Optimization: A Journal of Mathematical Programming

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On Pontryagin maximum principle for linear systems with constraints Phan Quoc Khanh

a

a

Nha B 15, P. 11 Khu Kim Lien, Hanoi, SR, Vietnam Published online: 27 Jun 2007.

To cite this article: Phan Quoc Khanh (1985) On Pontryagin maximum principle for linear systems with constraints, Optimization: A Journal of Mathematical Programming and Operations Research, 16:1, 63-69 To link to this article: http://dx.doi.org/10.1080/02331938508842990

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Optimization i6 (13S5j i , 53-53

Summery: The note considers a case in which POXTRYAGIN Mrtximum Principle for linear systems with constraints implies t h e closedness of the reachable set and shows a class of systems f ~-%hick r there is a n answer t o t h e .cuesti-r: I c;f S. ; i t c ~ . ~ win~[$I. a A Win

1980 SLzbjectC I E s ~ i f i ~ions sf :

Primary : Jii B 27, Sacundttry : 36 2 48

Xey words : Optimal coztrol, lineer systems, maximum principle.

-

x

Let a d Y he real 3 - a ~ a m spaces, let C!. -xi Y he a hounded operator, let P he a mntimm1s convex functional on X and let U c X be a closed convex set. We say that PONTRYAGM Maximum Principle ( P M P ) holds for the linear system Y* such that X ~ with Y the constraint U if for each y E CU there is inf {F(x) / Cx=y, xEU)=inf {F(x) 1 y(Cx)=q(y),XEU).

(1)

It is known that if Gr:=CRY:= C {x 1 P(x)S T ) are closed for all r , then P J f P without constraints ji.e. U = X j holds if and only if Qn* is closed j4]. The presence of the constraint U increases the possibility of the fact that PlMP holds. Therefore P M P does not imply the closedness of CX. Throughout the paper the notation Pr8G stands for the algebraic boundary of a set G. We recall that a point p EG is called an algebraic internal point of a closed convex set G if for every straight line L through p we have either L n G = (p) or L n G is a straight segment admiting p as its internal point. The set of all points of G not being algebraic internal points is said to be algebraic boundary of G and denoted by Pr,G. I n [6] S. J~OLEWTCZ proved the following Theorem 1 (S. ROLEWICZ) : Let Y be a reflexive &LVACH space and bet F be a unifcrmly cnnaen: f..Jnd?:onn.Zo n X ; _Jf th,er,(?rc: is a real r s?~.chthat Fr i-i (int Rr) a r > (2) ,?,

+

@

then PXP i m p l i u the closzdness of C X . He has posed the question whether assumption (2) is not essential. In the present note we consider a class of systems for which (2) can be replaced by a weaker condition, which is proved in suitable examples to be essential but not necessary. The case when the space X is nonreflexive is also considered in the note.

64

Optimization 16 (198.5) 1

Let r r i n f F(x). Then yEFraG, if and only if r =inf (F(x) j Cx=y) and, similarly, yEFraC ( U n h ; ) n C (G17Fr,h',) if and only if r=inf (F(x) 1 Cx=y, X E Eij. ?re have the equivalence between the inclusion y E Fr,G,

and the

W

---'-'Jill

equ&l:tleS

'

Fr,C (

t7 $Q 3

( c- :-n Fr,,k=r)

I . ,

r =inf (F(x) j Cx =y) =inf {F(s) I Cx = y, xE Ei) . f--A +Flat 7 i ~ W T- f A f l / T T T7 ;+ foiiOTffg L X S l&CL , , $5 ,2 f &.,Tri ! L '\ V , , Atj

LL-

\

A -

glr,c,?{ T P p (u I

77 \ n

iAr)i

i

1-! C ( U Ti FraK,). Indeed, denoting by Q a linear projection from 9 onto Ker C and setting P =IT -Q we h a m


C (Pr,PK,nP (UnK,))=FraG,nC ( U n K , ) .

Hence y E Ii'raGrTi C ( 0'i;K,j means that Therefore ji E F r a C ( U fl KT)fiC ( iJ n Fr,K,j. T h m the mentioned aboire equi-... ,-. .vaience show.; :ha: c:,m;t;on ( 2 ) means t h s t :he m e e consiciijrcd ji_ Theorem 1 & f h s iittie, in a certain sense, the ease without constraints. P J f P with constraint 0'can be interpreted $hat for all r each point of FraC ( U K,) C' ( U fl FraK,) is a point of support of the set C ( U rl KT).

n

n

Theorem 2 : Let X and P be as in Theorem 1. If the mentioned projection Q is continuous and int, P U I B

,

(3)

where P =I -Q and intp denotes the interior of a set i n the subspace P X , then PlMP implies the closedness of C X . The proof is based on two following lemmas.

