Necessary Conditions for the Optimality in Optimum Control Problems with Nonscalar-Valued Performance Criterion

NECESSARY CONDITIONS FOR THE OPTIMALITY IN OPTIMUM CONTROL PROBLEMS WITH NONSCALAR-VALUED PERFORMANCE CRITERION B. Lantos Department of Process Control, Technical University of Budapest, Budapest, Hungary

Abstract. The principial airr. of this paper is to give the necessary conditions for attaining otltimunl (minimum or infirr,um) for constrained nonscalar:-valued functions. The rrethod of DubovitskijMi lyutin is extended to the case in which the range of the function is a partially ordered linear topological space and the function has a minimum. The necessary conditions for the rrinimum (noninferior solution) and for the infimum (superior solution) are detai led for the case in which the function is Frechetdifferentiable and its range-space is a partially ordered (reflexive) Banach space. For dynarrical systems with finite-dirrensional linear norrred (not necessari ly Euclidean) state space and performance criterion range-space, the necess.:!ry condition of the infimurr is formulated in the form of a local supremum principle Iflhich can be applied to the analysis of dynamic vector estimation problems and to uncertain optimal control problems. Keywords. Optimal control; mathematical programming; nonscalarvalued performance criterion; necessary conditions. INTRODUcrION In many optimization problerrs the quality of the process cannot be characterized by a single scalar-valued optimality criterion. Optimization problems with nonscalarvalued performance criteria are studied in the paper. The rreaning of "better than" has to be defined, which is done by a partial-order relation. Simi larly to the minimization of a scalarvalued function, the constraints are approximated by their admissible er tangential directions (nethod of Dubovitskij-Milyutin). For the decreasing directions of a nonscalarvalued fUnction a modi fied de fini tion is introduced. The basic necessary ccnditions are summarized in 3 theorerrs that establish a re lation between the rrethod of Dubovitskij-Milyutin (Girsanov, 1970), the results of ~eus­ tadt (1969) and the results of Ritter (1970). The special case of the problem studied here, when the range of the performance criterion is a finitedirrensional partially ordered space, was ccnsidered by Athans and Ceering (1973). For dynamical systems, the necessary ccnditions of the infimum are summarized in the forms of a local and a

global supremum principle which establish a relation between the local maximum principle of Girsanov (1970) and the infimum nrinci~le of Ceering and Athans (1974). All the proofs of the theorems of this ;Japer can be found in the aut :, or's dissertation (Lantos, 1976). PARTIAL

ORDERn~C

Partial ordering cn a set is a reflexive, (antisymrretric) and transitive relation. If the set is a linear to?Ological space, then it will be supposed that tl:e partial ordering is gi ven by a closed and convex cone having a nonempty interior. Definition 1: Let (E ,"'0 be a linear topological space, and PC E be a 0 cl8sed and convex cone such that P ~ (p is the interior of p). We say that x~y if X,YEL and X-YEP. A linear topological space with a relation defined in this way is said to be a partially ordered linear topological space. ~a­ tation:(E,'T,2:). Since XEP~x2:0, cone P will be called the positive cone (defining the relation ~). If ±ZEP~ z=O, then ~ is antisymnetric,

1033

1034

B. Lantos

i.e.

x~y

and

and

y~x~x=y.

n

Exanple 1: Let R be the usually ndirrensional Euclidean spa~. If n P ={x=(x , ... ,X )£R :X 2::0, i=l, ... ,n}, l n i then P is a positive cone in Rn, an~ 50 P de fineR a parti al orde ring in R • l~otation: (R ,2::). Ifllxll=lIyll=l and x,y£P, then IIx+yll2:: 1. Example 2: Let 11 be a Hilbert space anc. B ( H~H) the space of bounded linear operatorc;. If E={AEB ( H~H): A is selfadjoint} and P={AEE: ( Ax,x)2::0 for all XEH}, then ECB (H~H) is a closed subspace ~ E is a Banach space and PCE is a p ositive cooe in E. ~otation:(E,2::). If IIAII=IIBII =1 and A,I3EP, then IIA+BII~1.

