Approximation of extremal problems. II

APPROXIMATION

OF EXTREMAL

PROBLEMS. II*

B. M. BUDAK and E. M. BERKOVICH

(Received

THIS paper is a continuation extremal extremal optimal

2 July 1970)

of [ 1 ] . Here new conditions

for approximating

the original

problem are considered. Various questions connected with the stability of problems are investigated. The general constructions are made specific for control problems.

1. More about approximation

“along the functional”

1. As in section

1 of [l] , we consider the necessary and sufficient conditions of of the extremal problem “along the functional” by a (E, A, Zi sequence of extremal problems (E,, A,, I,,), n = 1,2,. . . . We denote by (E, #A:Z) the problem of the minimum of the functional Z(U) on the set A c E. Everywhere in

approximation

what follows we will suppose that I,” = inf Z,,(u,,) =n=“n and the functional

A,, # >

0,

-00,

Z(U) is underbounded

n =

1,2 . . ..

on E.

Let ‘3 be a given set in a topological space with the first axiom of denumerability, ~9a fixed element of % . We suppose that for every a E Iz[ nonempty sets are specified. A n+cr c E,, n = 1,2,. . . Definition

1.1.

The sequence of problems

(E,, A,,, I,,),

n =

1,2,.

..,

is said to be A+ -stable (with respect to the family An+=, 13E+211, n=1,2 ,... ), a,+6 if for any sequence {Cz,} C YI, for which as n + + 00, the inequality

is valid. *Zh. vjkhisl.

Mat. mat. Fir.,

11, 4, 870-884,

1971.

71

B. M. Budak attd E. M. Bcrkovich

72

Theorem

1.1

(E,, An, In), n = 1,2, , . . , The sequence of extremal problems along a functional, if and only if, for every (E, A, I) approximates the problem a=$¶ nonempty sets A +a, 6” c E, LI,,‘~ C E,, n = i, 2,. . . , can be found such that the following conditions are satisfied. Dl. For every n = 1, 2, .. . elements u, E A,,, a, E QI u and I)&,, E Atan, a, + @ are valid as {Z,,(U~) - I,,*} --f 0, exist for which the relations n++

For every natural m elements o, E %[, @) E iam, s, E %, Q,&.J(~) E annr -+ 6 as n 3 + 00 for every for which the relations for every m = 1, lim {I, (Q,,P)) - I (u’“‘))
D2.

&yn

n =

1,2,. .)exist

m=l,2,... 2 ,..., inf u&

{Z(P,u,) -In(G)) GO. TL-b+o,

oo,lim

(Z(U(*') -lIlTjaJ20as m-,+.x,

Z(u)

for any set AC

E.

The problem (E, A, I) m = 1,2,. . , , that is,

is A--stable (see 11) with respect to the family

D3.

A%,

D4.

n=1,2,...,

The problem are

are satisfied, where Zi =

and the sequence of problems (E,, A,, I_), (E, A, I) A+ -stable respectively, with respect to the families ,A+a and

A n+a, a E 91, n = 1,2, . . . . Proof:

n = 1,2,.

A” and A.+a, a E 9f, The necessity of the existence of sets A+a, and . . , for which conditions Dl -- D4 are satisfied, was in fact established in the 1.1 of [I], where it was shown that if the sequence (E,, &, I,), along the functional, it may approximates the problem (E, A, I) We A -ta.= #A,,, a E a[, n = 1,2, . . . . that AL?L+a = AaF A,

proof of Theorem n = 1,2,. . . , be considered

prove sufficiency. Because of the condition Dl and the A+-stability of the problem we have (E, A, I) lim (I’-In*} = lim {r* - In&J}< lim {I* - ZI\+afi} + (1.1) n-ctm-+ lim {Ii+an ?Z++CO

n++CO

-

1 (P&J>

n4m

+

1’lm

V (KdbJ

-

L WI

<

0.

are taken from the condition Dl. Let Here the sequence {a,), {u,} and {P,,uTm E > 0 be an arbitrary number. Using conditions D2 - D4 we select and fix a number m so great that the inequalities

(1.2) are satisfied.

L&
I

(u’“‘) - Iiam < + .

13

Approximation of extremal problems. II

By condition
D2 a number

forall

exists such that Zn(Qnmrz(“‘)) -

n, = nl (a),

n>%(E);

I( u(“))

this together with (1.2) implies the inequality

From the A+-stability of the sequence of problems I’ < 38 / 4. I;,~+%,as )2 + +~eo a number nz@) > n,(e) and the relation unm ++ (&$L, In) exists such that I,,* - Zz*+a,, nZ(e). Therefore I,’ - I’ < a. for all n > n*(e), that is, Fyl the convergence

{I,,* -

I,’ + I’

I’} <

which together with (1 .I) implies

0,

n-t+co.

as

The theorem is proved.

2. We consider in more detail the case where for some pair of elements a’, ,a“ E % a relation a’ < cP is introduced, the convergence of any sequence {a,} c 51 to the element 13implying that for any there exists a number N(a) such that UC=% for all ?2 >

‘U*
N(a).

Theorem 1.2 (E,, A,,, I,,), n = 1,2,. . . , The sequence of extremal problems approximates the problem (E, A, I) along a functional if and only if for every ,A,,+= c E,, n = 1,2,. . . , can be found a e 9l non-empty sets A+= c E and such that the following conditions are satisfied. El.

