The problem's solution is constructed on the basis of the optima.lity conditions in local form, which a.re represented by special form of the one-sided ma.thematical physics problems. They a.re characterized by the presence of sets, on which the soln.tions loose the smooth.. Obtained and proved in the pa.per asymptotiC5 consider indica.ted peculiarities of problem. Keywords: optimal control, decomposition, asymptotics, singula.r pertuba.tion method, partial differential equa.tions.

1. INTRODUCTION.

where y(z, t) is the solution of variational inequality of parabolic type in (Barby, 1984, 1993)

In this pa.per a.n.thor presents some investiga.tions on the optimal control problem solution for parabolic singula.r perturbated variational inequa.lities. The presence of small parameter in the main part of differential operator allows to llBe asymptotic analysis methods and author's results in analogollB problems for solution parabolic equa.tions (Kapustya.n., 1996). Asymptotic solutions a.re constructed on the basis of necessary conditions of first-order optima.lity in (Ba.rby, 1984). Zeroth components of corresponding outer decompositions of optima.lity con.ditions break definition domain of problem on two classes of unintersectional sets: 1) th.e first one generated by solution's structure of varia.tional inequality; 2) the second one is defined by control which assumes outermost meanings its limita.tions. The indicated sets boundaries as the solutions a.re speci1ica.ted to a.rbitrary order of asymptotic exactness with inequalities optimality conditions preservation. Discovered asymptotic solutions &re substantia.ted.

(Yt(z, t) - ~ 6y(z, t) - g(z)v(t))(yCz, t) - ~(z)) a..e. in Q,

Yt(z, t) - ~ 6y(x, t) - g(z)v(t) ~ 0, y(z, t) y(z,o)

a..e. in

n,

y(x, t)

= 0,

-Pt - ~ 6p

= y(x, t) -

~;

z(z)

a..e. in {(z,t): y(z,t) > ~(x)},

p(z, t) = 0, a.e. in~; z(z))2dz+

a..e. in

n

T

= ~ J(J(y(z, t) -

(2)

a..e. in Q,

wg-

Consider such optimal control problem about obstacle: find ,,(t) E U = {v: vet) E Lz(O,T), I vet) I~ ~ for a..e. t E [0, TJ} such. that

o (1 +vv2 (t))dt _

= yo(x),

~ ~(z)

here Q = n x CO,T), ~ = an x (O,T), n E Jl"'has compact closure and smooth. (from COO) (n-l)measured boundary an, z(z) E .&.lcn), g(z) E L'1cn), 2 yo(z) E /'Io'I(n), !fJ(z) E H 2 (n), ~(x) ~ 0 a..e. on 8n, yo ~ ~Cz) a..e. in n, q >max(n,2), 0 < f <: 1, " =const> 0, 6 is Laplase operator. Problem (1 )-(2) has at least one solution u. Let (y,u) be a.n a.rbitrary pair in problem (1)-(2). Then (Ba.rby, 1984) exists a. function pE .&.l(0, T; HI(n)) BV([O, TJ; Y-), Y = H'Cn) nHl (n), $ > n/2 which sa.tisfies the equa.tions:

2. OPTIMALITY CONDITIONS.

[Cv)

=°

p(z, t)(g(z)u.(t) + ~ 6y) =

(1)

a..e. in {y = ~},

min, 31

°

(3)

p(x, t) = 0 a..e. in n, u(t)

-{, (g,p(" t)) - v{ > 0, _v-l(g,p(" t)), ~, (g,p(" t)) + v{ < 0,

={

where (g,p( ., t))

{

yo = -g(z)e, -Po = Yo - z,

(zEno, tE~) :

(4)

{

= Jg(x)p(z, t)d:c.

yo = g(z)€, -Po = yo - z,

3. FORMAL ASYMPTOTICS.

{

Find the outer (Na.za.rov, 1990) decomposition of the solution of (2)-( 4) in form

y(x, t)

= E cy.(z, t); i=l

E CPi(Z, t).

jj(x, t) =

i=1

ASSUMPTION 1. Suppose z(x), yo(z), ",(x), 0 ~

g(x) E CCO(n).