Fo:E~Eo

a mapping. We say that

hEE is a decreasing direction of the function Fo in point xoEE, if there exist a neighbourhood V of OEE and a real nurrber [0>0 such that F ( x )2::F ( x +£h) and o 0 0 0

(1)

Fo ( Xo ) *Fo(Xo+£h) for all hEh+V and O<£
Remark: If the partial ordering 2:: in the linear tq:>ological space (Eo,10l is defined by the positive cone Po' then (1) is equivalent to F (x )-F (x +£n)cP ~{O} . o 0 000

(2 )

Example 3: Notation is as in ExaITt>le 2. Let H_RIl (witH fAxed orthonormal basis). Then B (R -.R) can be identified with the set of nxn matrices and sirni larly E with the set of symnetric nxn matrices. 'l'hen P is the set of positive semidefinite symmetric nxn matrices. The positive cone P defines a partial ordering in the Banach space of symnetric nxn matrices. Notation:

The set of decreasing directions of F 0 forms an open cone.

(l"'~xn' 2::). l':~xn can be coosidered to

xo+t:hEQ

be a n (~+l) -

for all nEh+V and 0<£ <[0'

dimensional subspace of

De finition 4: Let (E ,1') be a linear topological space and let QCE and XoEE. We say that hEE is an admissible direction of set Q in point x , if there exist a-.neighbourhood V 8f OEE and a real number co>O such that

( 3)

2

the line ar n orme d sn ace Rn inner product). -

( wi thout

Renlark: If (E,t,2::) is a partially ordered linear topological space, then x~y and ZEE ~ x+z2::y+z.

xo=max Q, i f there does net exist

Definition 5: Let ( E,T ) be a linear topological space, let further QCE and x EE. We say that hEE is a tangentigl direction of set Q in point x , if there exist a real nUIl'ber E >0 O aRd a manning r: (0,£ ) ~E such that

any xE Q such that x 2:: x

x (E) =xo +Eh+r (t:) E Q

Definition 2: Let (E,1,2::) be a partially ordered linear topological space, QCE and xoEQ. We say that 1)

Remark: Let K be the set of admissible directions of set Q in point xo; then o K is an open cone. If XoE. Q , then K=E. If x.,AQ, then K~. (0 is the closure of Q).

0

and

x~x

0'

2)

x =min Q, i f there does not exist 0 any XEQ such that x ~x and x*x ' 0 o

3)

xo=sup Q, if x 02::x for all XEQ,

4)

xo=inf Q, if

x~x

0

for all XEQ.

In general, max Q and rrin Q are not unique, because partial ordering is generally not a linear ordering (Q may have elements which are not comparable ) . Sup Q and inf Q are always unique ( provided that they exist and the partial ordering is antisymmetric ).

-

0

(4 )

for all 0<£<£ , furtherrrore if V is an arbitrary Reighbourhood of OEE, then there exists a real number 0<'1[ CV) ~E such that o 0

i

(5 )

r (E.)EV

for all O<£ <"lo ( V), Remark: Let K be the set of tangential directions of set Q in point x ' then o K is a cone. Theorem 1: Let ( E, 1') be a linear topological space, let ( E ,T , ~ ) be a

o

SOME GENERAL NECESSARY CONDITIONS OF OPTIMALITY Definition 3: Let ( E ,t) be a linear topological space, (E ,T ,2::) a partialo o ly ordered linear topological space

0

partially ordered linear topological space and n+l Q = QiCE , XoEQ i=l and let F :E~E be a mapping. Let S 0 0 0

n

Optimum Control Probl ems

denote the set of decreasing directions of the fW1ction Fo' let Si denote the set of admissible directions of the set Q . , i=l, ..• ,n and S +1 be the set of taAgential direction~ Of the set Q + (W1derstood in point Xo reSDecn l tively ) .

i)

103 5

f EP~ i.e. o 0'

f ( P )2: 0; 0

0

fiE: ( Pi +Fi ( Xo )) ~, i.e. fi ( r\+Fi e XO )) ::: o,

ii)

i=l, ... ,n;

Suppose that

with the notation n+k 'P = f. 0 F~ ( x ) i=O ~ ~ ' o

1)

there exist open and ccnvex cones Ki such that 0,*,K CS for i=O, ... ,n, i i

the inequali ty

2)

there exists a convex cone Kn+l

'fle x) ~ 'fl( Xo ) iii)

such that q,*Kn+1CS n + l ,

3)

L

~

v) if the system R (F~ ( x

~

~

\

0

~

0

~

0

(10)