For every n = 1, 2,

. . elements

u, E An, a,, E %

and

P,u,

E Aian,

exist for which the conditions

are valid. E2.

For all natural 1, m, n elements U(~JE A, I?,& E E, al, m,RE for which the following relations are satisfied:

QI,Qnmz#

EA.,+%n", exist 1)

lim I(u(‘)) = I’, I++ 31

2)

lim‘ {I( R,u(‘)) TIl-++CC

3)

=

Tl++CC

{L(Qnmu(‘))

4) for any cz cs % natural m > m’(a, I) for 72 > n’(a, I, m).

I(U”‘) } d -

~(RnW)

for every

0 <

0

for every

1, 1, m,

exists such that for all and for any 1 a natural m’(o, 2) can be found such that aI, m, ,, < a n’(o, I, m)

B. M. Budak and E. M. Berkovich

14

E3. relation

n = n =

The system of sets a’ < a”

a E %, n = I,%. . . ,

A,,+,

implies the inclusion

A,,+Or’c A,+O

If

is chosen such that the for all n.

and the sequence of problems E4. The problem (E, A, Z) . .) are A+-stable with respect to the families A+= and

1,2,.

(E,, An, I,,), A,+,

a E 8,

1,2, . . . , respectively.

Proof: Let the sequence of extremal problems (E,, A,,, I,) approximate the problem (E, A, I) along a functional. We put A+n. = A. Anfa = An for every a E II, n = 1,2, . . . . It is obvious that conditions E3 and E4 are satisfied. For the satisfaction of the condition El it is sufficient to take as {u,} an arbitrary sequence of elements for which u,, E A,,, n = 1,2, . , . , {Z,,(u,,) - Zn*} * 0 as n + i-00, as {a,} to take an arbitrary sequence from %, converging to 8, and as {P,u,} to take an arbitrary sequence minimizing the functional I(U) on the set A. It is obvious that then lim {Z(P,u,) n-++m

-

Zn(un)}

=

lim {I* n++C=

In*} =

lim

{Z’ -

Zn*) = 0

We verify that condition E2 is satisfied. Let {u(r)> be some sequence minimizing the functional Z(U) on the set A; We put Z?,u(‘) = u(l) for every natural m. It is obvious that the relations (1) and (2) in E2 are satisfied. We put al, m, n = a, for all 2, m, and n, where a,+6 as n -+ + 00. Then the relation (4) in E2 is also satisfied. Let {u,> be some sequence of elements for which U, E A,,, n = 1, 2, . . . ) (Z,,(u,,) - In*} + 0 as n -+ +co. We put Q,,u(‘)F_ u, for all natural I, m and n. Then lim {Z~(Qnmu(‘))-Z(R,~~‘))~=lim lI-++CC

{Z&n)-I@(‘))}< n-+t m

< lim {Z, (u,) - z*> = lim {In* - I’} = 0. n*+m n-++c= In other words the relation (3) in E2 is satisfied, and this finally establishes

the validity

of E2. Necessity has been proved. We prove sufficiency. Just as in the proof of Theorem 1.1, the condition El and the A+-stability of the problem imply the in(E, A, Z) equality (1.3)

-KG-- {I* ?l-++m

In*} 4

0.

We show that for any a E % and any E> 0 be found such that the inequality Zi,*p < I’ -l- E We select and fix an 1 such that Z(U(‘)) < E2 we select an ml = m,(a, a) > m’(a, E),

a number fi (a, 8) can holds for all n > Z (a, E).

I’ + E / 3. By means of the condition for which Z(R,,u(‘)) < Z(0) -j- E/Z.

App~o~i~tion of exnemd problems.11

tit

fi(U, E)

n(a, all <&

E) >

Z~(~~*~~(~))

be such that

YL’(c1,1, 0%)

n > Z(o, s) for n>z(a,e).

for all n > and by the condition

This implies the existence of a sequence

15

and Z(ZL,~(‘~) < e I3 Then for I;(a, E). a& m,, TL
{a,f

C a,

un+

as

fi

Z*

n++co,

for which lim n++

(1.4)

(I* +a -Z*)
Indeed, let as k-++oo, eb > 0, a@ E 3, k = 1,2,. . . , Ed-+ 0 a(‘) +- 6 {r&f be an increasing sequence of natural numbers for which’ nk > iz(a(q, &A). We put an== a@) for nk
Z; -.+ ;i;

as

n++co.

1.2 is proved.

Note 1. A similar approach

difference

to the proof of convergence along a functional of for optimal control problems was used in [2 and 31. For every

approximations

net function U, a piecewise~onst~t continued function was taken as P,u, for the function U‘(r)E L?r [To, T*] the m-th element of a sequence of Steklov~obo~ev average functions was taken as R,,@ and as Qn,,# the “projection onto the n-th net” of the function R,r.0. 3. In this paragraph we will consider that non-empty

sets A,-”

c

E,,, n = 1, 2, . . . ,

are defined for every CCE % . Definition 1.2. The sequence of problems (E,, A.,, Z,), n = 2,2,. . . , is said to be A--stable (with respect to the family A,-“, a E %, n = 1, 2, . . . ), if the inequality lim

{ZL A_a, -

n4m

holds for any sequence Theorem

ZE] < 0

rn

{a,} c %,

for which a, -+ 6 as n -+ 00.

