Approximation conditions for max-min problems with connected sets

U.S.S.R. Comput. MathsMath. Phys. Vol. 18, pp. 75-85 o Pergamon Press Ltd. 1979. Printed in Great Britain.

0041-5553/78/0601-0075$07.50/O

APPROXIMATION CONDITIONS FOR MAX-MIN PROBLEMS WITH CONNECTED SETS* E. R. AVAKOV Moscow (Received 14 March 1977; revised 1 June 1977)

THE PROBLEM of finding a max-min with respect to connected sets is considered with fairly general assumptions. The problem is approximated by a sequence of simpler problems. In the light of a definition of approximation with respect to a functional, necessary and sufficient conditions for convergence of the approximation are obtained. A min-max optimal control problem, illustrating the general scheme, is examined. 1. Formulation of the problem Let U and V be arbitrary sets, and VI a given non-empty subset of U. Further, given any u E U1, let a non-empty subset M(u) of the set V exist. We consider the following problem: to find

sup inf 3 (u, v)=Y’. UGU,lEM(U)

where

Y(u,

r4

(1.1)

is a functional, defined and bounded in U X K

Such max-min problems with connected sets are most commonly encountered in the study of games with unopposed interests (see [l] and [2] No. 1). As a rule, the original statement of such a problem is fairly complicated, and it proved impossible to find Y’ exactly from (1.1). Hence, to find approximately a max-min with connected sets, the initial problem is approximated by a sequence of simpler problems. As a result, we arrive at the following family of problems. Given, for every integer m > 0, the sets Urn, V, and the subset VI, ~1, of the set Urn, dependent asaparameterel >O.Also,forany, [u]mEUm we introduce the subsets M,* ( [ u],,J of the set V, , dependent on a parameter e > 0. We wish to find

where 1, ([ulm,

[vImI

is a functional, defined in Urn X Vm.

Not so much attention has been paid so far to the problems arising in connection with. the approximation of problem (1-l) by a sequence of problems (1.2). In [2] , No. 2, sufficient conditions for the approximation of a class of max-min problems with connected sets are obtained. In [3] a difference approximation for differential games with a saddle-point is examined. In the *Zh. vjkhisl. Mat. mat. fiz., 18, 3,603-613,

1978.

75

76

E. R. Avakov

present paper a criterion for approximation with respect td a functional of problem (1 .I) by a sequence of problems (12) is derived. Definition A. We assume that the family of problems (1.2) approximates as m*m, ei* +O, e++O the problem (1 .l) with respect to a functional if: (1) there exists T > 0 such that, for any e, O 0 such that, for any Ed, O mg ; (2) we have the equation bounded in UimC1

lim lim lim Zn'(e,,e)=3*. s*+o r,*+o m-tw

(1.3)

We shall use a method which is a generalization and modification of the method of [4-- 61; these latter papers dealt with the minimization of a functional. 2. Approximation criterion Theorem 1 e>O, e,>O, m=1,2,..., to approximate In order for the family of problems (l-2), the problem (1 .l!in the sense of a functional, it is necessary and sufficient thtt, given any can be found such that, for any ei, OO,an e,=e,(e) SO there exist nonempty subsets nonempty subsets Ul”l of the set U, and for any UEU,Efi ME(u) of the set I/, such that the following conditions are satisfied: there (1) for any 6 > 0, we can find e=e(6)>0 such that, for every e, 0 m 1, there exists a mapping Q,(U) of ml=mi (e, ei) n&U;“2 there exists a mapping P,,, ( [ u],,J of V, into V, Uinto U,, and for any such that

(2.1)

pl,lM,"(Q,,(u",))=~2~(,,,~),

lim[~(n,,~,([~i,))-~,(Q,(~,), blm)la m-rm

for all

(2.2)

(2.3)

IuI~~~~(Q~(~~));

(2) given any 6 > 0, we can find -e--e(b)>0 such that, for any e, O m2, there exists_a mapping F,,, ([ u] ,) m2=m2(e, ei) there exists a mapping Q,(V) from V into V, such that from U, into U, and for [Fil,,,~U,,“l