~ 0,

!io,(z, t) - g(z)iio(t)

Yo(z, t)

Yo(z, 0) = yo(z) in -Po,

= yo(x, t) -

~{

n;

(13)

(14)

tE~ :

(g,Y020 = _v- -" 9 II~ (g,p:'0)o, { -(g,po)o = (g,yo)o, Yo = Yo - z. l

(15)

The problems (13)-(15) may be solved by dint of pha.se picture (Boltys.nsky, 1969) which allows to define the control structure. Then two ca.ses a.re possible: 1) pha.se point «g,yo)o,(g,po)) doesn't go on bounds, i.e. it belongs to set PI = {O < 1/ '). 11 g 110 (g, Po)o < (g, yo)o < oo} (g, Po)o ~ U{-ve ~ (g,po)o < 0, -00 < (g,yo)o < v-1/').X X 11 g 110 (g, Po)o}; 2) pha.se point goes on bounds, i.e. it belongs to set P'). = {(g, Po)o ~ ve, v- 1/2 I! g 110 (g,po)o ~ (g,yo)o < co} U{-ve > (g,po)o, -00 < (g,yo)o ~ v-l/'lll g 110 (g,Po)o}. In the ca.se 1) the solution ha.s a.n appearance:

(7)

Cg,po(" t)) + v€ < O.

Intrude sets

ve, v-

Qo = {(x, t) : y(x, t) Q+

no.

(6)

t)),

i=/:i,

(g,Yo)0=lIgIl5(, { -(g,po)o = (g,yo)o, (g,po)o + ve < 0;

"'(z) in Q,

-€, (g,poe-, t)) - v€ > 0, ~,

.=1

tE~ :

= 0 in n,

_V-l (g,po(-,

U1?, 1?n1j=0,

= 0 in Q,

= 0 a..e. in bio = "'},

Po(x, t)

T=

(12)

z,

e,

z(x) in {(x, t): yo(x, t) > "'(x)},

Po(x,t)g(Z)iio(t)

.. Ct)

~

= Yo -

(g'yo)o = - 11 9 II~ { -(g,po)o = (g,yo)o, (g,po)o - ve > 0;

Zero components of decomposition (5) are denned like the solution of problem

(Yo,(z, t) - g(x)iio(t))(yo(x, t) - ",(x))

-Po

where (., .) denotes the scalar products by Let's go over to the problems for (g,yo)o, (g,po)o for definition of zero components of the control switching moments tE~:

(5)

.

Yo ~ -v~lg(X)(g,po)o, 3

.

00

(11)

(g,po)o + ve < 0;

(z E no, t E ~) :

Cl

00

(10)

(g,po)o - v€ > 0;

= {(x, t):

= ",(x) a..e.},

y(x, t) > ",(x) a..e.},

Q = QoU~, QonQ+ =

(8)

e,

where y(x, t) is the solution of (2)-(4). Then Qo, Q+ are their zeroth-order approximatiollB which defined by conditions

(g,yo)o = (yo - z,g)och«v-l/'l !I 9 110 x x(T - t))(ch(v- 1/2 11 gilD T))-l, (g,Po)o = vl/'lll glia l (yo - z,g)ox xsh(v- 1/2 !I 9 110 (T - t))x x(ch(v- 1/'). iI 9 110 T))-l

I

Qo = {(x, t): yo(x, t) = 1/I(x)},

Q+

= {(x, t):

Q

= QoUQ+,

yo(x, t) > 1/I(x)},

QonQ+

(9)

=',

(16)

on condition that initiaJ data satisfy the inclusion

{(yo - z,g)o, vII'). 11 9 110 1 (yo - z,g)o x xth(v- 1/2 11 9 110 T)} E PI.