1.3

The sequence of extremal problems (En, A,,, I,,), n = I, 2,. . . , approximates along the functional if and only if for any a E % nonempty the problem (E, A, I) sets A-” c E A,-” c E,, n = 1,2,. . . , can be found such that the and following conditions are satisfied. Fl.

For every natural n elements

exist

B. M. Budak and E. M. Berkovich

76

for which the following VTI (%I) lim

n4m

relations I:, &J

are valid: -+ 09

{Z (P,U,,“_

%I**

n-++

as

001

I, (u,)} < 0.

F2. For every natural 1, m and n elements a(‘) E I[, 18) E A-a(‘), Z&,uV) E E, Qnm&) exist for which the following relations are valid:

E A,,

1) /i+” (I (24”‘) -cQ 2)

iz-

$_atlj } = 0, U(l)---f 6 as

{Z(R,u@))-Z(u(‘)))

& 0

I -+ + cc,

for every

4

VI-++-

3)

lim

{Z,(Q,,U(‘))

-

Z(R,U(‘))}

G 0

for every

4 7%

n++ar

F3.

2 ,*.*1 n=l,2,...

The problem

(E, d, Z) and the sequence of problems

are A--stable with respect to the families respectively. .

A-”

and

(E,, A,,, I,,) n = 1,

A,-a, a E a:,

RYX$ The necessity for the satisfaction of conditions Fl - F3 for the sets = A, AL\,-” = A.,, a=%, n= 1,2,. . . , in the case where the sequence (E,, LA-” A.,, I,,) approximates the problem (E, A, I) along a functional was actually established in the proof of Theorem 1.2. We prove sufficiency. By the condition 7 (1.5)

Fl and the A--stability

of the sequence of problems

(E., A,,, Z,,)

Iim {I’ - Cl < nim v (PTA) - I,‘} ,( *im {I (P,u,> +4-m * 7 - I,@,)) + zpw (1, (4) - 1: a-~,) + nFm (1, A-a, - &I G O. ’ n

’ 11

We now prove the inequality (1.6)

-G{In* -P}&O. n-++m

Let E > 0 be an arbitrary number. Because of the A--s\ability of the problem (E, A, I) and and the cor$ition F2 a natural number Z for which ZA_o(r) - Z’ < a/4 can be found. From the chosen 1 we find an m such that Z(u@) - Z,-a( 1) < E /d Finally, for the 1 and m, by condition (2) of F2 there Z(R,d’)) - z (u’“) < E / 4. exists a natural number n’ such that Z,(Qnmzdl)) - Z(R,u(‘)) < s/2 for all n > n’. - I'< E for all n > n’, which implies (1.6) Since Q,,u(‘) E A,,, we have I,' because of the arbitrariness of E > 0. The convergence from (1.5) and (1.6). Theorem (1.3) is proved.

Z,'+Z* as

n-t

+OO follows

Note 2. The condition Fl is satisfied, for example, if for any a E f?[ there exist elements u,& E An-=, P,una E E, n = 1, 2, . . , , and a number n 1(CY)such that Pnuna E A for n > nl(a), {Zn(zzna) -ln,L\;a 1 -to as n++q lim{I(P+,J - Zn(%a)) G 0.

n++m

Approximation

~lllk’d,

by

kt

k = 1, 2,

dk) f?Ea, &k > 0,

{n.{,) au increasing

. . . ,

sequence of natural

P*Una(k) E *,

Cdk) -+

6,

numbers

I (Pnunl(k))

n > ni, . We put

-

In

@$Jk))

<

k-+ +a.

We denote

<

‘k>

n

‘fc

for

n1, < n < ?tk+I.

satisfied.

A-stability

2.

as

0

a,, = ack), U, = u&k), P&,, = P,&&)

F I is obviously

Then, condition

ek+

for which

1, (u n&k)) - ‘:, *-&) ’

for

17

of‘ extremal problems.11

of the problem

(33, A,I)

I. In this paragraph we consider the A+-stability (see [I] , section 1) of the extremal problem (E, ,A. I) with respect to the given family A+a, a E Vf, of subsets of the set E. We first present a criterion of A+-stability the application of which to optimal control problems will be demonstrated below.

Theorem 2. I For the At-stability of the problem (E, A, 1) with respect to the family A+“, a E a[, it is necessary and sufficient for there to exist a subset BO, 6,, u E M, of the set E, for which the following conditions are satisfied: 1) I*’ S I* < I&*, 2)

@I,” a+0

< a

3) IaO*< the family

a,,

lim Zi+a, a-ta

limIaa* n--tu

(that is, the problem

(E, &,, I)

is &+-stable

with respect to

a E “u).

Proof The necessity of the conditions

of the theorem is obvious, since in the case

of the A+-stability of the problem (E, A, 1) with respect to the family Aba, a E 3, we may consider that ~3~= A, 6, = A+?. a E U. Sufficiency follows from the inequalities I** < IaO*< lim In’ < cl a+* Theorem

lim ]:+a. -. a+*

2.1 is proved.