Approximation conditions for max-min problems with connected sets

77

(2.4)

(2.5)

(2.6)

for all v, EJP” (P,, ( [ u ] ,,z) ) ; (3) we have the equation

inf 3 (u, 71). lim lim $e*,,e = LY*,whereYe,,e = sup -+o El++0 uErrlel ore tu) For the proof of Theorem 1, we require: Lemma 1 Assume that, for certain fixed e > 0, el > 0, there exists m,--mo(e, al) such that, for are nonempty, and moreover, for any UEU~V, UEU~~~, any m > mg, the sets UQ/’ U2Q P respectively are also non-empty; the sets k’8(f): $z(~), ~V,~([rz],) [UlrnEUlm8~, we then have the following four assertions: (a) there exist

zz,,,*~U*~‘~, m-m,,

m,+l,

...,

such that

-3 (urn*,vm)I GO lim[3s:,a.a.

(2.7)

m-o

for all v,&P*

(urn*) ;

(b) there exist

[rz],,,=YlmC1,m=mo, no-i-l,. . . ,

E[Zm*(ei, e>-L([ulm*,

such that

[vll;l) IGO

m-eon

forall

(2.8)

[vlmf=lf~e([ul,*); (c) for any

&,EU~$

there exists

v,, l4V

(a,,,)

lim 13 @m, urn’)-3ac,,r,a

such that

I GO;

(2.9)

In-COD

(d) for any [Z],EU,,@:

there exists

EL([Film,

nl-+oD

[v ],,,Ui&,‘( [ a],)

such that

[vlm’)-Zm’(ei, e) IGO.

The lemma is easily proved by recalling successively the definitions of lowest upper and greatest lower bound.

(2.10)

78

E. R. Avakov

Proof of Theorem 1. Necessity. Assume that (1.3) holds. Let us show that there then exist and for any UEUr=’ there exist nonempty sets Me(u) C I’ non-empty sets U,‘I cU such that all the conditions of the theorem are satisfied. In fact, we simply put U,+=U, for all el > 0 and for any u E UI we put @(u)=M(u) for all E > 0. Then, Y~,,*C=Y* and condition (3) of the theorem holds for all ar0, ai>0 Let us prove the necessity of condition (1). We fix some S > 0. From Eq. (1.3), on successively discovering the limit as E -++O and e,++O, we find that: there exists there exists isl, OCB~
I lim I,’ (et, e) -3’IGS

(2.11)

mlcDI

for all Om, (a, ai). for all E > 0 and el > 0, by virtue of their construction and because, by hypothesis, the sets iJ1 and M(u), where u E VI, are non-empty. We can therefore employ Lemma 1. For any such that (28) holds for all [ u],EM,,,” ([u],,,*). We put m>mg,we take [u],*EUt,‘l Qm(U) = [ulm’ for any u E U. It is a mapping from U into U, , constructed for all m~m,=mo. We fix an arbitrary fi,~U :I2 =U,. Then, Q,,,(&,) = [ zz],,,*~U’,,,,~~ and hence (2.1) holds. there exists u,*EJ!?~‘~ (Z,,,) =M (f&J such that (2.9) is satisfied. For this u”,dIJ,=U:E), =Y* for all O-Q%, OQkGh. We put Pm( [vim) =Umf Here, obviously, Y&Z It is a mapping from V, into V”,constructed for all mbm,=m,. for any [u],EV=. Hence,for any ~u]~~~~*(~~(~~))~ JL”{fu],‘) we have ~~([~]~)~~~*~~~(~~) and consequently, ~~~~~(~~(~~) ) =u,‘~n/r(fL) ==M2”(f&j ; hence (2.2) is proved. We shall now prove the necessity of (2.3). Using (2.8), (2.9), and (2.1 l), we obtain