Lemma I : Let ir" be a continuous convex junctional on X , kt Z be a BANACH space and B : X - Z be a linear operator, and let U c X be a closed convex set such that int BU* O. Set P=inf {r I B ( U n K , ) n i n t B U +

a}.

Then for all r W P we have L,: =lim {B (Ufl KT)flint B U ) =Z

.

P r o o f : Since l i r n s u ~( B ( U f l K i n ) n i n t BU)=int BU and limsilpL,=Zj

--

n--=

n-=-

where r, when n+-, it is sufficient t o show that if LT,EL,l, then for all r , P s r i - = r s r 2 , we have L,=Lrl. Let 2, 5e ar, arbitrary point belonging t~ B ( U n K J n i n t B U and not belonging t o LrI. Let z i € B ( U n K , ) n i n t BU. -- - We have z2= 3x2, x2E U n K T 2zl= Bxi, xi€ U f lK,,. The segment [xi, xz] intersects FraKr a t some point xo. Of course xoEK, fl U and BxoE B (KTfl U )n i n t BU. Since the linearity of Lrl, Lin ( B q , LrI)=Lin {Bxo,L,,)and L, = L,,l. rn

Lemma 2: Let F be a zcnifor~nlyconcex continuous functional on X , let 2, C and B satisfy the assumptions i ~ zLemma 1 and let, in addition, B be continzcozcs. Then there is r s w h that Fr,B < L ' I ^ K R~ {EVPFr,K,)*int )~ B p -

? . - - P

L

~

". _U,SSUW~ t ~ ~ at : f m r tha,t there is A

7:

~ 3F.

(4 1

-

sii:i&-;.ii;z -

F r a B ( C i'ifi-,l);-lint BL'+ i ~ ..

( 5j

-

Ex,,EFP,B ( t i !-!'!K,i:: q i n t B U , zoCFrn( C 7 K , I \ , Z C F ~ , z,@Fr,-Kr,, ~:~. then P ( q J= T' -=rl and we have ff


Bx,€Fr,B (FI-I_~',,) f i B (i;.i;Fr,ii',,j iiint B G , which is the required result in the lemma. Thus, to finish the proof i t is sufficient to get ( 5 ) . Suppose, by contradiction, thar for ail r satisfying Bi U nA*,.) riint B t T = =F @ x e have

Taking rl such that o-crr,-?-=a 2-

1

- IIBII

we can choose

XI, x2E UflK,',

jjx,-x2jJz

jjBx, - uDz2jjZ E , ? ~ B B ; x~ and ,~ BZ?belong t o 3 (li9K,,)P k t BlJ.Making USB

of (7) we see that

1.

-

(xi+x,) E K r , where r -=r,-6-c

r , which contradicts the fact

that

P r o o f of T h e o r e m 2 : According to Lemma 2 we take r satisfying (4) for the case B= P. Applying the lemma in [6] to the space PX, by the uniform convexity of P we obtain f E ( P X ) *= (Ker C ) I and a real u such that

A : = { x c P X 1 X E E( i i i i K r )f(x)scr)cin:, , Since PX is reflexive C*Y* is dense in (Ker C) C"P* and u, satisfying the irlclusiorl

PU

.

. Therefore

(8)

one can choose

A , : = { x E P X j xEP (Unl;,), f l ( x ) s q ) c i n t p PU . Since F r , C ( U n K , ) n C (UnFr,K,)=C ( F r a P ( U n K , ) n Y (UrnF~i'r~ri,)), PHlo dearly implies that each point of algebraic hoxndary of the set CA: is a point of support. By the reflexivity of X the set CA, is weakly compact and then closed. B y the continuity of F the set K, has interior, and then the set A l has algebraic 5

Optirn~zat~on 16 (1985) 1

66

Optimization I6 (1986) 1

internal points. Since being algebraic internal points is an invariable property with respect to h e a r operators, CAi has aigebraic internal points. Sow using ~ zin u ~[8! , we see that CAI has nonvoid interior in the theorem of J 7 . 7 0 ~ ~ - & s zP. space Yo= Lin CA Hence Yo =Lin C A i. Thus

,.


CX=CYX=CLin A,=Lin CAi= IT, is closed. gge Since int U + 0 implies int, PU + 0 (P is continuous), if condition (21 is satisfied, so is condition (3). But inverse imyiicatioi-Iis not true i.e. mnrlltion (3) is really weaker than (21, as shewn by

Example I: Let X be Euclidean space R3= {x j X = (xi, x2)), let C : Xi-iZ be defined by C(xi, 2,) = (0,x2),let F(x)=IIx/I, let U = (x / /]xsxvjj s 1) where z0= (2, 0). Then of course intp PU + a but for all r Fr22; !? C (int U T!Xr)= a . since int L7f!K,.=@, for rs1 and