Theorem ‘_.I enables the study of the At-stability of the problem (E, A, /) wit11 respect to the family A+a, a E U, to be reduced to the study of the cj,,t-stability (IV the problem (E, 6,,, f), where the set ho satisfies condition t) of this ,theorem with respect to the family 6,. a E %, which satisfies condition 2). For cxamplc, let the set IS, for every * E M consist of 3 single clement zz, G A+“, for which I(rc,) - Z14, < qa, qn-to as a - 6. It is then obvious that

78

B. M. Budak and E. M. Bcrkovich

I&= I(&), and condition (2) of Theorem 2.1 is satisfied. If the set &, is such that the A+-stability of the original problem (E, ,A, I) with respect to the z ao*= Z*‘, is equivalent to its A+-stability (or the &+-stability of the problem family A+=,a E i?f[, (E, &,, I))with respect to the family 6,, a E %. The sufficient conditions of this stability, presented in Theorem 2.2 and below, may be satisfied for the family 6,, and not satisfied for the family A+a,a E 8.

cz E a,

We consider the connection between the A+-stability of the problem (E, #A,I) with the (T - A’) -correctness of the family ,A+a, u E U (see [I] , section 4). We will consider that some topology T is introduced on the set E. Theorem 2.2 (compare

with [4] , Theorem

Let the following

conditions

7)

be satisfied:

(1)

the functional

I will be T-semicontinuous

(2)

the family ,A+a, a E 9l, is

(3)

for any sequence

{a,,} t

(IY-

downwards

A+) -correct;

21, a,+

I”, as n-t

-/-co

elements u, E IA+an, IZ= 1,2,. . . , for which {Z(zzn) has a r-limit point. Then the extremal problem (E, A, I) to the family

on the set A;

the arbitrary

sequence of

Zi40n} t 0

as n+ +m, will be A+-stable with respect

A+“, a E 3.

Then we can find a number e. > 0 and U, E ,A+=,, Z(U,) - Z* < --Ea, n= sequences {a,>c a[, (u,}t E, for which zz,,E ,A, which is 192, * * * 1. an+ 6 as n -+ +m. By conditions (2) and (3) an element Z(U,) the T-limit point of the sequence {u,} can be found. By condition (l), which contradicts the definition of I*. Theorem 2.2 is proved. < z* - Eo / Proof:

We assume the contrary.

Note 3. Obviously, with the strengthening of the topology 7 conditions (1) and (2) of Theorem 2.1 are weakened and condition (3) is strengthened. This must be considered in the choice of the topology

for the application

of Theorem

In the following theorem condition (3) of Theorem Theorem 2.3 (compare with [4], Theorem 8) Let the following (1) E is a metrical (2)

the functional

conditions

2.1 to specific problems.

2.2 is not used.

be satisfied:

space; Z(U) is uniformly

continuous

on the set

u A” U A; a&I (3) .B (I’“, A) --t 0 as a - ti, where p(,A+“, ,A) G= inf {E > 0: A+a 1san c-neighbourhood of the set A, in other words, for any sequence c Oe(A)},Oe(A)' an+ 6 as 71--t j-m and for any sequence {u,} c E, u,, E A+“vI, n = 1,2,. . . ,

79

Approximation of extremal problems.11

a sequence

{k}

c

A,

can be found for which

p(u,,

&)

-+O

as

n-t

i-00.

Here /, is the metric of the space E. Then the problem

(E,

A,Z) will

be A+-stable with respect to the family

A+“. a E 3. Proof

We assume the contrary.

Then a number

exist for which U, E A+an, Z(U*) {u,} (= E, as IZ --t +m. By condition (3) a sequence (un} c + 0 as n -+ + 00. By condition

-I(&,) 1< Ed, from Theorern

which

I(;,,)

0

-80,

and sequences n =

{a,,} c 3,

1,2, . . . ; a, -9

A,

(2) for sufficiently <

EO>

p <

can be found for which ip (u,, &) large values of n we have 1Z(u,)--

I*, which contradicts

the definition

of I*.

2.3 is proved.

Note 4. In the theorem just proved, instead of condition (2) we may consider a functional I(U) only semicontinuous downwards on the set A, provided that the set A is compact (in itself). 5. It is obvious that the relation fl(A+ ‘z. A) --+ I) as (Y+ 6 and the closedness of the set A imply the A+-correctness of the family A 1%.c1ESgfl. in the topology generated by the metric of the space E. Let F be an arbitrary set, u a topology on it, T a mapping of E into F, and r and 1“in, (L E Vl. - given subsets of F. We consider the conditions of the (T - A’) correctness of the family A+” for the case where the sets A and 11LSL% II E gI, have the form (2.1)

A=

Theorem

{u,EE:

TuE~},

A+a =

2.4

Let the following conditions be satisfied: -correct; (2) the mapping T is S,(T continuous. (t -

{II E E: l’rr E I-},

(1) the family 1”-2. CLE $1. is ((7 ~~- 1’ ) Then the family I\ 1*’ of (2.1) is

Aim) -correct. Proof

Let u be an arbitrary

T-limit point of the arbitrary

sequence

{J/ } c

k’. for

which L~,,EA-“‘~~. r&=1.:! ,.... u,,+Q as n + 00. Then the element Tu is a u-limit point of the sequence {TM,,}, Tu,, E r -‘lji, n = 1, 2. By condition (l), 7‘// E I’. that is, 11E ~1. Theorem 2.4 is proved. Theorem 2.4 ensures the satisfaction of condition (2) of Theorem 2.2 for the cast where the sets A, Y”. u E ‘2, are given by formulas (2.1). We now consider condition (3).

80

B. M. Budak and E. M. Berkovich

Theorem

2.5

Let {u,,} be an arbitrary

sequence of the set E, the sequence{T~L.}relatively

pact in F, lie in some u-closed subset of the set

T-’ (a).

stronger than the topology

TE c F

Then the sequence

u-com-

and the topology r be not {u,} has a r-limit point.