where yo(x, t) is the solution of (6)-(7). ASSUMPTION 2. Suppose that p(z, t) = 0 in Qo and let Qo be a. cylinder in JlR+l, its base be two-connected domain (n\no). Suppose that outer boundary is equivalent to an and inner boundary (an) possesses properties of outer boundary • Then the correla.tions a.re fulfilled in Q+

(17)

In th.e ca.se 2) systems (13)-(14) have the solution. in o ~ t ~ To (To is th.e moments of descent of control from limitation)

(g,Yo) = =F{ 11 9 II~ t + (g, Yo - z)o, Cg,po) = ±{Cv - 1/211 9 II~ x { X[Tg - t 2]) + (g,yo - .2o)o(To - t) .

(x E no, t E ~) : 32

(18)

On the segment t E (TO, T] system (15) has the solution

(g, Yo)o

= ±{/ll/l 11 9 110

by bounds Yo(x, t)

X

no

xch.(/I-I/l 11 9 110 (T - t»x

x(8h(/I-l / l 11 9 110 (T - TO)))-I, (g,po)o = ±{/l8h(/I-l/l 11 9 110 x x(T - t»(8h(/I-l/l1I 9 110 x x(T - TOm-I.

Let be a system of expanded sets which belong to n and contain (boundaries of the indicated sets have the properties of the boundary an). Then 00 is the solution of the optimization problem

(19)

no

~

At that the inclusion takes place

{(Ye - z,g)o, ±{(/I - 1/211 9 115 T~)+ +(g, Yo - Z)OTO} E Plo

= g(x)v(t), yo(x, 0) = yo(x).

T

J( J (Yo(.1:, t) -

060

z(z»2dz+

J (fjJ(.1:) - z(.1:»ldz+ 0/60 +/I( J g(z)po(z, t»2)dt -+ min +

(20)

=

Since the function (9, yo) has to be continuous by t TO, we obtain the equation for definition TO from (18)-

(26)

00 by bound (17),(25), Yo(z, t) is given by the representati:>n (24) and scalar products (',')0 are calcula.ted

(19) =F{ 11 g 110 (/l 1/lcth(/I- 1/l 11 g 110 x x(T - TO»± 11 9 110 TO) = (9, Yo - %0).

no

by in all terms. REMARK 1. Suppose that yo(z) > t/J(z) V .1: E n. Then n and it is enough that the condition (25) is fulfilled. It fulfiles always if (Yo - z, g) < a • q In case 2) the solution of (10),(12), continuous for t E [0, T} and smooth for x E is given by the couple (yo(z, t),po(.1:, t)). In particular, for t E [0, ToJ

(21)

no =

The equation (21) has the unique solution if the inclusion (20) is executed. Let's regard futher for definition tha.t the condition {(yo - z, g)o, e(/I - 1/211 g 115 T~)+ +(g,yo - Z)OTO} E Pl n{(g,po)o > 1 l V- / 11 9 110 (g,po)o ::; (g, Yo)o < oo},

lie,

no,

(22)

Yo(.1:, t) = -~g(.1:)t + Yo(.1:), for t E (TO, T]

which gua.rantees uniqueness of the solution of equation

-e 11 g 110

(/ll / l

cth(/I-I/l 11 g 110 x

x(T - TO))+ 11 9 110 TO) = (g,yo - z)o,

(23)

lS fulfilled. Thus in case 1) the couple (yo(z,t),po(z,t» is fined from (12). In particular

=

Yo(x, t) yo(x) - g(x) 11 g lI;l X x(Yo - z,g)0(ch{/I-1/211 9 110 T)_ch.(/I-l/l 11 9 110 (T - t»)x X (ch(/I- 1 / 2 11 9 110 T»-I.

