Invariance in linear abstract processes

INVARIANCE IN LINEAR ABSTRACT PROCESSES* FAM HYU SHAK (Hanoi)

A GENERAL definition of invariance is given, and various necessary and sufficient conditions are obtained for the invariance of processes taking place in functional abstract spaces. Examples are discussed, Invariance conditions in processes described by differential or difference equations have been discussed on many occasions. The aim of the present paper is to indicate conditions for invariance in general linear abstract processes, i.e., processes described in Iinear topological spaces. In Section 1 we give a general definition af invariance and obtain some necessary and sufficient conditions for invariance. fn Section 2 we examine some examptes.

1. Invariance conditions Let U be a linear space, Y and Z Iocally convex Iinear topofogicaf. spaces, and L)cU, Mcy? McZ non-empty convex sets. Let T : D+ Y, S : II-+Z be linear operators, and t=Y, ~45 given points. Let Ti and Si denote the operators defined by T,u=Tui-t, Sju=Su-l-s, UED, Re~~jrj~~ I, The operator Sr is catfed M-invariant for the constraints

if, for any u satisfying (1) we have &“&EM. If Itif= (Of f where 0 is the zero dement (of space Z), any-inva~ant operator $1 wiil simply be called invariant. If Z is a linear normed space, and M is a sphere, radius c centre the zero point of space 2, then, instead of saying that the “operator Si is M-invariant,” we shall say that it is 8 4nvariant. Consider some examples, Given in ~-Dimensions Euclidean space R” the process s=dx/dt=Ax+

bu

and the linear functional

where ZER*, u=R’, A is an (rr X ~1)matrix, b and d are n-dimensional vectors, and (,) denotes the scalar product, We shall assume that the function u(t), 0 < I < tl , belongs to the space

01

Invariance in linear abstract processes

21

L,[O, ti], where 1

2, . . . , n, the solutions of the systems dq/dt=-A*$

drpldt =Acp,

with the initial conditions (pk(0) =$“(O) = ek, where ek is the k-th unit coordinate vector of space Rn, and the asterisk denotes transposition. It is easily seen that weak invariance is equivalent to the fact that the operator 11 Siu =

s

(q(d, t,,

T), b)zz(z)dT:

L,[O, ti]+R’=Z,

0

where $(d,t,r_)=

n E (4 %(t))$k(r), A-l is invariant (in the sense of Definition 1) for the constraint u(t)d=U=L,[O,

ti].

Strong invariance implies that the operator S,u=

s

($(d,t,

T), b)u(T)dc

is invariant for the constr%nt u(t) =D=U=L,

L,[O, S,]-C[O,

fi]=Z

[ 0, f, I.

Parametric invariance may similarly be reduced to invariance in the sense of Definition 1. Now let N and M be convex cones. We put N’= {ZJ’EY’ : ( y’, n) 20

bdv},

A!r={Z*EZ*

vr?z4!f>,

: (z’, m) 20

where Y* (Z*) is the space adjoint to Y(Z). We shah require the following lemma, proved in [ 1 ] . Lemma 1 Let M be a convex closed cone, and let ~2.

Then z&I

if and only if (z*, z) > 0 for all

z*=w. Let N be a convex cone of space Y. For points yi, yz=Y

we shall write

yIQ

( or ih&i),

if

yz--yiEN,

yI
( or

if

Y~GY~and yzfyl.

y2>yI),

The relations <, 2, <, and > are defined similarly in space Z( Y*, Z*) by means of the cone M(N*,M*).

i;am Hyu Shak

22

Definition 2. Let N and M be convex cones. The system

l&ED,

o
O~Tll.2,

(4)

is said to be compatible if there exists at least one point u=U satisfying (4). Compatibility is similarly defined for the system OGT,U.