Z+oof. By hypothesis, the sequence (Tu,} has a u-limit point u E F, and an element u E E, for which Tu = u. The element u is the T-’ (a) -limit point of the sequence (u,}, and consequently, its r-limit point, since the topology r is not stronger than T-’ (cr) . The theorem is proved. Note 6. As a rule it is inadvisable to use a topology T weaker than T-‘(o), since this violates condition (2) of Theorem 2.4 (the t, a -continuity of the mapping 7’). Hence we mostly put r = T-‘(a) when using Theorem 2.5. We conclude

this paragraph by considering

the case where the A+-stability

problem (E, A, I) with respect to one family of sets implies its A+-stability respect to another family.

of the with

Let E, F be metrical spaces, T a mapping of E into F, A,, a specified set in E, and r a specified set in F. We put A =

Ao fl A,,

A,=

I’uE~‘}.

{uEE:

Let % be a set of pairs of non-negative real numbers, ti = (0, 0). In the set E let there be defined a family A+‘, E > 0, of “extensions” of the set A,, and in the set F let of strong “extensions” of the set r, in other words, for there be specified a family P certain positive constants Car Cr’, Cr” < + 00 the inclusions (2.2)

A,, c

are satisfied, where

A? c

OcAe (Ad 9

0, (A)

oc;,

F) c

is an e-neighbourhood

r+c c

O,;,(r)

7

of the set A. For every a =

(a’, a’)

E 5% we consider the sets (2.3)

A:“’ =

{u E E: Tu E l-‘+a’}, A,;

=

A:“‘fl

A:“‘,

A&;=

Ao

r-l Ai+,’

It is obvious that the A+-stability of the problem (E, A, I) with respect to the family implies its A+-stability with respect to the family Acrl), a E U. A(r)+ a, a E%, Theorem

2.6

Let the mapping T satisfy a Lipschitz condition with the constant LT, that is, for Let the functional Tu”) < LTpE(u’, u”). any u’, uN E E we have PF ( Tuf, co = const > 0. Then I(U) be uniformly continuous on the set U (Ao+~ I-JAi’“), the A+-stability of the problem (E, A, I) “z%? respect to the family implies its stability with respect to the family ,A& a E M.

A,$,,

a E 3,

81

Approximation of extremal problems.11

Proof.

Let {CL} be an arbitrary sequence of pairs of non-negative real numbers as n-++oo,i=O,L Weshowthat I>% 4-p an= ( a,, o a ‘), ani++ as n--+-too. For every natural n we consider an element u, E At;> fo~‘~~~h I (U,) - IL +a < l/n. “(1; Because of the relations (2.2) and (2.3), for every n an element u,, E A,,-, be found for which @(u,,, &) < Cauno. Then pE(Tun, Tu,) < L#Zaann, n = I, 2, . . . , and consequently,

(2.4)

can

in other words +*n

V,Ehi

*

En=-

CrNan‘ + L&au, Cr’

’

n = 1,2,. I, *

Weput “a,= (0,8,), n-1,2,... . Obviously, k-+6 Because of the uniform continuity of the functional I(U) fl(U,)

Z(G)

-

1 +O

as

n*

as n+

j-m.

+ O”-

Taking into account (2.4), we have

od-z’+a

]+fz*- -I@,)]+ AtET i’(u,,)] + [f(u,,--I* 1< UZ’ - cl 1 + $2 4% =[r’-I*+,

AtIIY

Amn + [I

(u,) -

+ fl(%f

e

-fob)l+

that is, IL+an -+ I* as (0

[~@TJ-

u-+0 f*a Ad

as

n-++-7

12-+ + CQ. Theorem 2.6 is proved.

We give For every a = (a”, al) E 91 we consider the set A& = Az%nAI. with respect to the conditions on which the A+-stability of the problem (E, A, I) family A& UE ‘3, implies its A.*atability with respect to the family A$ u E 9f. Theorem 2.7 Let there exist an i = 0, 1, for any

EO

>

0, such that for any a = n (a*, a’), 0 ,C a’ ,(

ear

there exists an element v G Ai, for which b (u, v) < r(a), where r(a) -+ + 0 as a -+ 6. Let the functional I(U) be uniformly continuous on the set O<&$Ao+e II At+“), and the family AO+=,E 2 0, generate a family of strong extensions of the set do, that is, the inclusions U E

Ai”l”;

Oc,* (ho) c A:’ c O,al

(A,),

are valid for some positive constants C,‘, C,“.

820

82

3. M. B&k and E. M. ~er~~iek

Then the A+-stabihty

of the problem

a E %, implies its A+-stability Proof.

Let a, =

(E,

A, I)

with respect to the family

Ati<, a ~“2.

with respect to the family

(‘a,,‘, a,‘),

al-+

f

0,

+co,.i

as n-f

A&,

0, 1, be an arbitrary

=

sequence of the set Q[. We show that (2.5)

I;+a,+I* (1)

as

n-++

bo.

