JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 19, No. l, MAY 1976

Weak Lower Semicontinuity of Integral Functionals

C . O L E C H 1

Communicated by L. Cesari

Abstract. A lower semicontinuity theorem for integral functionals is proved under LI-strong convergence of the trajectories and Lrweak convergence of the control functions. An alternative statement is also proved under pointwise convergence of the trajectories.

Key Words. Lower semiconfinuity, integral functionals, convexity, measurable maps, measurable set-valued maps, strong convergence, weak convergence, pointwise convergence, epigraph, lower closure.

1. Introduction

The purpose of this paper is to give a proof of the following result announced in Ref. 1, Theorem 4.

Consider an integral functional

I(x, u) = f~ f(t, x(t), u(t)) dt, (1)

where G is a bounded region in R n, f maps G × R k × R l into R u { + co}, and x : G ~ R k u : G ~ R I are assumed to be integrable,

Theorem 1.1. Assume that f(t, x, u) is measurable with the respect to the o--field ~ x N, where ~ is the I ~ b e a g u e field in G and N the Boral field in R k x R l, lower semicontinuous (1.s.c.) in x, u for fixed t and convex in u for fixed t, x. Assume, further, that there are constants M, MI and an integrable qJ : G ~ R such that, for each integrable x : G ~ R k, there is measurable p: G-~ R ~, Ip(t)l<~M, such that

sup,(-f(t, x(t), u) +(u, p(t))) <<- ¢(t) + Mitx(t)l (2)

a Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland.

3

© 1976 Plenum Publishing Corporation. 227 West 17th Street, New York, N.Y. I00t l~ No part of this publication may be reproduced, stored in a retrieval systerr~, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.

4 JOTA: VOL. 19, NO. 1, MAY 1976

almost everywhere in G. Then, the functional I is lower semicontinuous with respect to L 1-strong convergence for x and L~-weak sequential convergence for u. []

If M~ = 0 in (2), then the strong L~-convergence of x in the statement can be replaced by pointwise convergence.

This result generalizes a few known sufficient conditions for that kind of lower semicontinuity of integral functionals. In particular, Berkovitz (Ref. 2) proved recently this result in the case that f is continuous in x, u and everywhere finite.

Also recently, Cesari (Refs. 3-5, particularly Ref. 4) considered the case when convergence in x is in measure, which is the same in this case as pointwise convergence, and his additional assumption is that f(t, x, u) is bounded if l ul <~ 1, which together with the convexity of f in u implies (2) with M1 = 0 and ~0 constant. When this paper was completed, the author learned of two other papers by Cesari and Suryanarayana (Refs. 6-7), appearing in this same issue. In Ref. 6, the boundedness condition in Ref. 4 has been considerably relaxed; and, in 7, a further generalization of the lower closure theorem is given (Ref. 7, Theorem 4.2). The assumption of this theorem, when specified to the case of Theorem 1.1, says that, if x, (t) -~ x (t) in measure and un ~ u weakly in L1, then f(t, xn(t), u , ( t ) ) - f ( t , x(t), u~(t)) tends to zero in measure and f(t, x,(t), u,(t)) is bounded from below by a weakly convergent sequence of integrable functions. If f is assumed to be continuous in x, then Theorem 1.1 is contained in the result mentioned in Ref. 7.

Poljak (Ref. 8) gives a result analogous to Theorem 1.1 assuming, besides (2), a kind of H61der-type estimate for the differences f(t, x, u ) - f(t, xo(t), u). In particular, his assumption in the case that

f(t, x, u) = a(t, x)u + b(t, x)

can be satisfied only if a does not depend on x, while our theorem applies also to the case when a dpends on x but is bounded.

Morozov and Plotnikov (Ref. 9) obtained a theorem analogous to Theorem 1.1 in the case where f is continuous in all variables. They did not need assumption (2), because they worked in the space of bounded functions for the x-components.

The theorem applies to the functional

t(z) = I~ f(t, z(t), Vz(t)) dt

when 1.s.c. is considered with respect to weak convergence of z in H~(G). Indeed, weak convergence of z in H~(G) means Ll-strong convergence of z

JOTA: VOL. 19, NO. 1, MAY 1976 5

and Ll-weak convergence of the first derivatives of z, hence of the gradient Vz (see, for example, Morrey's book, Ref. 10).