UED,

(5)

Definition 3. The points ul, . . . , uk of space U, where k is a positive integer, are said to be admissible if zz$sD, i=l, . . . , k, and the system

is compatible. Here and throughout, we denote by [ ul, . , . , ZQ,]the convex hull of points ul, . . . , uk, and by U the collection of all sets of the form {I&,,. . . , uk}, where ul, . . . , uk are admissible points. Let

(u i, *

0

*

,

uA}&L We put

where Z is a linear normed space. Throughout, unless specially stipulated, it will be assumed that U is a linear space, and Y and 2 are locally convex linear topological spaces, where Y is separable, and N and M are convex closed cones, while the system (I) (i.e., (5)) is compatible. Lemmil2 Let Z be a linear normed space. 1. Let the cone M be such that there exists at least one point mle%, If the system (4) is incompatible, then

for which -m,zM.

2. Conversely, let (7) hold and let inf{(z*, m):z’BO, for any rn~M\

f(z’II=i)>O

(0). Then the system (4) is incompatible.

Note I. If Z = R’ , condition (8) is satisfied.

Proof: 1. We take an arbitrary point mo=M, for which -m&M, pointsur,. . . , uk. We consider the space Y X Z. Clearly, the set H={(y,

z)EYXZ

r&U,,

. **, &I}

: g=T,u,

and arbitrary admissible

z=Scu-n20 for some

Invariance in linear abstract processes

23

is the convex hull of a finite number of points of the separable space Y X 2. Hence it is convex and

compact [2] . The set K= {(y, 2) EYXZ : y=N, z=fI!} is convex and compact. Obviously, KM=@. For, there would otherwise exist a point UE [ u,, . . . , uk] CD, such that T,uEN and Siu~M+m,. On the other hand, S,u#O, since -m,EM. The system (4) is thus compatible, which is impossible. By the theorem on the separability of convex sets, there exists a pair (y’, z’), y *EY*, z*&*, such that the functionsy* and z* do not vanish simultaneously, and such that sup

[

(y’, Tiu) + (f, Siu) 1-c (y’, n> + b*, mfmo)

(9)

ua[uI.....ukl

holds for all nMVand m=M. The inequality (9) shows that Y*EN*, z*EM’ by virtue of the fact that N and M are cones. Setting m = 0 and n = 0 in (9), we get sup UE[UL,

..‘Uk

[

(y’, Tiu) + (z*,SIU)I< (z*,mo>.

(10)

I

It is easily seen that z’Z0. For, the inequality (10) would otherwise take the form sup (Y’, Tiu) -= 0, uP[ur,...,uhl

which contradicts the fact that the system (6) is compatible. (see Lemma 1). It can therefore be assumed that 1Iz*11=1.The inequality (10) gives

(f, mo). f(Ul, . * * Uk)-=c

(11)

7

Since mo~M is an arbitrary point with the property --m,EiV, and ul, . . . , uk are arbitrary admissible points, (7) follows from (11). 2. Conversely, let the inequalities (7) and (8) be satisfied, and let the system (4) be compatible, i.e., there exists a point u. which satisfies the constraints (4). Since {uo}EU, we obtain from (7) inf (y’, Tiu,) + inf (z*, SIuo) < 0. Z’W,IIL*l[=l Y’>O

(12)

It is clear from Lemma 1 and condition (8) that inf (y’, Tiuo) = 0, U.>O

inf (z*, SIuO) > 0. z*>o,Ilz‘(l=l

(13)

Note 2. If D is the convex hull of a finite number of points of space U, and we assume as before that the system (5) is compatible, then (7) takes the form inf

sup [ (Y’, TiU) + (f, SiU)

y*>o.r*>o,[lZ+=.IUED

1Q 0.

(14)

Let us state another lemma, the proof of which is similar to the proof of Lemma 2. Lemma 2’

Let Z be a linear normed space, U a linear topological space, DcU S linear continuous operators.

a compact set, and T and

24

Fam Hyu Shak

1. Let M be a cone having the property mentioned (4) is incompatible, then the inequality (14) will hold. 2. Conversely, if the inequalities

in Paragraph 1 of Lemma 2. If the system

(14) and (8) hold, then the system (4) is incompatible.