E and Ei are Banach spaces,

can be satisfied, then fi Vi)

Pi CE i

i

F ( x ) *0 for at least one i; o

0

specially i f R( F{ e x )) =E " i=n+l, ... ,n+k aRd th~ system F ~ ( x ) ( x- x )=0,

is a closed and convex cone,

0

~

i =n + 1 , ... ,n +k ;

0

OfP i *q, and Pi ( as a posi ti ve cone )

iE{l, ... ,n) and -Fi ( xoHP~

de fines a partial ordering 2: in the Banach spa~ E , i=O, .•. ,n; i

- F ~ ( x ) ( x- x ) E Pc:' + { ;'F . ( x ) : ~

Fi:E~Ei is

~

0

1)

i=n+l, ..• ,n+k space ) ;

fo

2)

0

Fi ( x ) =O,

(8 )

Then there are continuous linear fW1ctionals fiE: i=O, .•. ,n+k such that

El..,

0

F' ( x ) ¥J.

o

0

P iCEi is a closed and convex cone

I

otP~*Q> and Pi ( as", positive cone ) de fines a partial ordering 2: in the Banach space E , i=O, .•. ,n; i furthermore there is a real number 6> 0 such that for all Ilylll = IIY 2 11 =1

XEA}

0

(11)

i=l, ... ,n+k, Eo is a re flexi ve Banach space;

is a constraint, XoEQ and there exists a neighbourhood U of Xo such that

o

0

1) E and Ei are Banach spaces,

( R is the range-

min{F ( x ) :XEQn U}=F ( x ) .

~

Theore m 3: Suppose that the following conditions are satisfied.

~

ACE is a convex set and AO~;

i=n+l, •.• ,n+k;

~

has also a solution x, then

is closed in E . ,

6) 0={XEE:-F ( x )2:0, i=l, .•• ,n; i

0

7I> 0}; XEA

a mapping which is con-

which R ( F~ ( X

0

=l

o

Fi :E--tEi is a mapping which has a

tin uous ly Frechet di fferenci able in a neighbourhood of Xo and for

5)

0 ;

fi *0 for at least one i.

Frechet derivative Fi( X ) in point x ' i=O, ... ,n; o o 4)

0

~

o

3)

~

R ( - F ~ ( x ) \; n ( pc:' + {'AF . ( x ) :

i=O, ..• ,n+k; 2)

)) =E . , i=n+l, ... ,n+k,.

iE{l, ... ,n} and -Fi ( xo ) *P~ ==*

(7 )

fo+f l +·· .+fn+l=O

0

R ( -F' ( x )\ npo*Q>,.

Theorem 2: Suppose the following conditions are s atis fied: 1)

fi *0 for at least one i,

then fi =0;

( K~ is the set of continuous linear fulictionals nonnegati ve on K ) . Then i the~ are continuous linear fW1ctionals fiEK , i=O, ... ,n+l such that i

ii)

(9 )

iv) i f i E {l, ... ,n } and -Fi ( xo )E P~,

there is a neighbourhood U of Xo such that min{Fo ( X) :x EQnU}=Fo ( Xo ) ' (6 )

Let K~ denote the polar cone of K ..

i)

holds for all x EA;

and Yl'Y2EPo i t is 3)

IIY1+Y~I~5;

Fi :E .... E is a mapping which has a i Frechet derivative F~ ( x ) in point ~

0

x ' i=O, .•. ,n and for which o

1036

B. Lantos R ( F~ ( x ))

i=l~ .. r:;,n; 4)

Fi:E~ Ei

is closed in E , i

is a mapping which is con-

tinuously Frechet differentiable in a neighbourhood of Xo and for \>,hich R ( Fi ( X )) o i=n+l, ... ,n+k;

is closed in E , i

5)

ACE is a convex set and A°¥fJ;

6)

Q={X EE:-Fi ( x )~O,

general than his bacause the spaces are Banach spaces, the mappings are differentiable, and the constraint A is convex and has a noneIll>ty interior. However, it is also more general in certain respects because the constraints given by equations can be differential equations. Theorem 3 is essentially the same as Ritter's result ( 1970, Theorem 3.1, pp. 143-147 ) . Its proof makes use of SOIre results of Ritter (1969, Theorem 4.9, pp. 203-204; and 1970, Lemma 2.2, Pl? 137-139 ) . The local supremum principle mentioned below can be derived from it.