The proof of Theorem 1.1 given in Section 4 is rather straightforward and elementary. It is based on a simple lemma from convex analysis (proved in Section 2), which may be of some interest by itself, and on a representa- tion of a convex, closed, set-valued measurable map by the intersection of denumerably many closed half-space-valued maps, which is presented in Section 3 together with some basic properties of measurable maps. Finally, in the last section, we discuss briefly lower closure theorems.

2. A Lemma from Convex Analysis

Let g be a function from R ~ ~ R w { + oo}. The function

g*(p) = sup ( - g(u) +(u, p)) u E R n

is called the conjugate function of g. Again, g* may assume +oo values, and it is known to be 1.s.c. and convex. It is well known also that

(g*)* = cl co g,

where the latter stands for the closed and convex envelope of g, that is, the largest convex and l.s.c, function h(u) such that

h(u) <. g(u).

By gN, we denote the function defined by the formula gN(u) = g(u) for lu[ ~< N and +oo otherwise. One can prove that gN,(p) satisfies a Lipschitz condition with constant N. In fact, for g convex, gN, is the largest function h which is convex, satisfies a Lipschitz condition with constant N, and bounds g* from below. For the theory of conjugate functions in the above sense, we refer to Ref. 11. For the lemma which follows, we need to allow also - 0o as values of g. However, note that, if g(u) = - 0 o for some u, then g*(p) - +oo. On the other hand, if g(u) =-- + oo, then g*(p) -- - oo. If g is allowed to assume - e e values, then the relation

may not hold.

(g*)* = cl co g

Lemma 2.1. tions. Then,

(lim inf gi(v))*(p) = lim lim sup g~*(p). i -~oo N ~ o o i - ~ v -~u

Let gi : R" -~ R w { + oo}, i = 1, 2 . . . . . be arbitrary func-

(3)

6 JOTA: VOL. 19, NO. 1, M A Y 1976

P r o o f . By definition, the left-hand side of (3) is equal to

sup (-~lim inf g,(v)+ (u, p)) = sup qim sup(-g~(v)+(u, p))), u

We note here that

and that

(4)

sup lim sup hi(v) = lira sup sup h~(u). (6) t,N<N i-,~o i-,~ tut~n

v-~u, lvl~<t¢

To prove (6), we note that both sides can be written as

sup (lira sup h~(vl)), (7) {v,} i~oo

where, for the right-hand side, the sup is taken for all sequence {v~} bounded by N, while, in the case of the left-hand side, only convergent sequence {v~} are taken into consideration. However, one sees that, also for the case of the right-hand side, the supremum in (7) is attained by a convergent sequence, which proves (6). Relation (5) is obvious. It is clear that (4), (5), and (6) imply (3).

and

Clearly,

Remark 2.1. If one assumes that there is a bounded sequence {u~} such that {gi(ui)} is also bounded, then one can prove a stronger conclusion, namely that

(lim inf g~(v))*(p) = lim lira sup ( inf g*(q)). (8) i ~ e o e > O i~o~ lq--pt<e l ) - -~u

However, for our purpose, (3) is satisfactory. Relation (8) has a nice geometric counterpart. Namely, put

Pi = epi gi = {(u, x)[x >I gdu)}

P~* - {(u, y)[ Y ~ g~(u)}.

P* = {(P, Y)[ - x + (u, p) ~< y for each (u, x) E Pi}.

Now, (8) is equivalent to

(lim sup Pi)* = lira inf P*, i---}oO i ~ o o

(9)

sup h(u)= lira sup h(u) (5) u N ~ JuI~N

JOTA: VOL. 19, NO. 1, MAY 1976 7

provided there is a bounded sequence {z~} such that z~ ~ P~. Above,

lim sup Pi = {z I lira inf d(z, P~) = 0} i--~oo i~oO

and

lim inf Pi = {z ]lim d(z, P~) = 0}.

Note that, in (3) as well as in (8), the right-hand side may be equal identically to + oo and that both sides of (9) may be empty. This is the case (for example) if, for some u,

lira inf gi(v) = -oo, i-*oO v--~U

or if cl co(lim sup P~), the closed convex hull, contains a line

{(u, x) lu = const}.

3. Measurable Set-Valued Functions

The regularity assumption in the theorem concerning [ implies that, for each Lebesgue measurable x : G-->R k and u : G - ~ R t, [(t, x(t), u(t)) is ~-measurab le as well as f(t, x(t), u) is ~ x ~ measurable for each x(t) ~-measurable . These properties, in general, are not implied by ~ - measurability in t and lower semicontinuity in (x, u) of f. However, if [ is continuous in (x, u) for fixed t and ~-measurab le in t for fixed (x, u), then it is ~ x ~ measurable (see, for example, Ref. 12).