+liIu(9*-Z,‘(el,e))+~[Z,*(e~,e)--lm(~u]m*t ?n+co

[ulm)]<6

m*m

Let us show that condition (2) is necessary. For this, we again return to Lemma 1. Given anym>mo,wetake h*=U?lz =Ui such that (2.7) holds for all ~,~kf~~(rf,,,‘). for all O-CAT, O-Ce&&. We put Pm( [ulm)=&* Here, obviously, 3 &SC fY for any [u] m E U,. It is a mapping from U, into U,constructed for all m>rztz=mo. We fuc an arbitrary ff.Z]=n~U~,,,~‘. Then, Pm( [El,) =EC,‘EU,SU~ and hence (2.4) is satisfied. such that (2.10) is satisfied. We put For this [.z]~EU~,,,~~ there exists fu],*~M,“([~],) @,, (0) = [u] m* for any YE I’_It is a mapping from V into V, , constructed for all m > m2 = mo. Hence,forany u~~l;~~~(P,([~]~))~=Me~~(u~*) wehave @,,(u,,,)=[u],*EIM’([~~]~) and consequently, Q,JIP2 (P, ([ ~1~) ) = [ u],,,*EMC ([u”] ,,,). Hence (2.5) is proved. Let us prove the necessity of (2.6). Using (2.7), (2.8), and (2.1 l), we obtain

Approximation conditions for max-min problems with connected sets

79

lim[z,([nl,,Q,(V,))-YtPm(r~Im)r~m)l m-tm

=‘iz[Z,([u”]m,ru]m*)-Y(U,*,Vm>l m-m

&c[z,([~],,

[ul,‘)-fm*(Et,e)l+~[I*‘(e,,E)-Y’l

?n*m

+El

[Y.-Y

(urn*,u) ]a

m-cm

for all u~EJ!P (u,‘) s&P” (u,,,*) =&P2 (I’, ( [u”] m) ) . The necessity is proved. ~~f~c~~ncy. For any E > 0, let an ;I > 0 exist such that non~rnp~ sets Ulel exist, for ah and for any u E Ul El, there exist non-empty sets me such that conditions Et, O 0. It follows from (2.1) that the sets Ulrnel are non-empty for all e,, O mo,the ==max(m,, mz). sets in Lemma 1 are nonempty. We will use this lemma. We take u,*EU~‘~ such that (2.7) for [Zlm=Qm (~~*)~U*~~l we take [~]~*~~~~~[~I~) holds for all v,&P” (urn’) ; ~~*$(~~(~*))such that(2.lO)holds.Fu~her,weput u,= P,([u],*); since [Y]~‘EM,* and we finally find from (2.2) that u,4P (a,*). (Q,,,(G,*)), then G@JL~(Q,(~*)) and inequalities (2.7) and (2.8), we obtain Now recalling (2.3) for u”,= CA”,*, [ ujm= [ u],,*

s iimfY:,,2& -Y{U,‘,U,)l+limrY~u,‘,~~(r~t~*)) m+m

w&-Pm

Further, we take [u],~“EU~~~~ such that (2.8) holds for all [u],,,~ M,,,“([u],*), such that (2.9) is and for G,=Pm ([ ~1,‘) ” we take u,*EMC’*(P, [u],*) -JP2(~,,,) satisfied. Next, we put [u] ,,,=Qm (urn’) ; since u,,,*EM~‘~(P, ( [u] m’>) , then [ u],,,EQ,M”* and we finally find from (2.5) that [v],EM,“( Lu]~*). Now, taking (~mruJm*)) account of (2.6) for [qm=[z&*, v,=v,* and inequalities (2.8) and (2.9), we get

iz(I,*(E,, In-c9

E)-Y;~,,,,z)
--~?n([~lm*, ~v3~>l+lim~z,~~ul~‘,Bm~zl~‘~~ m-m -Y(P,([u],‘),v,‘)j+lim[Y~~,,u,“~-Y2~,,,,~1~6. m-+m

80

E. R. Avakov

We have thus found that, for any 6 > 0, there exists i? > 0 such that, for any a, O
On passing in turn to the lower limit in the first inequality, and to the upper limit in the second, as el+i-O, e-t+0 and recalling that 6 > 0 is arbitrary, we get

(2.12)

By condition (3), the repeated limit

exists and is equal to 9’. Then, lim lim Y,,+i& iiii Y~,,~==Y* -a-w+0 *c-c+0 l++ocrpto and we conclude from (2.12) that

exists and is equal to P.