i=l, •.. ,n;

F . ( x ) = 0, 1. i=n+l, •.. ,n+k; XEA}

is a constraint, x £Q and there exists a neighbourRood U of Xo such that

It can be shown that

if

R (-F~ ( Xo ») n p~~ and R (-Fi ( xo»)n(p~+{ I\ Fi ( Xo ) :'II> o} ) ¥/), then

Then there are linear mappings T.EL ( E .--.E ) , which are continuous on Rl'F~ ( x1. )) i=O, ..• ,n+k and for which 1. 0

?

i)

ToYoEP o Le. ToYo~O for all YoEP o ; TiYiEPi Le. TiYi~O for all 1.0

Ki={hEE:-Fi C Xo) hEP~+{ ~ Fi ( XO ) :~ >O}C CS i '

(15)

1.1.0

THE SUPREMUl-l PRINCIPLE

i=l, ... ,n, ii)

Condition ( C ) :We say that(n,r,m,T satisfies condition ( C ) , if

with the notation n+k T = ~ T.oF~ ( x ) i=O 1. 1. 0

O
SC[ O,T J and

Le. T ( x-xo) EP o

holds for all xEA,

(13)

iii ) Ti~ for at least one i,

1) there exist 4>x(x,u,t) u

{4>x(·'· ,t ) :tES} ,{4>u C·,· ,t ): t ES}

v ) if t h e syste m ( la ) can be satisfied, ~hen Ti 0 Fi e xo ) *<) for at least one 1. ,

Th eorem 1 is an extension of the rrethod of Dubovitskij and Milyutin to the case of the nonscalar-valued performance criterion. Its proof makes use of a lemma of Dubovitskij and Milyutin (Girsanov, 1970, LemmaS.ll, p. 37 ) . Theorem; 2 and 3 can be deduced from it. Theorem 2 is a result analogous to that of Neustadt ( 1969 ) . It is less

and

4> ( x,u,t ) for all t ES, and

then Ti=O'

specially if R ( F~ ( x J) =E., 1. 0 1. i=n+l, ... ,n+k and the system (11) has also a solution in x, then To=I is the identity operator.

A( [0 ,T)\ S ) =0

(A is the Lebesgue measure), furthermore ,

iv ) i f H {l, ... ,n } and -Fi ( xo) E P~,

vi)

,4»

4>:RnxRrx[O,TJ~Rm is a mapping,

the inequali ty T~TXOI

(14)

and K and Ki are nonerrpty open and conve~ cones.

y .E R (-F~ ( x )·)' .]( P.+F. ( x )) ,

1.

K ={hEE:-F' ( x ) hEpo}CS , 00000

are equicontinuousninfx,u) on all compact sets FxGCR xR ; 2) 1> ( x,u,. ) ,4>x ex,u,. )

and 4>u ( x,u,. )

are measurable functions in t

for

all fixed eX,U )E RnXRr; 3)