Consider the set-valued mapping

Q(t) = epi f(t, x(t) , . ) = {(u, a ) l a >~f(t, x(t), u)}, (10)

which, for each t, is the epigraph of the function [(t, x(t) , . ). We shall say that a set-valued map Q is ~-measurab le if its graph is ~ x ~ measurable. If values of Q are dosed, then the latter is equivalent to the various other definitions in the literature (see Ref. 13). In our case, when Q is given by (10) and the assumptions of Theorem 1.1 hold, Q is measurable if x(t) is ~-measurable . Indeed, it is easy to check that the epigraph is ~ x measurable iff the graph has this property. Because of 1.s.c. and convexity of f in the u variable, Q(t) is closed and convex for each t.

The aim of this section is to prove the following proposition.

Proposition 3.1. I f f satisfies the assumptions of Theorem 1.1, x : G R k is ~LP-measurable, and Q is given by (10), then there are Pl : G ~ R I and

8 JOTA: VOL. 19, NO. 1, MAY 1976

/3~ : G ~ R measurable, i = 1, 2 . . . . . such that

O(t) = (~ {(x, y ) l - y + (x, pi(t))<<- fli(t)}, i = 1

where

(11)

fl,(t) = f* ( t , x(t), p,( t )) .

Moreover, if (2) holds, then the functions p~,/3i can be so chosen that the p~ are bounded and the/3~ are integrable.

The proof of this proposition will be based on the following rather deep result concerning measurable set-valued mappings (see, for this and other results, Ref. 13).

Proposition 3.2. Let F be a mapping from G into closed subsets of R". F is measurable iff

T={t~ GIF(t) ¢ 0 }

is ~-measurable and there exists a countable set {zi} of 5f-measurable functions zi: T ~ R " such that

F(t) = cl{zi(t)} for every t 6 T.

To prove Proposition 3.1, we need to show that the conjugate function f*(t, x(t), • )(p), which we shall denote in the sequel simply by f*(t, x(t), p), is ~? × ~ measurable.

Note that

f*(t, x(t), p) = lim fN*(t, x(t), p), (12) N---> oo

where

fN*(t, x(t), p) = sup (-f(t , x(t), u)+(u, p)). (13) tul~<N

Thus, f* is measurable if f N , is such for each N. Since f is l.s.c, in u, therefore the supremum in (13) is attained. Hence,

for fixed p,

iff there is u such that

lul N Thus, the set

fN*(t, x(t), p) >!

and -f(t , x(t), u) +(u, p) ~> a.

{tlfN*(t, x(t), p ) ~ }

JOTA: VOL. 19, NO. 1, MAY 1976 9

is the projection on G of the set

{(t, u) l - f ( t, x( t), u) +(u, p) >i o~}.

The latter is ~ × ~ measurable, therefore the first one is Lebesgue measura- ble, which shows that fN*(t, x(t), p) is measurable in t for fixed p. It is finite and convex in p, thus f is continuous in p for each fixed t. As we noticed the latter two conditions imply ~ × ~ measurability of f N , and by (12), of /*(t, x(t), p) also. Hence, the graph of [*(t, x(t) , . ),

S(t) ={(p, fl)lfl = f*(t, x(t), p)},

is 5f-measurable. Applying Proposition 3.2 to S(t), we obtain a sequence {p~,/3~} of measurable functions such that

cl{(p~(t),/3z(t))} = S(t) for each t. (14)

Note that

Q ( t ) = ~ { ( x , y ) l - y + ( , p ) ~ / 3 } foreacht . (15) (p,~)~S(t)

Therefore, (11), with Pi, fli satisfying (14), follows from (15). Now, by the assumption (2), there is p(t) bounded such that

fl(t) = f*(t , x( t) , p( t ) )

is integrable. Put

(pl.m (t), fli,m(t)) = (pi(t), /3,(t)) if Ipi(t)l <~ m, ]13i(t) I <~ m,

(pi,m(t),/3i.m(t)) = (p(t), fl(t)) otherwise.

It is clear that

cl{(p,,m(t), fli, m(t))} = cl{(p~(t),/3~(t))} = S(t),

P~,m are bounded, and fli, m are integrable. This completes the proof of Proposition 3.1.