The theorem is proved.

3. Difference approximation of a max-min problem of optimal control Given the functional Y(u, v)=jz(T,

% +---IIta,

C3.1)

where x(., U, v) = x is the solution of the Cauchy problem qo)

U=U(t)~U={U(t)el;,(r’~to,

~=~~t)~~u{~(t~~~~~

=zl,

Tl, u(t)=P almost everywhere in [to, 2’1),

[to, 2’1, v(t>EQ ahnost everywhere inEto, Tl),

(3.2)

(3.3)

(3.4)

where t is the time, z= (~9, . . . , sn) are the phase coordinates, u- (u’, . . . , u’), v - ( v’, . . . ( up) are the control parameters, P and Q are known, closed, bounded, and convex sets of Euclidean spaces E, and E4 respectively, A(t), B(t), c(f), and f(t) are piecewise continuous nXn, nXr, nXq, nX1 matrices, respectively, specified in to < t Q T, and the instants tu, T are also assumed to be given, where G is a known closed set of the and the points x&r@, y&‘, phase space E,.

~(a)

We next introduce the set M(U) = {22: ZYEV,s(t, U, u)EG, t,
We assume that We have to find

(3.5)

sup inf S(u, v) =Y’ UEUlSM(U) under conditions @.I)-(3.4). To approximate problem f3.1)--(3.S), we take a sequence of difference meshes (t;jm= ---do and a constant Cu > 0 exists, such that (to==&,m< . , . -++,,, m=Z’) such that lim iI&-m-too d,,,= max

(~w,~-L)

OCf*;Nm-i

We fix parameters ~30, to find

st>O

co

4 - , N,

(3.6)

and consider the following probfem in the m-th mesh:

under the constraints

[u],,,d,,,={[u],=(uo,.

. . , u~+,,_J:u~~P~ i=O, 1,. . . , Nm-l),

(3.10)

obviously, if the set Ulmef is non~mpty, then the sets &&‘I ( iu]J, and hence also the sets ~~~~~u~~) are likewise non~mpty for all [ufm dJimex and ES&, The proof that the set Ulmel is non-empty for any EI > 0 is given below. We wish to see if the sequence of problems (3.7)-(3.1 l), E=% 820, m='i,2,..., approximates the problem (3.1)-(3.5) in the sense of a functional in the sense of definition A. Below, we shall also require the sets Mefu), where u E U, and U,“*, O
82

E. R. Avakov

l”heorem 2 Under the conditions listed above on the initial data of problem (3:1)-(3.9, lim lim Y,:,, =Y’, e-e+0et++0

zde

Y,:,, = sup inf Y (u, v). ud:” Vdd’(U)

F’ruojI Since, for fured E > 0, the quantity Y:,, e ai, a>eiZO, decreases, and is lower-bounded by 3,*=

we have

decreases monoton~c~ly as

sup inf Y(u,v), asUl Odf’(U)

then there exists the limit

lim Ye~e=3t5,e23s+.

(3.12)

*I-++0

is monotonically increasing as E > 0 decreases and is upper-bounded Similarly, the quantity 3,’ by 3’9 so that there exists the limit limY,*=Y+o~3*. E--L+0

(3.13)

Let us show that lim lim 3$B3*. t-tf0 as-+!-5 Given any 6 > 0, we take ua’eU,

such that

(3.14) Next, given any E > 0, we can find a control

u~*EM’ (u<)

inf 3 (a&*,v) 23 &fU(V’af

such that

(a,*, 27;) --E.