(24)

(k= 0)

The question about choice of domain 00 is solved this way. Let be a set from n, which sa.tisfies the condition YO(.1:) > fjJ(.1:) and suppose that for any z E the inequality

h. - Ykn =

no

11

= - 2: Lj(!, t, 8/8!,8/8t)x

no

Yo(x, T)

> fjJ(x)

j=I

xY.-j(t,!, t),

P., + P... + y. =

(25)

= 2:~=1

is fulfilled, at that the function ch(lI- l/l 11 g 110 T)d(/I-I/l 11 g 110 (T - t» increases monotonica.lly. Systems (10)-(12) a.re the conditions of optimality in the optimal control problem: find iio(t) E U such that

= ~ / (/ (Yo(x, t) -+

(28 )

XPl_j(t, s, t),

=

where t -f(, ( is a distance to aOo along outer normal, s is local coordina.tes on surface 800; Lj is differential operator not exceeded second order on varia.ble s and not exceeded first order on variable t; their coefficiens smoothly depend on coordinate s on 8fio and

z(x»2dx+

o 00

+v,?(t))dt

Lj(!, i, 8/8!, 8/8t) x

Yl(O, 8, t) = -Yl(O, 8, t)~ PlCO, 8, t) = -Plica, s, t); Yl(t,!, to) = Pl(i, s,T) = a,

T

Io(v)

ev

=

yo(x, t) -{g(x)ro + yo(x) + l / 2 11 g 11;1 x xg(.1:)(8h(/I- 1 / 2 11 9 110 (T - To»)-1(ch(/I-I/2X x 11 9 110 (T - t) - ch{/I-l/2 11 9 110 (T - TO»), (27) The question a.bout choice of domain 00 is solved by analogy with preceding case with next changes: the function (26) is minimized by bounds (22),(23),(25) and yo(x, t) is given by representation (27). Let's supplement the solutions (yo( x, t), f>o( x, t)) on 8fio by following boundary layer functions yo(t, 5, t), poCt, s, t) (Vasil'eva. and Butuwv 1990), defined from problems

min 33

polymially depend on t. The problems (28) are solved by single way in parabolic boundary la.yer functional class (Vasil'eva a.n.d Butuzov 1990). REMARK 2. Note that inequality (g,po(., t)) > 0, t E [to, To) will be fulfilled with. exactness O( E), if po(z, t) is replaced by sum po(z, t) + Po . The inequality (25) is fuInlled by replacement yo(z, t) on Yo( z, t) +yo a.t the expense ofinitial da.ta. ch.oice only. Th.ereby the solution of (6)-(7) is constructed completely, i.e. zeroth components of decomposition (5) are fined . ASSUMPTION 3. Suppose the problem's da.ta. such. tha.t the moment of control switching TO E (0, T) existes a.nd = an • Show further the algorithm of specifica.tion of the control switching moment. Let T be a. moment of control descent from bou.nd in initial problem. Let's find it in the form of asymptotic series

(g, yd = -[(g, ojjo(., TO+ )/&t)x XTI + al(To)}v-I/~llgllch(v-I/21Igllx x(T - t))(sh(v-1nllgll(T - To)))-l + V-I X xllgWch(v-l/~llgll(t - ro))(sh(v- I / 2I1gllx x(T - ro)))-l J~ alCt)x X$h(v- I / 2 1Igll(T - t))dt - ,,-ll1gI1 2 al(r)x xch(v- I / 2 I1gll(t - r))dr,

ve

1:0

(35)

(g,pd = -[(g,8po(.,ro+)/8t)x XTl + al(To)]sh(v- I / 2 I1gIICT - t))x x ($h(v- I / 2 I1gIlCT - ro)))-l - v-I/2I1gI18hCv-I/~x xllgll(t - ro))(sh(v- 1/ 2 I1gll(T - To)))-l x x J~ al(t)$h(v- I /2l1gll(T - t))dt+ +v- I/ 21Igl1 o al(t)sh(v- I/ 2 x xllgll(t - T))dr. (36) Hence, (g,Pl(.,t) = -[(g.&po(.,To+)18t)Tl +al(To)}, (g,YI(" t)) = 0, t E [0, TO} and t = TO it can be ob-

ono

J:

co

T

=

E

(29)