for each fixed bounded set r FxGCRnxR , there is a real number k such that 1i4> ( x,u,t )li< k, l1 4>x ex,u,t )lIu(x,u,t)lI
Theorem 4 ( local supremum principle) : Assume that ( n,r,n,T, ~) satisfies

~~~~;~~n~~ ) ~~~~xc~~:~ ~tRfu0s~~

that Po *rJJ and -z EP ~ z =0, and let the

Optimum Control Probl e ms pos~tive ~onerlo dfifine anfartial orae rl.ng 2: ~n R ( H and R are not ne~ss~rily Euclidean spaces). Let F:R-+R be a differentiable mapJ?ing and t!1e constraint Q be

Q= { (X,U ) E: c ( n ) ( O,T ) xL ( r 1( O,T J : 00

dx-Ct )_ \.r" '- x( t ) ,u () t ,t ), dt

for a.e. t E r O,T J ,

x ( 0 ) =c },

(16 )

and ~ xo' u o ) EQ. Suppos: that there is a nel.gnbourhood V of \ x ,u ) such that

o

0

( xo'uo \ is a solution of the problem inf{F ( X( T ); :(x,U ) E0nV}.

i)

' If-' (t )

~= -\.jJ Ct )\(:>x ( xo ( t ) ,Uo ( t ) ,t )

(18)

H ) \jJ (T ) =-F' ( Xo ( T ) ) ;

(19 )

Hi)

( 20 )

fo r a Imos t every t E [ 0 ,T J ; iv)

Remark: If also the right end-point of the trajectory is fixed and the r control is constrained by U( t )E ECR , an analogous local and global supremurn principle can be derived froll', the necessary condition of the infimum gi ven previous ly for Banach-spaces. Condition CC ) and the essentially bounded control space make it possible to derive the global suprernum principle from the local supremurn principle. A

all trajectories starting from c stay entirely in Z eRn;

2) \f :RnXRrX[ O ,T] ..... Rn is continuously differentiable and \jl (x ,u,t ) is convex in u with respect to P ; o there exists inf \jl( x,'U,t ) , and u

specially if m=n and F' ( x ( T '\ an inverse, then \ 0 J,

clas

lfl u : xo Ct ) ,uo Ct ) ,t) =0

(21 )

\jl u ( x,u,t ) = 0 has a

for almost every tE.[O,T].

3)

(It

y~O~Ov~~,t )

u

~ 24 '

y2: 0

'r! (X,t ) E Zx[O ,T

1.

If toe control uo Ct ) satisfies the system dxo ( t ) dt = lj) (xo Ct ) ,uo ( t ) ,t)

ift E [O,T],

x ( 0 ) = c, o

uo ( t ) =uo ( xo ( t ) ,t )

( 25 ' 'rItE[O,T],

then u ( t) is the unique of theOcontrol problem

dX~~ l = \{J ( x (t ) ,u (t ) ,t ) , u (.)

( 22 a )

inf lj) ( x,u,t) =0

V ( x,t l =x,

inf

r SUP{H ( XoCt ) ,u,4-' Ct ) ) :UER }=

'O x

if (x,t ) EZx[O,T ] ,

(22 )

Clobal supremum principle: The generalization of Pontryagin 's maximum principle in the form of a global supr~ll'um principle can be derived from the local supremum principle with the sane technique as used by Girsanov (1970, pp. 83-92 ) for autonomous systems. Instead of ( 22 ) ,

solution uo=uo Cx,t ) ,

n there is a function V:Zx[O,TJ ..... R wiich is continuously differentiable in both of its argUlrents and satis fies the system Cl V (x,t ) +Qv Cx,t)

Remark: Let H( x,u, 4.i ,t ) =\.jJI¥ (x,u,t ) , then the function H ( x e t ) ,u, \flC t ) ,t ) satisfies by Theorem <.l

the necessaryrcondition of the local supremum on R for alr.ost every tE[O,T) in the point u=u e t ) . Hence Theorem 4 is a local sup~errum ::>rinciple.

unique

for all (x,tJEZx[O,T];

If F is a convex function and the dYnamic system is linear, then the local infimum in ( x ,u ) is also a qlobal infimum on Q, °fu
H ( Xo ( t ) ,uoCt ) ,\.VCt ») =const

SUFFICIENCY RESULT

Theorem S: Consider the problerr in Theo re m 4 \-li th F=I. Suppose 1)

for almost every tE:(O,T ] ;

\j!( t )IPu Cxo Ct ) ,uo Ct ) ,t) =0

must be satisfied in the global supremurn principle.

( 17 )

Then there exists a rnxn rratrix function 'fI ( t ) such th at

10 3 7

x (T ) .

solution

x ( O) =c,

( 26 :

1038

B. Lantos

Rerrark: It is well kno.m that, if f ( x,t ) = inf \jJ(x,u,t ) and fx(x,t )

for all ueU there is a bounded linear operator A having closed range-s~ace in H2~ Then UoEU is the

u

solution of the problelT'

are ccntinuous, t:ten