(3.15)

Since Yis weakly compact in L’c,P)[1,, T], we can extract from the sequence (u,*(t) } a subsequence {~~~*(t) ), weakly convergent in Li’,a)[to, T] to control U*=U*(t) EV as ek++O. Then, SUP IQ, u6*, v,;)--z(t, ud*, u’) i=r,+O as k-+w. bCfCT

EG,~, then J: (t, u6*? u’) =Gck+Yk. Hence, as k + 00we obtain x (t, zz6’, u’) EG for all t E [to, T] , and hence V’@kf (Ub*). Next, in view of (3.15), we obtain the chain of inequalities Since

2 (t,

u<,

uclr *)

Hence, recalling (3.14), we obtain Em 3,‘= lim Y,;> lim[3(rz,*, u,;) --en] R+m C*+O k-+m =3 (ud*, u*) Z= inf Y (r&9*,u) >3’--6. v&M(u*b )

Approximation conditions for max-minproblems with connected sets Next, recalling

83

relations (3.12) and (3.13), and the fact that 6 > 0 is arbitrary, we finally obtain

lim lim Ye:,,> lim YYE*== lim T,*=P*. e-t+0 I-t+0 cr-*to e-t+0

(3.16)

Let us show that, on the other hand, lim Em P,;, 0 and consider the family of controls

S,:,,<

{u.,‘(t)}

inf 3 (%,*, 0) fe,. O&fs(U;,)

such that

(3.17)

It belongs to the set U, which is weakly compact in L’t [to, T]. Hence we can extract from {ut,* (t)} a subsequence {u~,x*(t)}, weakly convergent in ~5’;’[to, T] to a controi It is easily shown that U* f U1. But then, a control u~‘EM((u’) exists, u’=u’(t)EU. such that Y (u*, ua’)G inf Y (U’, U) +6 UEM(U’)

(3.18)

sup 1z (4 w*, h’) - 2 (6 s(t, u*, ua*)EG. Then, ~(4 %,x8, v,*j =GTfit where %3 0 exists, such that Sk < e for any k > IV, and vo*) f=yk-tO as k--a. Eebce u6*EWk (Z&k*)GM” (&,K*), Hence, in the light of (3.17), we obtain the chain of inequalities and l

for all k>N.

But then, using the inequality (3.18), we obtain, ask 400, lim P,,,Z=lim Y,,kf,Glim 3 (u,,< u,*) Slim eik er-+a R-rm k-t-7 k-tm f inf P(u*, u) f6< sup inf Y (U, V) 4&=Y’+6. v5%f(u*) UEU,UfM(U)

Hence lim Y&
for all

&>O.

Hence, passing to the upper limit as E + + 0 and recalling that 6 > 0 is arbitrary, we finally obtain lim lim Y,z,
(3.

9)

E. R. Avakov

84 Theorem

3

Let the sequence of difference meshes satisfy condition (3.6). Then, lim lim lim ZNm(E,, E) =Y*, e-++rJ e,-t+rJm-PC0 there exist For the proof, it is sufficient to show that, given any E and E,, O 0; but in that case, since 2e > ?4el, the sets @e(u) will also be non-empty for any UEU T1’p. For an integer m 2 1, we construct a mapping Q,(u) of Pinto U, as follows. We take an arbitrary mesh {tt},,,={tO=tO, ,n< . . .
Q~(u)=[u],=(u~,

J u (.t, dr, ’ tr

i=O, 1, . . ., N,,,-1. For every integer m > 1, we construct a mapping P, ( [v] ,,) Obviously, [u],=U,. from V, into V as follows.. In the same {ti}, , for any [u] ,,,= ( uo, . . . , uNm_.,), we put P, ([u]~) =v(t) =u,,,(t) =u; for tiGtCt<+i, i=O, 1, ...,N,-1. Obviously, the control and show that Q,,, ( Em) EU$, . For this, we vm(t) E V. We fix an arbitrary E,,,sU p’2 take an arbitrary u,~Mt~‘~ (u”,,,) and consider

I~Itn-Qrn (04= (~0, .m ., UN,,--I), vi=

-

1

where

‘i+i

Ati J

u(T)dz,

i-0 , I ,.*a, N,-1.