Theorem 1 An operator Si is M-invariant

for the constraints

(1) if and only if

sup g (z*> < 0, z*>0

(15)

where

hoof:

It is clear from Lemma 1 and Definition

the constraints

(1) is equivalent

1 that M-invariance

of the operator $

for

to the fact that, for any z*EM* we have (z*, Siu) > 0, if UED

and 0 < Ti u. In other words, the operator Si is M-invariant

if and only if, for every z*EM’ the

system

ED, is incompatible,

oc-

OGTiu,

(z*, S,u)

i.e.,

(16)

g(f)<0 (see Lemma 2 and Note 1). The inequality

(16) will be satisfied for all z*EM’. Hence (15) follows. The theorem is proved.

Notes. 3. Theorem 1 may be proved by means of Lemma 2 for any convex set D and any linear operators T and S. If the assumptions made in Note 2 or in Lemma 2’ are now satisfied, the necessary and sufficient condition for the operator S1 to be M-invariant will be given by SUD inf SUD [(y',Tiu)-(z*,S~u)

Z’S0u*>oUED

I< 0

4. We can investigate M-invariance by using the theory of duality in mathematical [3]. In fact, let Tz : D + Y be a convex operator (with respect to the cone N), i.e.,

programming

Tz(au+~u)GaTzu+i3Tzii a, l3>0, a+p=l and U, ii=D, where, as before, the < relation is defined by the cone N. Let the convex (with respect to the cone M) operator 5s : D + 2 be M-invariant for the constraints UED, Tzu>O (the definition of the M-invariance of ,S, is similar to Definition 1). This is equivalent to the fact that, for any z*EM* the number

for all

sup [-(z’, must be non-positive.

Szu) : u=D, TzzeO]

(17)

Let the duality theorem hold for problem (17) for any z*=M* i.e., sup[ - (z*, &u) :u=D,

T+>O]=

inf sup v’>O

u=D

[ (y’,

2’2~)

-(z’,

SZU)

11

Invariance in linear abstract processes

Then the necessary and sufficient

condition

25

for S, to be M-invariant

is that

sup inf sup [ (y’, Tzu) - (z*, &u) ] G 0. r*>O

u’,O

Let us emphasize that the inequality

(18)

u=D

(18) holds under the condition

that the duality theorem hold

for the problem (17) for all z*GIP. Several existence theorems for the duality relation in Banach spaces are to be found in [3] . From them may be obtained

conditions

for M-invariance

in the form

(18), as we have shown above. Now let the space Y satisfy the first axiom of countability. 1 that, for any

sup

inthe

It can then be shown by Theorem

z*EM’ the weak duality theorem will always hold for the problem

[-(z*,S~U):U~D,O~T~U]

sense that, given any admissible points

uI ,

. . . , uk, we have

u”sup[-(z’,s~u):u~[u,,...,u~], =

For, Theorem

O
sup inf [ (y’, T,u) - (z*, Slu) 1. UEIU1,...,Ukl Ii*>0

1 and Lemma 2.1 of [3] show that inf sup 1 (y’, T,u) - (z*, As&) 1G u I/**0 US[Ul,....U/J = sup inf [ (y’, T,u) - (z*, S,u) 1. uar1r,,...,I$] y*>o

The required expression follows from this. If we assume that the conditions

mentioned

in Note I

or in Lemma 2’ are satisfied, we obtain the usual duality theorem. Note 5. Let us show that an invariance condition linear inequalities

can be obtained by using the theory of

[4]. Let the convex set D IcU be specified by a compatible

system of inequalities

and equations (fj, U)2czj,

i=I={1,

We shah indicate the conditions

u,

==fikr

andgk are linear functionals,

find conditions

E},

(fit

U)=Ui,

2 ,...,

iEZ={l,

m}.

under which, for every u satisfying the constraints h?k,

(here,fi,fi,

2,...,

k=K={l,

(19)

(19),

2,. . . , n}

and oi, Cyi,and & are numbers).

for the operator $ to be invariant for the constraint

In other words, we shall

USD,, where Si : Di-+Rn is

the operator defined by

&u=[(gi,

u)-f%, . . . , (gn, u)-p,]=R”.