Example 2: Let X = Y = 1, and C be defined by C{xB}= (x,Jn). Let P (x)= Ilxll. I t is easy to see that CX is not closed. If U is a finite dimensional subspace of X then of course f Nf holds. Without the reflexivity and the uniform convexity, the closedness of CX yields the validity of P M P if int, PU =I 0.Indeed, by Lemma I we can choose r such that int, P ( U n K , ) I.0.Hence each point of Fr,P (UnK,) f l P ( U n nFr,K,) is a point of support. I t follows from thk equality (PX)*=C*Y* that for each support functional f of P ( U n h ' , ) there is q E Y*, f=C*q, such that ip is a support f-aiictiona! of C j tT flK?). This each p&i"lt"fi,C LTTn IT:)fl C (UnPr,K,) is a point of support, i.e. P H P holds. Of course neither condition (3) nor other ones uppon only U and F are necessary in Theorem 2. If either X or Y is finite dimensional P M P always holds and CX is always closed. The following example shows that even in infinite dimensional cases condition (3) is not necessary.

n

.

Example 3: Suppose that S;, is a set of continuum cardinality and Chat X = Y = =1,(9), where 12(8) denotes the space of all functionals x(.) defined on Q with satisfying conciition countabie support JZ,, Ix(t) . jjxl/ : = (

/y2< t : ,

tw,

Suppose that C is the identity operator, that U = (x 1 x(t) wO for all t -

2

ltllU

L LllltL

!

1' \A

I

1 -[/All.

A D L T LUG11 MILp 1 U =11IL V

1

"

-

11x11-= 1)

- 2 : L : 2 T lt11U b l l l t i t j tjitUll p U l I l L U I U V

TT = LJ

is a point of support [5], so is each point of FraC (U flKT)n C ( U nFraK,) for all real numbers r , i.e. P M P holds.

PHANQcoc KXAYE: On PONTRSACIN 31axim~111 Principle for Linear Systems

67

It is known that if X is reflexive then C* Y* is dense in (Ker C)I. The following theorem shows that P X P warrants this density in the case of nonreflexivity. Theorem 3: Let X be a BAXACX space? kt projection & be co.ni.inuous and iei in:, PC +. g ,Let 7 be &;Pergy&&~e a?:d ktLif3r9itlg i . e n . ? ~ ~~-*~ ~ . ' I ~ - ~$ ~p i.,tzpiic.s e ~ ,-\Aj . that .J b l b J i ; %n - ;Pa.. w i, I v i

c*,*

,

--.-

,-..A

V

P r o o f : By Y X P there is j ~ ( B e C' r )' and a real n such that each point of alaehraic Enundarv of the set !iii


A = { x E P X j X E P( U C I K ~f(z) ), s ~ j is a point of support. At each point of Er, A there is only one support hyperplane. Indeed, suppose that at xi E Fr,P ( U nK,) there are in ( P X ) * two support funciiVn& f1 aI:d f.,, fi+kf,, Since -D* is en injec:i.r;n thera are in 9"twc; ~ 3 p p c . d fi1nc.t-innah 9; and F a j y 1 + E p , a t the point x,EF~, jUfiK.;~TiFr,X7, P ~ , = X ; , which c ~ n t r a & c the t &Eerenti&iiity of "F. -L.- -. --'L;- -v..-T.j.;nOn accountof ail siipport iiii2ctioi-iajs A , .;-;*iL l l iild sf suppert' f1~2ctinc& of the f n r ~ 11 is a, real, heinnu17 t o cy*P*. But s u ~ ~ o i - t f-wxticzds =f the bosnded closed cnnvex set -4 are dense in (Uer C)l [I]. Thus C*Y* is dense in (Ber C)'. IP~ The results above have evident applications to controi probiems for &strikuted parameter systems. Fer example, we consider a, control system described by the linear parabolic partial differential equation in one space dimension W l

~ ~ O U L V IV UA V U ~ LIVAL

Ii)i +7 1

-~

A

with the boundary conditions

where p(.) is a positive twice continuously differentiable function defined on [ 0 , a], q(.) is continuous on [ 0 , a] and b(.) E L [ O , a] ; f ( . ) , g ( - ) , h ( - ) are controls, admissible if f , g, h lie in L,[O, TI. I t is known that given an initial state w(-, 0 )=wo(.)E LEO, a], the initialboundary value problem has a unique solution w(z, t ) , 0 z z s a , (3 t s T in the following sense : (a) For each t E [ O , TI, w(., t )E L J O , a] and for t P O ,

w., t) --- and ;iz

a2w ( . , t )

--- exist ;ir?

in the sense of the theory of distributions of L. SCHWARTZ and both lie in L,[O, a]. (b) The function w(-, ti : LO, Tj -.L2[0, a] is continuous on i0, a] mid continnoizsly differentiable on (0, a ] , in both cases v i t h respect t o the norm of L 2 [ 0 ,a]. In particular the initial condition is satisfied in the sense that l$ llw(-,t ) -wo(.)ll4[0,.~=,0 . 5'