11

La

[~klm=bo,

[ulm=Qm(am)

. . . , SN,> be the solution of system (3.9), corresponding to the controls and

[u]~=Q~(u~).

Using the method of [6] ,pp. 75-77, we have

max Is(t,, ii,, urn)-xkI=ym+O W&N,

as m-00.

But in that case, an M > 0 can be found such that rm < %el for any m > M, and hence But [xk] m is a discrete trajectory, for all m>M, k-0,1, . . . , N,. x&L,~+r,,,=Gt, ;rrTponing to the controls Qm(k,,) EU,, and Qm(u,) EV,, and hence Q,” (u”,) E Qm(um)~Mm*~(Q,(Eim)) for allm>M. Hence (2.1) is proved. Next, recalling is non-empty for all m > M. We fix an thze1 GE, we conclude that the set M,‘(Qm (n,)) arbitrary control [u],,EM,~(Q, (cm)) and construct a control u,=u, (t) =P, ( [u,] ) E V. Let us prove relations (2.2) and (2.3). Let x(t) =x( t, fi,, P, ([VI ,,J> be the solution of system (3.2) corresponding to the controls I”~,EU:J~ and P, ([u],)EV, [rklm=(50, ..., xNm) is the solution of system (3.9), corresponding to the controls Q,,, (E,,,) EU,~~~ and [u],E M,8(Q,(&J). Since

max OCICN,

lx(tk,G,, P,([u],)-x,l=y,+O

as

m+m

(3.20)

~pproxi~t~on conditions for rn~~-minpro#~emswith connected sets

uniformly

with respect to the choice of

E,,,f L’~”

[17 I “,~:f!f,,~’ (Q,,:(E,,,>>,

and

lim ts(~~,~~(E~l~>)-~,(Qln(~m), m-*a and [u]~GN~“(Q,(~~)) for all r^i,,f: ueJ2 (2.3) is proved. It follows from (3.20) that x(t,

and for any fured

IV] fpt iS ally

In the problem under consideration,

condition

to condition

(1) of Theorem

from condition

hence, for the proof of condition In conclusion,

Ed, O-=C+be. forallm>ml

Hence (e),and

control, belOnging t0 the set i?+f,~’(Qni (g,)

(1) of Theorem

(2) of Theorem I is in a sense symmetric

1 that the set VI, el is non-empty (2), we can certainly

),

1 is proved.

1 and a separate proof is unnecessary.

(1) of Theorem

then

W&=0

u”,, P,([t’Jm))4L,

E Ilif’” (n,n). Since hence P, (fvl,) then (2.2) is proved, and hence condition

8.5

Notice only that it follows for any ~1 > 0, m ZM, and

(u”] m6rlmB~.

choose

f sincerely thank my scientific director F. P. Vasil’ev for his assistance.

1.

GERMEIER, Yu. B., Games with unopposed interests (Igry s neprotivopolozhnymi interesami), Nauka, Moscow, 1976.

2.

EEDOROV, V. V., Methods of calculatinga max-min (Metody vychisleniya maksimina), Nos. 1, 2, Izii-vo MGU, Moscow, 1975.

3.

BUDAK, V. M., and IVANOV, A. I., On d~f~~enc~appro~~ations mat. I%..., 10, No. 3,630~643, fP70.

4.

BIJDAK, V. M., BERKOVICH, E. M., and SOLOV’EV, E. N., On the convergence of difference approx~ations for optimal control problems, Zh. $khisl. Mat. mat. Fiz., 9, No. 3,522~547,196P.

5.

BIJDAK, B. M., and BERKOVICH, E. M., On the approximation mat. Fiz., 11, No. 3,580~586,1971; No. 4,870-884.

6.

BUDAK, B. M., and VASIL’EV, F. P., Some computational aspects of optimal control prablems {Nekotorye ~chislitel’nye aspekty zadach optima~nogo upravleniya~, fzd-vo MGIJ, Moscow, 1975.

for differential games. Zh. ~~~~s~ &fat.

of extremal problems, 2%. vj%rhisZ.Mat.