(20)

Theorem 2

j=J,

The operator (20) is invariant kEK,exist such that

for the constraint

(19) if and only if constants

a?, bp,

c,k,

d,k,

i=I,

(21)

26

Before proving, our Theorems 3 and 4, we require

Proo$ Lemma 3 shows that ~-i~var~an~eof the operator SXis eq~~a~ent to the i%ct that, &err any z*f&* the functional (z*, Sr u) is e-invariant for the constrdnts (1). The latter is in turn equivalent to the ~CQmpatibiIit~ of the systems t;eG& O
Invarinnce in linear abstract processes

27

ZED, OGT,u, O< (z*, Siu) --E. To complete the proof, we only have to use Lemma 2.

From the proof of Theorem 3 we obtain Theorem 3’

The operator Sr is invariant for the constraints (1) if and only if aup

aup

inf

sup

[(Y’, T+) - (A

SUP

aup

inf

sup

[(Ye, T,u) + (3’7&u)l

Z’EZ’@I,...,u&a Y’>pUE[UI I...,u)J

WI Q 0,

z*Gz* @I,...,u@I IPal UE[U, ,...IU/J

d 0.

Notice that the Z in Theorem 3’ is not necessarily a linear normed space. Theorem 4

Let U, Y, and Z be linear normed spaces, where U and Z are reflexive, E and 1 are given positive numbers, T : U+Y, S : U+Z are linear continuous operators, and D,={uEU : Ilull
if and only if we have the inclusion S’Zi’c {T*NI},,l,

(29)

where S* and T* are the operators adjoint to S and T, and {T’N’},,~={~*EU’

: p(u*, T*N’)+Z}

(p (u*, T’N’) is the distance from the point U* to the set T*N* in the linear normed space u*

adjoint to U). Proof: Sufficiency.

(28), and show that that, given any 00

Let the inclusion (29) hold. We take an arbitrary point UEU, satisfying

IISull
(30)

From (30) we have (z*, Su)=(S’z*, > (S’z*-T’y’,

u)=(y’,

Tu)+(S’z’-T*y’,

u) ~-IIS’Z’-T*~*IIZ>-E-~~.

u) (31)

Since 6, is arbitrary, the inequalities (3 1) give (z*, Su) k--E.

(32)

The relation (32) holds for any z*c%*. Since z*6Zi* if and only if -z*EZ,*, it follows from (32) that 1(z*, Su) 1Ga for any z*E&*. By Lemma 3, we have IISr.zll<~, which is what we had to prove.

28

Fam Hyu Shak

Necessity. Assume that the operator S is E -invariant, but that (29) does not hold, i.e., s’z”G {TIN’),,lcU” for some z*~Z,*. It follows from the theorem on the separability of convex sets and the reflexivity of the space U that there exists an element UEG?*=U, such that

(if, Su) = (SY, 24)> (u, u)

(33)

for all UE (T’N’) s,~. Since IV is a cone, it follows from (33) that (Y*, 2%) < 0 for all y*&V, i.e., Ty-U) > 0 (see Lemma 1). It can be assumed without loss of generality that [lull=Z. By the Hahn-Banach theorem, there exists ii*~U’ such that IIU’ll=1 and (U*, U) = Ilull =Z. Clearly, (e/Z) ii*= (T‘J’V}*,[. The inequality (33) implies that (-z*, S (-U) ) > (a/Z) (E’, U) =E. This means that I/S (-u) \/>a (by Lemma 3 and the condition z+E.&* ). Hence the point -U satisfies the constraints (28) and the condition IlS( -u) II>&, which contradicts the a-invariance of the operator S. This completes the proof. The following can be proved by means of Lemma 1 and the theorem on the separability of convex sets. Theorem 5 Let U, Y, Z be linear normed spaces, where U is reflexive, and T : U-Y, S : U-tZ are linear continuous operators. The operator S is ~-invariant for the constraint 0 < 2%if and only if