For every natural n we consider an element (I (r&J -

(2.6) Without

loss of generality

By hypothesis, n =

1’~~rjJ-f+O +a

$2,.

as

for which

n++ffi* a’ <

we may consider that

for every n there exists an element

80, i =

u, E A,,

0, 1, n =

Q(v,,

u,)

c

1,2, . . . . r(a,),

. . . Since U, E Aian

C OC,*A~~O (Ad,

we have U, E OC~A~~~.+~(~~)(A,) C Aicn, n =

where e, = (1 /CA’) [Ca”ano + r(a,)]) n = 1,2,, . . . Then (2.7)

un E A$,

n=1,2,...,

u,, E A;%,, Jl(U,)

-

1,2, . . . .

a,-+6

I(ZJn) 1 -+ 0 as

n--f

as

We put

ik =

(en, 0)

n-4+00;

+ 00.

From (2.6) and (2.7) it follows that O<

lim n-c-+

+iG&*

[Y--I*,,

]< *(rY

^

lim[IL-I*+a ?I-++03

-Wn)l+

?t-+CCAtan (IU

which implies (2.5). Theorem

]< *(I;

lim[/*--I* n-+-m

h 1-k A;Iz?,

lim W+-~(~,)l+ Tl-?+CU

2.7 is proved.

2. In this paragraph we consider some conditions

of the A--stability of the problem A-“, a E %[, of the set E

(E, L!L I) with respect to a specified family of subsets (see [I ] , section 1). Let the topology

r be given on the set E.

Definition 2.1. The family A’, a E 3, is said to be (t - A-) -correct, if for any sequence (a,,,) c 5u, converging to 8 as m + + 03, any element u E A is a r-limit point for some sequence (u,,}, for which 1~, E A-“=l, m = 1, 2,. . . .

Approximation of extremal problems. II

Theorem

83

2.8

Let the following

conditions

(1)

the functional

(2)

the family

be satisfied:

I(U) is r-semicontinuous

,A-“,

a E U,

is

is A--stable with respect to the family

upwards on the set A;

(r - A-) -correct. Then the problem A-“, a E I[.

(E,

A, Z)

Proof We assume the contrary. Then a number &o > 0 and elements u. E A, m= 1,2,..., can be found for which Z * _oLm- Z ( uo) > ‘CO,m a, E % U, E ,Aeam, m = 1, 2,. . . . By condition (2) there exists a sequence of eltments = 1,2...., for which the element u,, is a r-limit point. By condition (1) we obtain +&o/2

fromthis Z(uo) >Z(u,,) --o/2BZ;_a,-~E0/2BZ(u,) values of m. This contradiction proves Theorem 2.8. Note 7. It is obvious that the strengthening of Theorem 2.4, but strengthens condition (2).

of the topology

forsome

weakens condition

(1)

Let there be specified an arbitrary set F, a topology u on it, a mapping T : E + F of the set I’, P” c F, a E ‘3. We present sufficient conditions for the (T-‘(G) - A-) -correctness of the family of sets A-“= {U E E : TL:.8~ I’-“}, a E II, where thesetahastheform

A=

{uEE:TuEI’}.

Theorem 2.9 Let the following (1)

the family

(2)

TA-”

=

conditions

be satisfied:

I’-“, a E II

is (a -

r-”

for every

JY) -correct;

a E 3.

Then the given family ,A-“, a E U

is

(T-l ((J) - A) -correct.

Proof. Let a,-+6 as m-t+ w and u. E .A. By condition (1) there exists V, E r-s, m = 1,2,. . . , for which the element v. = Tu, a sequence (u,} c F, E I’ is a u-limit point. By condition (2) this implies the existence of a sequence of elements U, E A-am, m = 1, 2,. . . , point. Theorem 2.9 is proved. Notes 8. Obviously, a theorem A--stability of the problem (E, A, I).

for which the element

similar to Theorem

u. is a

2.1 is formulated

T-‘(o)

-limit

for the case of

9. In the case where E is a metrical space, the relation p(3, A-“) -0 as aimplies the (r - x-) -correctness of the family A-a, a E %, where r is the topology

fk

B. M. Budak and E. M Berkovich

84

generated

by the metric of the space E.

3.

Stability

We consider the optimal which, in particular,

of optimal control problems

control problem

the notation

in section 3 of [I]

formulated

used below is taken):

minimize

(from

the functional

tP I(u) = Z(w,zo,to,t,, . . .) 1q)= s g(z[t; ul, w(t), f,G!,q,; 111,.. . to . . . )“p,; ul, to,. . . ,t,)dt + go(G,2[b; ul.. . . ,5[L,:Ill, toI... ,4)

(3.1)

on the set

{u =

!I = E

(w, zo, to, t,, . . . , t,z) : 11'E

G,,(t,) c

R*", j=

L,“[T,I. T,], zll,; 111E

14 c

0, 1. . . . , q, xl/;

111E G(l)

c N”, I,, G 1 G

< l,,, T, < t, < 1, & . . . d I,, < 7’1). [TO, T, ] is a fixed bounded

Here q is a given natural number, I+‘, Gi(t), G(t), t E [To, ‘/‘,I the Cauchy problem corresponding (3.2)

.f[l; X[k

segment of the real axis,

are given sets, .l=s[l; /I; is the solution to the element 1~.= (I(‘: .I’,,, L,, tl, . . . , I,) :

u] =

f(.r[r;

rll3

II] =

x,,:

,i. p, g’,*

rl’(f),

l) for almost all

t E

[to,

of

&I,

are specified functions

Let

(3.3)

W’ = ! E

{r/9(/)

E

N(t)

c R’

for almost all

IT,,. T,]}.

where M(t) for every Euclidean

L,’ [To. T,] : w(t)

E

1E

[ 7’,,, T,]

is a specified closed convex set in an r-dimensional

space Rr, situated in some fixed (independent

of t) bounded

ball

K c

R’.