68

Optimization 16 (1985) 1

(c) For each t (0, TI the equation ( 9 ) is satisfied for almost all zc [O, a], ZW/?Z and 22w/8x:!being defined as in (a) while &/at is defined as in (b). id) The boundary conditions 110) are satisfied for t E (0.TI. The control probleix is to steer the state of the system fram 7r'tJ(.) = G t o a t r y minal state to(-, T'I =zr,(.!F L2[0,a ] , The problem is reduced to a moment pmFlem '," *,-.]]ows [ 2 ] . a3 I U Let A be an operator of L2[0, a] to L,[O. a ] defined by


with the domain of definition consisting of y(-)EL,[O, a ] having y', y" in the sense of the theory of distributions as noted in (a) and

-

that the expansions of biz) and wijx) are the foliowing: b ( z ) = Z %pr,(2j7 , L L ' ~ (= ~)

2~

??==I

~ ~ ~ Then ( 2the ) contro! .

prnblerr, is reduced t o the mameat prshkm :

n-1

where GA +G;

ul(t) =f ( t ) , u2(t)=g(t), u3(t)= h(t) +i: = v, with

and v: =?:/a,,

v i = $J/ls, vi = $Jd,,

Formula (11) defines a linear bounded operator C : (L2[0,T])3-(12)3 by C(u1, u', u" = ((v;), (v:}, (vi}). Set u(t) = (ul(t),u2(t),u3(t)). The optimal control problem consists in finding ai? optima! control a,(-j i U c jL2[0, Tjj 3 steerkig the system from zero t o wi(.) with a minimal norm liu,(.)ll, where U is a closed conves set. the srr?oothing I t is B n n w ~[2: 3, ?I that C(L2[0,T!)3 is dense in ( k -., ) 3 . F r ~ m properties of solutions of parabolic equations it follows that the reachable set C(L,[O, T]j"ffers froill the whole space (E2j; and hence is not dosed. Since ( L 2 [ 0 ,TI)>is a H ~ B E Rspace, T & the orthogonal projection onto Ker C is continuous. If int, PU* 0,then making use of Theorem 2 we see that P M P does not hold. For instance, if U = {u I jj~(-)jj~,[~,,~,,, sM}, then int, PU I 0 because of t h e continuity of P. But under the assumption that U is a closed conves set in

(L2[0, T ] ) 3such that CU is finite dimensional we have the validity of P M P because every boundary point of a finite dimensional convex set is a point of support.


Ref erenees [ I J BISROP,H.; P. PHELPS: The support functionals of a convex set. i n : Convexity. Proceeding of syrnpositt in pure rtiathernatics, Voi. TI,Rhode Island, 1963, 27-35. [ 2 j FATTORINI,H. 9.; RUSSEL,D. L.:Exact controlability theorems for linear parabolic equations in one space dimension. Archive for rational mechanics and analysis, 43 (1971) 4, 272-292. [3] X A C C A I ~ 3. , C.; MIZEL, V. J. ; SEID'MAN, T.I. : Appresimote haundsw coatm!sbi!ity for t h e beat, e ~ u a t i o n J. . of Math. Anai. and Anpi.: 24 (1968) 3; 899- 703. .;*; - -%s'EmlCz, a.: m ! ~ l l k . ; r l ~ ~ l & UXLU ~ ~,~! ; >u :~ ~y ~: ~ ~~ ~ l .l r 3pr1;g~r. ~ g . ~ ~ f ?'l-.Keg, ~ ~ ~ I~ !~u., [5] R o ~ r w ~ c Sz.,: On convex set containing only points of s q p o r t . Commentationnee &th. ssr. Spec., (1978), 279-2gi. ~ & ~ L E ~ I C ZS.: : Vn PONTRYAGIN Maximum Principle for systems. with non-one-point ,>

-

_

-

.

I

,

.

1

1

.

0 ,

, 7

.-.

-7

7

-

A,%--

target sets and systems with additional constraints. Math. Oper. Forsch. Statist. Ser. Optimization, i O ji979) i , 97-100. [7] SAKAWA, Y. : Controlability to partial differential equations of parabolic type. SIAM J. Control, 12 (1974), 389-400. 181 W o ~ ~ ~ s z c zP.u :sA, theorem nn convex sets re!sted tc? abst,ract, P G N T ~ Y ~9hxixmm C-IN Principle. Bull. Acad. Polon. Sci., s6r. Sci. Ma&. Astron. Physics, XM (1973) 10, 93-95. Received July 1382, revised September 1983 Phan Quoc Khanh Nha B 15 P. 11 K h u Kim Lien Hanoi SR Vietnam