where the bar denotes closure in the linear normed space Lr* adjoint to U. To conclude this section, notice that Lemmas 2 and 2’ can be used, not only to obtain invariance conditions, but also to investigate conditions for optim~ity in linear extremal problems. In fact, we shall say that the point U~EU is a solution if u. satisfies the constraints (5). We call the point uO=U an optimal solution for the operator S if it is a solution and there exists no other solution ii=U, such that SUCSu,. Here, as above, N and M are convex closed cones, and T : I)+ Y, S : D-+.2 are linear operators, where 2 is a linear normed space. Theorem 6 1. Let the cone M satisfy the condition mentioned in Paragraph 1 of Lemma 2, and let u. be the optimal solution for the operator S. Then, given any a>0 and arbitrary admissible points uk, there will exist function~s y,‘EN’, &,*Ew( ilz,*/ =I), dependent on e, 7&, . . . , ak %,..., such that - (yo’, Tu+t) +(zOf, Su) >-- (y’, Tu,+t)

forall 22~ [ rk, . , . ,QJ

+ (zo*, Su,) --E

(34)

and y*=N”.

2. Conversely, let the inequality (8) be satisfied and let the point u. belong to the convex hull of certain admissible points. Further, given any EXl and arbitrary admissible points [lz,‘II=‘f, dependent on e, ui, . . . ul , . . . , ztk, let there exist functionah yO*EN’, &*Ew, such that (34) holds. Then z+,is the optimal solution for the operator S.

, uk

PLoof: 1. Let u. be the optimal solution. Then the system UED. O< T,u, OtSu,-Su is incompatible. By Lemma 2 and the inequality (y’, Tu,+t) 20 for all y*eN* the statement of

Invariance in linear abstract processes

29

the theorem readily follows. 2. Now let the conditions

mentioned

in part 2 be satisfied. Let u1 , . . . , uk be arbitrary

admissible points, the convex hull of which contains the point u,. For all UE [ ul, . . . , uk] we have

-

G/o*,Tu) + (za’, Su))-

(Yo’,%I) + (zol,Sue)-&,

(35)

OG (yo*, Tuo-tt)
(36)

= 0, u = uo.

We now set u = u. in (34). We then get - (yO*, Tu,+t) Consequently, (y’, Tuoft)

-(y*,


T&t)

for all y*EN*. Since E>O is arbitrary,

20 for all BEEN*, i.e., O
optimal solution,

i.e., SD < Sue for some solution

on E, ul,

and the inequalities

. . . , uk?

for all y*EN*.

we have

UE li. Clearly, ur , . . . , uk, ii are admissible

.fjo=N*, &*df*, IIZ,,*II =I, are functionals

for which an inequality

of the type (34) holds, then condition

(8)

of the type (3.5), (36) give us Ocinf <-

which is impossible,

U

-e

Hence u. is a solution. Assume that u. is not the

points, the convex hull of which contains u. . Hence if dependent

> - (y’, Tu,+t)

{ (z*, Su,--SE) (&I*, Tu+t)

: z*>O,

+ (&*, TU,+t)

~~z*~~ =I}<

(Fo*, Su,--SE)

+&<2E,

since E is arbitrary.

From Theorem 6 and its proof we have the corollaries: Corollary 1

Let the cone M satisfy the condition

indicated in Paragraph 1 of Lemma 2, and let u. be the

optimal solution for the operator S. Then, given any

~-0 and arbitrary admissible points ul, . . . ,uk

the convex hull of which contains the point uo, there will exist functionals I/z,,*ll=l,dependent

on E, ul,

. . . , uk

yO*~N*, z,,*~fl~,

such that (35) and (36) hold.

Corollary 2

Conversely, let uQ be a solution and let (8) hold. Further, given any es0 and arbitrary admissible points ul, . . . , uk, the convex hull of which contains the point u. , let there exist functionals yo*=N*, ZO*EM*, Ilz~*ll =I, dependent on E, ul, and (36). Then u. is the optimal solution for the operator S.

.

.

.