~ o't' --

f ( y, ,..", L ) ,

y ( t ) =x

(27 )

inf{AU ( Ql) :UEU}=A

u

(Ql ) '

(30)

o

has a locally unique solution and the solution ( the characteristic function ) y (T) =y ( 't,t,x ) satisfies

where ~ is the partial ordering defined by set inclusion (set-valued cost), if and cnly if A is the solu-

i)

tion of the problem

y ( t , t ,x ) =x

ii)

I

Yx (T ,t,x ) is the solution of t h e followin g problelT':

aY d~ =fx ('t' ,Y i i i)

('t' ,t,x)

) ~Y,

(28 )

can be chosen as

(29)

V ( x,t ) =y Cl',t ,x ) .

At app lication it nlust be tested ...,h ether ( 2 7) has a solution on [ O ,T)x [ O,TJxZ, and whether the solution of ( 28 ) is a p ositive operator. APPLICATIO:'-JS fvlatrix-valued performance criteria can be applied in sorre dynamic vector estirration p roblems and in sone uncertain optin'al ccntrol problems. n Theorem 6: Let QCR and A be a posi ti ve semidefinite nxn matrix. Let (D.,A, p) n be a probabi li ty s p ace, let JC{x:n...,R d random variable: Ex=O and there exists E (xxTJ} ana let

~={X€1':p ({W:X (W ) EQ})=l}. Use the following notations ( x p ose of x ) : F: '?-HI nxn 1

T

is t h e trans-

T

Fl ( x ) =E ( x

F2:1-4Rl,

F ( x ) =trace E CxxT); 2

l F :1...,R , 3

F ( x ) =det E (xxT) . 3

~

where ~ is as defined in Exarrple 2. Specially, if H2=Rn, then the cost in

Theorefll 6 is a generalisation of the well-known fact that the Kalman-Bucy fi Ite r is optimal for a variety of performance criteria. The dependence of the cost functional was examined by Athans and Tse ( 1967 ) using the maximuIl' principle/which is only a necessary condition of the optimufll. The ?roof of Theorem 6 is a pure ly alg,~hraic one. The proof of Theorem 7 makes use of the properties of the support functional and Banach's open mapping theorem. It is inportant that the uncertainty Q of the parameters ( or l some signals ) x of the linear system A is qi ven by an ellipsoid ( energy dSns traint ) Q in the Hi lbe rt space of the paraJreters, otherwise the cost space usually is not finite-dimensional. Exarrple 4 ( rederivation of the KalmanBucy fi Iter from the local supremUfll principle ) : Consider the system ) u (t ) ,

y ( t ) =C (t ) x ( t ) +z ( t ) ,

If F(x ) ~F ( y ) , Le. F ( x ) -F (y) is a positive semidefinite symrretric matrix, t h en F . (x ) ~F .( y ) , i=1,2,3. If x Er. ~

( 31 )

x (O) ,

Ax ) ;

and inf{F ( X ) :XE:~}=F (Xo ) ' then

inf{A A-IA~:UEU}=A A-IA~ u u U U o o

x( t ) =A ( t ) x ( t ) +B Ct

F ( x ) =E ( xxT ) ;

Fl:1~R,

o

the second problem is IT'atrix-valued.

y t (T , t , x ) +y x (';:' , t , x ) f Cx, t ) =0

Hence V( x,t )

U

0

?

min {F i ( x ) : XE:~}=Fi ( x ) , i=1,2, 3. o Theorem 7: Let E be a Banach space and UCE. Let H 1 and H2 be Hi lbe rt spaces. Let A:H~H be a positive definite and se!fadjoint q:>erator and suppose t h ere exists a retl number 0(>0 such tilat (Ax,x>~ ()(lIxll for all XEH l · Let Qf'{XEH 1: ( Ax, x> ~ I}. Suppose,

( 32)

where the n-vector x ( t) is the state, t h e r-vector u ( t ) and the m-vector z ( t) are ....,hi te stochastic processes, and the m-vector ve t ) is the ohservati o n. It "" i l l b e assumed that x ( O ) is inde p endent of u ( t) and z ( t) ,

(x(O)-xo)TJ=~,

EX (O) =Xo '

E [exCO l -xo)

Eu(t ) =0,

T E [u (t) u ('!') ] =Q(t)o ( t-7),

Ez ( t ) =0,

E [z ( t)zT('T) ] =R(t ) 6 ( t-7), E [u(t) zTC'Y )

1 =0,

(33)

...,here L ~O and Q(t ) :::O, and R(t) is a positivg definite IT'atrix. It is desired to find the linear filter

Optimum Control Problems

1039

X(t)=F(t»«t) +G(t)y(t), X(O)

(42) (34)

-x (T»)(x (T) -~ (T)) TJ--+ inf imum.

(35)

1={xCT)-~(T): there are J{(O),F(·) and G(·)such that