Let positive constants u*‘: j = 0, 1,. . . , y + 1 be specified. We put 9l= {a = (f”. II’, . . , CL”’ ‘) : C)< (A.’< u,‘, j = 0, 1, f . . , q -j ,I}, r is the coordinate origin in Rqf2. We suppose that the set A(: = {U = (w, x0, to, . . . , tql) : w E W+a O ~[t,:

:I] E G;l(/,),

j =

1,. . . , (I,

x[[t,,; I~]E G,(t,,),

z[t;

U] E G+” “l(t),

is defined for every B = (a’, . . ., te < 1 < t,. ‘I’,,< I,, < 1, < . . . < l,, < T,}, 11; I X*I’-- {IC E L,‘[T,, T,]: w(t) E ~f+~“(t) for where . . . . I!” ‘) E % almost all ! 65 I7’,,, ?‘,I}: NIfA”(t), 0 < u” < u”*,G ;,j (t), 0 < (Lj & a,j, j = I.. . . , (1, G+““+’ (I) ~ I) < (L”’’ < a,“! ‘, are specified families of extensions [l] of the corresponding sets. We give some conditions of the A+-stability of the original optimal control problem

Approximation

(6, A, I)

85

of‘ extremal problems.11

with respect to the family

IA;:, ?I E %.

T: (L,’ T,] x RN x [To, T,]q+l, associated with Tu = (z[t; u], each element 1~ = (w, zO, to, . . . , tq) a corresponding element is the trajectory of the Cauchy t .,tq), where s[t; u],t,
(4) and (5) of section 3, [l] be satisfied. Then the mapping