Remark 3.1. For the first part of Proposition 3.1, only measurability of closed convex set-valued Q given by (10) is needed.

Assumption (2) was used to prove the second part. On the other hand, if we assume that there is p ~ Lq such that f*(t, Xo(t), p(t)) is integrable, then (11) holds with all Pi from Lq and/3i integrable.

Remark 3.2. Let

K = { w c Ll lw( t ) c Q(t)},

Q an arbitrary set-valued mapping. If (11) holds, then K is convex and

10 JOTA: VOL. 19, NO. 1, MAY 1976

weakly closed. The opposite is also true, in the sense that, if K is given above, then, for the weak closure/(" of K, there is 0 of the form (11) such that

/~ = {w ~ Lllw(t) e 0(t)} .

That essentially shows that convexity in u is necessary for weak l.s.c, of I(x, u). Similar results hold for weak* topology. But, in this case, p~ in (11) have to be continuous. For a detailed discussion of this and other results, see Ref. 14.

4. Proof of the Theorem

Let x~, Xo, u, Uo, i = 1, 2 . . . . . be integrable functions, and assume that

(16)

(17)

Since

A (N, k) = {t I sup fN*(t, x,(t), s(t)) >1 ¢~(t) + 1}.

fN*(t, x,(t), p)<~ f°v+l)*(t, xi(t), p),

N - - ~ o O i --~ oO

Put

x,(t)~Xo(t) a.e. in G and IIx,-xollL,- 0, ul ~ Uo weakly in LI,

ui) = So f(t, xi(t), ui(t)) dt-~ a < +oo. (18) I(x~,

To prove the theorem, it is enough to show that, if (16)-(18) hold, then I ( Xo, Uo) <- a.

From the assumption (2), there are p~(t) measurable, ]ps(t)l ~<M, such that

- f ( t , x,(t), u,(t))+(u,(t), pi(t))<~f*(t, x,(t), p~(t))<<- ~O(t)+M1lx,(t)l. (19)

Let us fix s(t) measurable and bounded and such that

riO) =f*(t, Xo(t), s(t)) is integrable.

Because of the lower semicontinuity of f and (16),

lira inf f ( t, xi ( t ) , v ) ~ f ( t, Xo( t ) , u ). i --~ oO

19. -~u

The latter, together with Lemma 2.1, implies that

lim lim sup fN*(t, x~(t), s(t))<~f*(t, Xo(t), s(t)) = fl(t) a.e. in G. (20)

JOTA: VOL. 19, NO. 1, MAY 1976 11

therefore

and since

thus

A ( N + 1, k ) ~ A ( N , k);

sup fN*(t, xi(t), p) >1 sup fN*(t, xi(t), p), i ~ k i ~ k + l

A ( N , k + 1) c A(N, k).

From the above and (20), it follows that

m([') A(N, k)) = 0 k

where m stands for Lebesgue measure. Put

Clearly,

for each N,

B(i, N) = {t[lui(t)[ > N}.

m ( ( ~ B(i, N) ) = O;

(21)

(22)

therefore, to each • > 0, there is N(• ) such that

m ( B ( i , N ) ) < • for N~> N(•).

Because of (17), N(E) can be chosen to be independent of i. Let us now fix a sequence of positive reals ej decreasing to zero. We

choose Nj/> N(Ej), so that Nj+I > Nj + 1. For each Nj, we choose ij so that m(A(Nj, ij))< ej. Such/1- exists, because of (21). We shall require also that ij+l > i j + 1. For ii<~i<ij+~, put

si( t ) = s( t), /3~( t) = max(/3(t), sup f u* j ( t, Xk ( t), s( t) ) ) k ~ i

if t¢ B(i, Nj) w A(Nj , ij)= D~, (23)

s,(t) = p,(t) , fli(t) = ~(t) + MllX,(t)[ if t ~ D~. (24)

By (19), (22), (23), (24), and (14), we have the inequality

;t,( t) = - f ( t, x,( t), u,(t)) + (u,( t ), s,(t))-/3,(t) ~< o. (25)

We shall study now the convergence property of different terms of (25). Let us write

(u, (t), si (t)) = (u, ( t), s ( t)) + (ui ( t), si ( t) - s ( t)). (26)

12 JOTA: VOL. 19, NO. 1, MAY 1976

Because of (17), (ui(t), s(t)) is Lx-weakly convergent to (Uo(t), s(t)), while the second term either is zero or

if t ~ D~. Thus,

[(u,( t), s,( t) -s( t))] ~ ]u~( t )]2M

f I(u,(t), s,(t)-s(t)) I dt<~2Mfo I lug(t) I dt.