,

uk

with the properties (35)

2. Examples

1. Invariance in linear continuous connected processes Consider two processes, the behaviour of which is described by the systems xt=AiXifb,u, where Ai are n X n matrices, the bi are n-dimensional a scalar function: uER’.

i=l,

27

vectors, GER”,

(37)

and the input signal u is

Fam Hyu Shak

30

Boblem 1. To find the constraints on the right-hand sides of the systems (37& under which every input signal u (t) , O
We shall assume that at least one input signal exists having this property. We know that

OGtGt, k-1

where ~i=(ai’,

. . . , Uin)

i=l, 2,

(38)i

0

;

(pk,

and $“, i(t) , k= 1, . . . , n, are solutions of the systems

i(t)

@/dt=Aiq,

d$/dt=-Ai'll,

(39)j

under the initial conditions (pk,

i (0)

=$“’

=ek.

i (0)

It is clear from (38)j that the problem 1 is equivalent to Problem 1’. To find the constraints, imposed on Ai and bi, for which the expression

CJ k=,

(pk,i

(t)

(T),

(d”

h)

u

(7)

da=“,

(40)

0

i T,s

which holds for some u ( .t) , O< 6

t, implies that

n

k=i

qk.Z(t)

($k’2(dr

b’2)

u(a)dz=O.

(41)

0

It is clear from Theorem 2 that the expression (40) implies (41) if and only if a matrix C exists such that C

k=l

=

(pk,l(t)

CJ

($“‘%), b’)“(‘t)dz

0

CJ

(42) (pk.2(t)

($“*”(T>, bd U (7)d’G.

Denote by P the (n + 1) X n matrix whose rows are the vectors bl, Aibi, . . . , A,‘%, and by Pi the (n t 1) X (n + 1) matrix whose first n columns are the same as the columns of P, while its last column has the form ( (bz)i, (A*bz)ty . . . 7 (A,“b2)i)*,

where (Azjbz) i is the i-th coordinate of the vector A-/'bz . Denote by
. . =r(P,).

Proof: If the condition indicated in Problem 1 is satisfied, then, as we showed above, (42) holds for any U(Z) , where C is a matrix, whose i-th row will be denoted by cj. Then, (42) gives

(43)

Invariance in linear abstract processes

31

(44) for all T, O
where

@Cd!T)=

c (4 qk.,(t))p(T),

i=l,

O&&l,

2

k-i

(d

is an n-dimensional

$‘(d,

vector).

It can easily be seen that $’ (d, z) is the trajectory of the system (39)i with the end condition f) =d. On successively differentiating

resulting expressions,

the relations (44) with respect to r and setting

r=t

in the

we get

(Afkbt, Ct)=(Atb*,

k=O, 1,. . . , TZ, i=l,

ei)=(A?Ab?),,

2,.

. . , ?Z.

(45)

The system of equations

(A,kb,, x) = (A,kb*), thus has a solution Ci. By the Kronecker-Capelli

k=O, 1,. . . n,

(46)

(

theorem, relations (43) must hold.

Suppose now that Eqs. (43) hold. Then, by the Kronecker-Capelli

theorem, for every i = I,

. . . , n, the system of equations (46) has a solution, which we shall denote by x = Ci. To prove that the condition indicated in Problem 1 is satisfied, we only have to show that the matrix C, whose i-th row is the vector ci, satisfies Eq. (42. Denote by m(
m, exist such that

m il;+‘b,

=

h,A,‘b,.

c j-0

(47)

and hence it follows from Eqs. (45) that

i=l,

. . * n. 9

i.e., m

A,“+&=

c

hfli,‘b,.

(48)

j-0

We set qi”(T)=(\r;‘(ci,

z), A,“b,)-_(+‘(ei,

z), A?bz),

k-0,

1, . . . . m-I-1,

(49)

0
Expressions (47)--(49)

give

(z),

k=O,

1, . . . m.