[T,,, T,] x R” x [To, T,]q+‘) +P[T,,

~~~~bl~~~‘(3.2) (continuously extended by a constant to the whole segment [To, T,]), satisfies a Lipschitz condition on the set A:$ , a, = (a,“, a.‘, . . . , usq+‘). Let conditions (10) - (12) of section 3, [l] be satisfied. Then, obviously, the functional (3.1) is uniformly continuous (because of the topology of the Hilbert space

Lzr[To, T,] x h? x [T,, T,]q+“)

on the set 4tI4’.

As in the case of Theorem 2.6 it is easy to show that the A+-stability of the optimal control problem considered with respect to the family A$, B E 8: is equivalent to its A+-stability with respect to the family of sets (3.4)

A+= = =

{U =

(W, X0, 20, . . . ) tq) : W E IV, x[t,;

1 I. **, q, s[to;

u] E G,,(b),

zlt;

U] E Gj+ai(tj)j=

u] E G+” ‘+’ (t), to < t < t,,

To
CZ= (n’, . . .,aq+‘)

ES,

% = {a = (a’,. . .,cP+‘):

0 <~‘
i=l,...,q+l}. In what follows we will in fact investigate the A+-stability of the problem considered with respect to the family (3.4) A+, a E %. Let the assumption section 3 of [l] be satisfied. As in [3, 5, 61 it is easy to prove the following

(7) of

result.

Lemma 3.1 Let the function f(x. w, t) have the form f(Z, W, t) = f,(~, t) + where the functions f, (x, t) and f2 (x, t) are defined for all z E: A”,

+ f2 (2, t) w,

and all t E [To, T,] , are measurable with respect to t on [To, T, ] for all z E R”, constants A,‘, i = 1, 2,. . . (5, exist such that for all 5, z’, z” E RN, t E [Z’,, T,], the inequalities 1f, (x. t) 1 < ‘.1,‘[3.j + AZ’, If+, t) 1 < Aa’, 1/&T’, t)t)-fT(xl”,t) I
bounded, weakly closed set in L,‘[T,, T,]. Then the weak conT,] of the sequence {w,} c W to the element w and the relations

Ini--+ ti, i = 0, 1, . . . , q,

imply that for any

t E [To,

T,]

we have

fi

B. M. Budak and E. M. Berkovich

86

a: u,t.]*x[t; u]

(w,,, xn, t,,,, 2,,, . . . , tnq), u and x[ ., U] are continuously

as n--f + 00, where U, = lp) and the trajectories z[ -, u-1 = (20, x0, to, . . . , extended by a constant to the segment [TO, T1 ] . As in Theorem

2.4, this implies that the family (3.4) is weakly A+-correct.

If, in addition to the assumptions of Lemma 3.1, the functional (3.1) is weakly semicontinuous downwards (in particular, is weakly continuous, see [3-51) on the set A, then as in Theorem 2.2, the original optimal control problem is A+-stable with respect to the family (3.4) [3] . The weak semicontinuity downwards on the set A of the functional (3.1) on the assumptions of Lemma 3.3, can be proved if g(z,

w, t, 50, 51, * * . , &I, to, 6, . . . , tcJ=

. . . , xq, to, . . . , tJ +gz(x,t,xo,...,x*)W and the functions gi(x, t, x0, q, . . ..a~~.to,...,

gi (2, t, 50, 21, . . .

tJ,

g2(x,

t,

XO,...,~~,

to,. . . ,tq),

to,..., tq) are semicontinuous downwards with respect to r, x0, . . . , for all t E [To, T ] i , and gr and gz are integrable with respect to t xq, to, . . . , t, t, and are underbounded by some on [TO, TI I for ali 2, x0, . . . ,2,, to, . . . , integrable function of t. 90(x0,

xi,...,xq,

We denote by u the strong topology of uniform convergence) in the space C” [To, &I x RN X[To, T,] q+l. As in the case of Theorem 2.4 it is shown that the family (3.4) is (T-‘(o) - A+) -correct. In order that the functional (3.1) be T-’ (0) -semicontinuous downwards on the set A, it is obviously sufficient that g(s, w, 1, x0, xi, . . * * t xc?,to, ti, . . . ,tqI) 3 gt(x, t, 50, . . . , 2,, to, . . . , tq), and the functions g, and go satisfy the conditions formulated above. Thereby conditions (1) and (2) of Theorem 2.2 are satisfied. The greatest difficulty Theorem

{Tu,}

is caused by condition

(3) of this theorem.

By

2.5 it is sufficient to show that for any sequence (u,} c A+a the sequence is relatively u-compact and lies in some u-closed subset of the set TE. The

relative u-compactness of the sequence {Tu,} in the case where the set W has the form (3.3) is ensured by condition (4) of section 3 in [l] . The condition of u-compactness of the subset, containing TA-tn, of the set TE will be satisfied, if closedness in the topology corresponding to all possof uniform convergence of the set of trajectories Z[ ., u], ible controls

w E w

is satisfied. Similar questions

[7-91.

We give a sufficient

Lemma

3.2

condition

are investigated,

for example,

in

of such a closedness.

Let the set W have the form (3.3) condition (4) of section 3 of [l] be satisfied and the function f(x, IV, e) be continuous with respect to t for all x, w. Also, let the following condition be satisfied: f (z, M( 2) , t) is a convex closed set in RN for all x, t, semicontinuous upwards with respect to inclusions (of x and t).

Approximation

Let {w,}

CI

87

W, {x,,~} c RN, {t,,,} c [To, T,] and

as n-t+

=x”(t)

of extremal problems. II

00, where

L,(t)

=f(~,,(t),

x,~*~o,

Til, z,,(t) = x,0, T, 4 1 < t,,, n = 1, 2,. . . . exists for which

io(t)

=

f(z,(t),

w,,(2), 1)

[3,

r,,(t)

t E [tno,

w. (t) E W,

[to, I”,].

of the proof of [lo]

and is not

here.

From this and Theorems A+-stability

and

Then a function

for almost all Z E

The proof of this lemma is a slight modification reproduced

L~*to

t) for almost all

m,,(t),

of an optimal

2.2, 2.5 we obtain

control problem

another

sufficient

condition

with respect to the family

for the

A+=, a E % (see

1II). We mention

that another

classes of optimal

control

which is the establishment

approach

to the establishment

problems. connected of condition

of the A--stability

with Theorem

3) of this theorem,

of various

2.3, the main difficulty

is explained

of

in [3, 11, 121.

Tkanshted by J. Berry. REFERENCES I. Zh. v?chisl.

1.

BUDAK, B. M. and BERKOVICH, Mat. mat. Fiz. 11, 3, 580-596,

3-,

BUDAK, B. M., BERKOVICH, E. M. and SOLOV’EVA, E. N. Difference approximations optimal control problems, in: Computing Methods and Programming (Vychisl. metody programmirovanie). 12, 115-134, Izd-vo MGU, Moscow, 1969.

3.

BUDAK, B. M., BERKOVICH, ;;r;ximations for optimal

4.

LEVITIN, E. S. Correctness of constraints Ser. Mat., mekh. 2, 8-22, 1968.

5.

LEE, E. B. and MARKUS, L. Optimal control for nonlinear processes, in: Kibernetich. sb. 2, New Series, 86-117, “Mir”, Moscow, 1966. (Original in: Arch. Rational Mech. 8, 36-58, 1961).

6.

PODINOVSKII, V. V. The question of the existence of solutions of optimization fvted time. Izv. Akad. Nauk SSSR. Ser. tekhn Kibernetiki 3, 41-45, 1967.

7.

GUINN, T. Solutions 1968.

8.

CESARI, L. Existence 517-552, 1968.

9.

LASOTA, A. and OLECH, C. On the closedness of the set of trajectories of a control Bull. Acad, Polon Sci. Ser. Sci. Math., Astron., Phys. 14, 11, 615-621, 1966.

10.

FILIPPOV, 1959.

E. M. Approximation 1971.

A. F. Some questions

problems.

in i

E. M. and SOLOV’EVA, E. N. Convergence of difference control problems. Zh. vj%hisl. Mat. mat. Fiz. 9, 3, 522-547,

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theorems

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problems.

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II. V&n.

MGU.

problems

with

J. Comput. System Sci. 1, 227-234,

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type.

control.vestn.

SIAM J. Control

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BUDAK, B. M. and BERKOVICH, E. M. Difference approximations problems with moving ends in the presence of phase constraints. mekh. 1, 39-47,197O.

for optimal control II. Vesrn. MGU. Ser. mat.,

12.

BUDAK, B. M., BERKOVICH, E. M. and SOLOV’EVA. The convergence of difference approximations for optimal control problems, in: Computing Methods and Programming (Vfchisl. metody i programmirovanie) 12, 135-142, Izd-vo MGU, Moscow, 1969.