But, because of (17), ui are equniformly integrable, therefore

fG (l~li(t), s~(t)-s(t)) dt-~O,

since m(Di) ~< 2~j if ij ~< i < ij+~

and tends to zero as i -~ oo, which shows that

(ui, si) -~ (uo, So) weakly in L1. (27)

Passing to/3~(t), we notice that

+Io , ]O(t)+ Malx,(t)]-fi(t)] dt.

By Fatou's lemma, the first integral tends to zero because of (20) and the definition of A(N, k), while the second tends to zero because of (16) and the fact that m(D~) tends to zero as i ~ m. Therefore,

/3i (t) ~ /3 (0 in Ll-norm. (28)

From (27), (28), (18), and (25), it follows that ~.~]f(t, x~(t), u~(t))l dt is bounded; hence, {f(t, x~(t), u~(t))} is compact in the weak * topology of the conjugate space C* of the space C(cl G, R) of continuous functions from cl G into R, and so is {A~}, the left-hand side of (25). Without any loss of generality, we may assume that f(t, xi(t), u~(t)) converges weak * to a measure/z; that is,

fG fi(t, ui(t))q~(t) dt-, Ic ~(t) ritz(t) xi(t),

for each continuous q~ on cl G. Of course, (27) and (28) imply that (u~(t), s~(t)) and/3~(t) are weak * convergent to (u~(t), s(t)) and/3(0, respec- tively. Therefore, A~ is weak * convergent to a nonpositive measure ~, and

JOTA: VOL. 19, NO. 1, MAY 1976 13

both absolutely continuous and singular parts of the limit measure are nonpositive also. But the singular part ~s of v is equal to -/Zs, where/z, is the singular part of/x. Hence,/Zs ~> 0. The density of the absolutely continuous part v~ of ~ is equal to

- dtz. ( t ) /d t + (Uo(t), s(t)) - fl(t) <~ O,

where ~ stands for the absolutely continuous part of p~. Thus, we have the inequality:

- d l ~ ( t ) / d t + ( u o ( t ) , s(t)) < - fi(t) =f*(t, Xo(t), s(t)) a.e. in G. (29)

Since s(t), /3(t) in (29) are arbitrary, but such that seLoo and f l ( t )= f*(t, Xo(t), s(t)) is integrable, therefore (29) holds for each Pi, fli given by Proposition 3.1, and this implies that

But

dtxa(t)/ dt >~ f(t, xo(t), Uo(t)). (30)

f I(x,, ui) = Jc f(t, xi(t), u,(t)) dt -~ tx(G) = ~s (G) + t~a(G) = a.

But/xs(G) ~> 0 and, by (30),

txa(G) >i I(xo, Uo).

Thus, I(Xo, Uo) <~ a, and the lower semicontinuity of I(x, u) at (Xo, uo), in the sense described in the theorem, is proved. This completes the proof of Theorem 1.1.

Remark 4.1. Note that we used convexity assumption only through Proposition 3.1 and only for x = Xo(t). Therefore, we can restate the theorem in such a way that convexity of f(t, x, u) in u is assumed only for x = Xo(t), but then l.s.c, of I(x, u) in the conclusion can be claimed only for x = xo. On the other hand, convexity and closedness of the sets O(t) defined by (10), thus convexity and 1.s.c. of f(t, Xo(t), • ) is a'necessary condition for weak 1.s.c. of I(xo," ) [see Poljak (Ref. 8) and also Cesari (Refs. 3-5)].

Remark 4.2. The assumption (4) implies that

f ( t, x, u ) ~ - O ( t) - M l [ x I - M l u l .