(50)

32

Fam Hyu Shak

Qi‘n+’(IT)=

(51)

ihjq?(T).

j=O

The functions qi”(7)) qii (T) , . . . , qim(a) thus satisfy a system consisting of m + 1 linear homogeneous equations (see Eqs. (50) and (5 1) ), and the conditions qik(t)=(Cit

Aikbl)-(ei,

Azkbz)=O,

k=O, 1,. . . , m

(see Eqs. (45) ). Using the uniqueness theorem to solve linear differential equations, we find that qik( T) ‘0,

k=O, 1,. . . , m,

0<1St.

We have thus proved (44), i.e., Eq. (42) is satisfied. 2. Invariance in linear discrete connected processes Consider, in the linear normed space X, the processes described by the difference equations k=O, 1,. . . , m-l,

a(lc+l)=Ais(k)+b,u(k), xi

(0) =ai,

i=l,

2,

(52)i

(53)i

and the linear functional

(x*7x>

(54)

7

where Xi(k) EX, u(k) ER’; A; : X+X are linear continuous operators, bi and ai are given points, and m is a positive integer. Let the points ai, a’i be given. Problem 2. Let the signals

{i(k),

k=O, 1,. . . , m-l}

(55)

move the phase point of the system (52)i from al to ai. To find the constraints to be imposed on Eqs. (52)i and the functional (54), such that the following condition is satisfied: given any signals {u(k),

k=O, 1,. . . , m-l},

(56)

moving the phase point of the system (52)i from aI to atI and satisfying the inequalities lu(i)--i(i) l
. . +b,u(m-1).

(57)

We shall denote by U = Rm the m-dimensional space of elements u= (u (0), . . . , zz(m-l) ) withthenorm Ilull=maxlu(i) I. WedefinetheoperatorsT: U+Y=XandS: U+Z=R’ as i follows: Tu=A:-’

b,u (0) + . . . +A,b,a(m-2)-l-

b,u(m-1), (58)

Su = (z*, A:-‘bz)

u (0) + . . . + (d, A,b,) u (m-2)

+ (z*, b,) u (m-l),

33

Extremal valuesof game problems

and denote by N the cone of space X, consisting of the single point 0. It is clear from (57)i that the condition indicated in Problem 2 implies that the operator S is E-invariant for the constraints Ilr.zll
c I cz*,Ai’b,) - (X*,Aib,)

I < E/l.

iC0 Translated by D. E. Brown REFERENCES 1.

DUFFIN, R. .I., Infinite programs, in:Linear inequalitiesand related systems, Princeton U. P.

2.

BOURBAKI, N., Espaces vectoriek topologiques, Hermann, Paris, 1955.

3.

GOL’SHTEIN, E. G., Theory of duality in mathematicalprogrammingand its applications (Teoriya dvoistvennosti v matematicheskom programmirovanii i ee primeneniya), Nauka, Moscow, 1971.

4.

KY FAN, On systems of linear inequalities, in: Linear inequalitiesand related systems, Princeton U. P.

EXPANSION WITH RESPECT TO A PARAMETER OF THE EXTREMAL VALUES OF GAME PROBLEMS* V. F. DEM’YANOVand A. B. PEVNYI (Leningrad) (Received 15 May 1973)

THE EXTREMAL value of a parametric problem of mathematical programming is considered as a function of a parameter. The directional derivatives of different orders of this function are investigated. Consider the function cp(z) = max f (x, Y), v=Q(r)

where a(x) are closed sets of Er , dependent on the parameter x:“E,; E,, EY are finite-dimensional Euclidean spaces. We are interested in the first directional derivative acp(x0) /dg. This problem was discussed in [l-lo] . If &p (x0) /ag exists and is finite, then the limit

d2q(x0) Bg”

acp (x0> t =,li$+ [q(xo+&J--q(xo)_ al? I

will be called the second directional derivative. If the limit (0.1) exists and is fmite, then

cpb”+tg) = cp(x0)+

%J(x0)

Zh. vjkhisl. Mat. mat. Fiz., 14.5, 111&1130, 1974. U.S.SR.- C

dg

&.

+

1% 2

be> t2+o dgZ

(P) .

(0.1)