On the other hand (27), i.e. the matrix Riccati differential equation, has a solution overall on [O,T]x[O,T]xZ, and the solution of (28), i.e.

dY (.)=U{[AC7)-

d7

- L (T, t , LO) R -1 ('r) C (T)]

[x (t ) - ~ ( t )] =0

'lftE[O,T],

(43)

x(0) =x o .

The cost is matrix-valued and the infimum has to be found, depending upon ~(O),F(·) and GC·). Let

E

;

;... " T R-1 (y-Cx), " X=AX+LC

E[ x Ct) -l«t)] =0 for all tE: [O,T], E[(x (T)

L(O)=L

O

with

(36)

Y('r) C·)},

yet) (.) =i ( . ) ,

there exists is the positive operator (44)

then the problem is inf E [ (x (T)

where ~(T,t) is the solution of

-l< (T »)(x (T) -~ (T») T] ,

x(T)-J{(T)E1.

(37)

By Theorem 6, the solution of (37) is optimal for a number of scalar-valued cost criteria, too. The following notations are used: L(t) =E [(x (t) s , U:M --;M nxn nxn , T:M --l M nxn rnxn

-x (t))(x (t) -~ et») T], U(x)=:C+x T , T T(Y)=Y.

(38) (39)

(40)

Then F (t) =A (t) -G (t) c (t) and (37) is equivalent to the deterministic problem infLCt) , tCt) =U {[A Ct) -G Ct)

c (t)] L(t)} +

+B (t) Q (t) BT (t) +G (t) R Ct) G1t), L(O) =2:"0.

(41)

The following correspondences are valid:

Z

positive semidefinite matrices

u

G

~(x,u,t) ~u(x,u,t)

T T U{[A-GC)2}+BQB +GRG U{[GR-UT]TC·)}

f(x,t)

U(AL)+BQB T -LC T R- I C2

fx(x,t)

U{fA-LCTR-IC] (.)}

c

LO.

Since U{YTe':)l}=oC·)~Y=O, hence by (21) G=2C T R , and the solution of the problem is

dg>~~,t)

=[A(T)-

-L(r, t,LO ) R-IC'I) c('r)]~('I,t) ,
(4 5)

(iC') is the identity operator). Hence by Theorem 6, the unique global solution of the filtering problem is (43).

CONCLUSIONS This paper has been concerned with optimum control problems under a nonscalar- valued performance criterion. The range-space of the criterion had to be partially ordered by a convex positive cone having a nonempty interior. The main contribution to the theory of optimal control is a generalization of the method of Dubovitskij-Milyutin to the case in which the range of the function is a partially ordered linear topological space and the function has a minimum. The necessary conditions for the minimum (noninferior solution) and for the infimum (superior solution) were detailed for Banach spaces. The necessary condition of the infimum was formulated for dinamical systems with finite-dimensional state and cost spaces in the form of the local supremum principle. In the theory, the problem of the existence of an infimum solution is still open to further research. It is unclear what reasonable assumptions are needed in the problem statement to obtain the existence of infimum rather than the existence of minimum solutions.

1040

B. Lantos

ACKNOWLEDGEMENT The author wishes to thank the Head of his Department, Professor A. Frigyes, for his interest and help in the research of the necessary conditions for the optimality. REFERENCES Girsanov,I.V. (1970). Lectures in Mathematical Theory of Extremal Prob lems (in Russian). Moscow Univ. Press, Moscow. Neustadt,L. (1969). A yeneral theory of extremals. J. Comput.& Syst. Sci., 1, 57. Ritter,K. (1969). Optimization theory in linear spaces I. Math. Annal., 182, 182. Ritter,K. (1970). Optimization theory in linear spaces Ill. Math. Annal., 184, 133.

Athans,M., and H.P.Geering (1973). Necessary and sufficient conditions for differentiable nonscalar-valued functions to attain extrema. IEEE Trans. Autom. Control, 18, 132. Geering,H.P. and M.Athans (1974). The infimum principle. IEEE Trans. Autom. Control, 19, 485. Lantos,B. (1976) . An application of functional analysis in control theory (in Hungarian). Ph.D. (Candidat) Dissertation, Budapest. Athans,M . and E.Tse (1967). A direct derivation of the optimal linear filter using the maximum principle. IEEE Trans. Autom. Control, ~, 690.