Ei Tj •

i=O

tained that

TI = (g, 8po(., TO+ ))-1 x x [v- 1/2l1gll(ch(v- I / 2 I1gll(T - TOm-Ix x J..~ a] (t)8h(v- 1/ 2 I1gl1 x x(T - t))dt - al(To)}

Find the first coefficients of decomposition (29). Assume TI = To + ETI, TI > O. Then YI (z, t) = 0, PI = P(ZI), t E [0, TI), and if t E (rI, T], then, according to (15),

{

~l PI

+ V-~g(z)(g,px) = -v-lg(z)al(t),

= -YI,

and at that the right side of (37) has to be positive. Return to the inequality (32). It accepts the form

(30)

Cg,po(., t) + !PI(., t)) + Ea] (t) -

al(t) = J800 D(z)/ D(O)(g(O, $ ))fio(e, $, t)d$rJ.e, poet, 8, t) is the solution of (28) for k = 0; D(z)/ D«(, 8) is Jacobia.n of transformation z - (-(,8) in environs

J;;

8no.

(g, epo(., TO + a(t - To))18t)(t - TO)+ +E{g,PI(.,rO))

(31)

> o,a E (0,1),

which ca.n be written with exactness O( ~ in the form

accepts the form

(g,po(., t) + Ep(., t)) + WI (t) -

(38)

For t E [0, TO) the inequality (38) is fulfilled by virtue of (22) a.nd for t E [ra, Tl) it accepts the form

At tha.t the inequality

ve > 0

ve > 0,

tE [O,rl).

where

(g,p(., t) -

(37)

(g, 8po(., TO+ )18t)(t - TO - ET,),

ve > 0, t E [0, Tl).

that

Form (32) obtain the condition by t = ness O(~)

TI

rl

with exact-

(9, opo(., To+ ))/&tTI + al (ro) + (g,PI (., TO))

pla.ce

by

virtue

of

(19):

< O. At tha.t C8.S€ reasonings sta.y to be true for

TI > 0 except such. momen.ts: in equality (33) ch.a.nge (g, 8poC., ro+)) - (g, 8po(., ro- )); needs to be done; qua.ntity rl is defined by (37) with the same ch.ange. The inequality (38) doesn't fuInl on (TI, TO). Really

= 0,

(33) and according to (30) the function (g,PI(" t)) is defined by

it can be accemped by the form

(g, opo(., ro- )18t)(t - To - ETl) > 0, but (g, &Po(., '1'0- )18t) = ~llgll2ro - (g, Yo) < 0 by virtue of (18),(22) a.n.d t - ro - Erl > 0, hence inverse inequality ta.kes place. Thereby the first step of algorithm for definition of coeficients of decomposition (29) is constructed completely: if

(34)

The condition (33) shows that the functions Yl, t TO. System (34) with boundary conditions (g,PI(" To)) = -[(g, 8po(., TO+ )/8t)TJ + al (To)l, (g,PI(" T)) = 0 has the solution

PI will have only one break point

tea.ke

(g,8po(.,ro+)18t) < 0; t-ro-ETl < O. Assume

(32)

=

Al = v- 1/2l1gll(ch(v- l /2llgll(T X

34

TO)))-l

J~ al (t)"h(v- /2l1gll(T - t))dt< 0,

-at (TO)

1

x (39)

then

> 0 and it is defined by (37) otherwise

Futher using the results of papers (Kapustyan 1993, 1996; Nazarov 1990) obtain that n.ext theorem is true. THEOREM. Suppose that the conditions of lemma. are fulfilled. Then the functions (43)-(45) are the asymptotics of initial problem solution and next valuations take place

°and it is defined by the same form with change (g, 81'0(" TO+)) ...... (g, 81'0(" TO- )); if A = ° then = But functions Yl(X, t), ji(x, t) are fined -Tl

Tl

<

Tl O. from the system (34) and completed by the boundary la.yer functions YI> PI> which are defined from (28). Easy to check by induction tha.t next lemma. is true. LEMMA. Let's suppose tha.t the assumptions 1-3 are true and (22) takes place. Then series (29) is constructed simply and if Tl > 0, then

IIgTad(y - yeN) )IIL,(Q)

+ IIgTa.d(p + lip -

11(1£) - l(u(N»)1 $

Ai = Rt+(To,.oo, Ti-I) + v 1/ 2I1gl1- 1x

(41 )

> 0,

then Ti > 0, otherwise - Ti < 0; if Tl < 0, then the form (40) together with inequality (41) preserve their sense with change (9, 8po(., To+)/8t) ...... (g,8po(.,TO-)/8t),

Ra+(TO, 00., Ti-l)

......

Ra-(TO, 00', Ti-l)

•

Decompositions (5) a.re completed by the boundary la.yer functions defined by (28). 4. SUBSTATIATION OF ASYMPTOTIC

DECOMPOSITIONS. Assume that the segment of asymptotic series (29) is constructed, i. e. N

N T

~

.

= L..J CTi ·

(42)

i=O

Consider the a.symptotic decompositions N

y(N)(X,t) = L(Yi(X,t) + Yi(t,s, t)) ,

(43)

j=1J N

p(N)(x, t) = LCPj(x, t) + pj(i, s, t)),

e~(N+I) .

Barby V. (1984). Optimal control Of variation inequalitie&. Pitman, London. Barby V. (1993). AnlllY8i8 and control of nonlinear infinite dimen&ioMl ,y&tem8. Academic Press, Ins. Boltya.n.sky V. G. (1969). MathemlltiClll method8 of optimal controL Nauh., Moscow. Kapustya.n V. Y. (1996). Asymptotics of locally boundedcontrol in optimal pa.ra.bolic problems. UJ:r. math. J. 48, N1, 50-56. Kapustyan V. Y. (1993). Asymptotic:s of control in optimal singular pertyrbated pa.ra.bolic problems. Global bounds on control. DocL AN (RlUlia), 333, N4, 428-431. Naza.rov S. A. (1990). Asymptotic solution of variation inequalittes for linear operator with small pa.rameter by senior deriva.tives. lzv. AN USSR. Serie, of math., 54, N4 , 754-773. Vasil'eva. A. B. and Butuzov V. F. (1990). Asymptotic method, in 8inguillr perturhation, theory. Vysshaya Shkola., Moscow.

where Rt+( To, ..., Ti-l), Cli(t) are known. functions; for = 1 they are puted a.bove and if

t))-

$ ell+ 1 ,

REFERENCES

i

x(th(v- 1/ 21Igll(T - TO)) J;O(g, ~Yi-2(" t))dt + (ch(v- 1n llgll(T - TO)))-1 x

p(N)IIL,(Q)

111£ - u(N)IIL2 (O,T) $ e~+I,

(40)

x IIgll(T - t))«g, ~Yi-2(" t)) - 1I- 1 11gW X 1 2 X Cli(t)) + v- / I1gll(g, ~pi-2(" t))x xch(v-1/21Igll(T - t))Jdt)J, i> 2,

-v- 1 I1glFl Cli - (t)) + 1I- 1 / 2 I1gll(g, Llpi-2(., t))ch(v- 1/ 2I1gll(T - t))Jdt)

$ ell,

Ily - yeN) II L 2(Q) +

Ti = -(g, 81'0(" TO+ )/8t)-1 x x [Rt+( To, ..., Ti-l) + v1n llgll- 1x x (th(v- 1/2I1gll(T - TO)) J;O x x (g, ~Yi-2(" t))dt + (ch(v- 1 / 2 1Igl1 x x(T - To)))-1 J~[&h(II-1/2X

x J;'[&h(v- 1/ 2I1gll(T - t))«g, ~Yi-2("

peN) )II L l(Q)

+

(44)

j=O

35