On the other hand, the latter inequality plus convexity in u imply (2). In Ref. 8, it is proved that this is a necessary condition for lower semicontinuity ol I ( . , u) with respect to strong convergence in L1. In Ref. 8 and in Ref. 2 also, convergence in Lp-topology is considered if p >/1. Such theorems do not differ very much from the case considered here as far as the proof is

14 JOTA: VOL. 19, NO. 1, MAY 1976

concerned. If x, ~ x in measure, then a necessary condition of 1.s.c. of I ( . , u) for fixed u is inequality (2) with M1 = 0. If xn (t) is uniformly bounded and convergent pointwise, then, instead of (2), the following inequality is enough:

f ( t ,x , u)>-q,(t,p) if Ixl<<_p,

where qJ(., O) is integrable for each fixed O. If x, -~ Xo in Lp-topology, then, in the assumption (2), Ix~(t)l should be replaced by Ix(t)[ p. If the weak convergence in Lp is considered for u, then, in (2), instead of bounded p, it is enough to require that p is from Lq, where Lq is the conjugate space of Lp. Again, this change does not require any essential change in the proof.

5. Lower Closure Theorem

In this section, we shall formulate Theorem 1.1 in an equivalent form as a lower closure theorem for orientor fields in the sense of Cesari (Refs. 3-5).

Let Q(t, x), t ~ G, x ~ R k, be a set (possibly empty) of points (q, r) R m X R with the following property:

(i) if (q, r) ~ Q( t, x) and ro> r, then ( q, ro) ~ Q( t, x) too. By an orientor field, the following relation is meant:

(u(t), v(t))~ Q(t, x(t)) a.e. in G, (31)

where u : G ~ R " , v : G ~ R and x : G ~ R k are integrable functions on G. The term orientor field was originally introduced by Wa~.ewski for the

generalized differential equation of the form dx/dt ~ Q(t, x), in this case Q(t, x) c R k and G an interval. By analogy to this, following Cesari, we use the same term for relation (31), since in applications u is in general the value of a differential operator on x.

The orientor field (31) has lower closure property with respect to strong convergence in x and weak sequential convergence in u if, for each sequence (u., v., x.) of integrable functions such that (u.(t), v.(t)) ~ Q(t, x.(t)) a.e. in G, u. ~ Uo weakly in L1, x. ~ xo strongly in L1, and

there is VoC L1 such that

I v.(t) d t ~ M < + o o ,

(Uo(t), Vo(t)) z Q(t, Xo(t)),

IG v°(t) dt~<liminf I~ vn(t) dr.

JOTA: VOL. 19, NO. 1, MAY 1976 15

Theorem 5.1. Assume that the map Q(t, x) of G × R k into convex subsets of R "÷1 has property (i) above. Assume also the following:

(ii) Q(t, x) is measurable, in the sense that the graph of Q, or {(t, x, w) ~ G x R k x Rm+l[w c Q(t, x)}, is ~ x N measurable;

(iii) Q is lower semicontinuous in x for fixed t, in the sense that the graph of Q(t,. ) is closed for each fixed t.

Assume further that there is ~0 : G ~ R integrable and constants M, M1 such that, for each integrable x : G - ~ R k, there is a measurable map p : G-~R m such that Ip(t)[<~M and

-r+(p( t ) , q)<~ ~b(t)+Ml[X(t)l for each (q, r)~ Q(t, x(t)). (32)

Then, (31) has the lower closure property with respect to strong convergence of x and L~-weak sequential convergence in u.

Notice that, putting

f(t, x, u) = inf{r [ (u, r) ~ Q(t, x)}, (33)

we obtain a function which, because of (iii), is l.s.c, in x, u for fixed t, because of (ii) is ~ x N measurable, and because of the convexity of Q is convex in u. Assumption (32) means inequality (2) for f, while lower closure property of (31) is equivalent to l.s.c, of (1), with f given by (33). For more detailed discussion of lower closure theorems, we refer to Cesari's papers (for example, Refs. 3-5).

Remark 5.1. The measurability assumption (ii) is the least which we need to assume so that measurable u, v, x such that (u(t), v(t))~ O(t, x(t)) exist; also, assumption (i) is rather technical. Thus, essentially convexity of O and (iii) form the sufficient condition for lower semicontinuity in question. It would be interesting to know whenever they are necessary as well.

Note Added in Proof. Both Lemma 2.1 and Remark 2.1 are implicitly contained in a recent paper by J. L. Joly, Journal de Math6matiques Pures et Applique6s, Vol. 52, pp. 421-441, 1973.

References

1. OLECH, C., Existence Theory in Optimal Control Problems--the Underlying Ideas, Proceedings of the International Conference on Differential Equations, Los Angeles, California, 1974, Academic Press, New York, New York, 1975.

2. BERKOVITZ, L. D., Lower Semicontinuity of Integral Functionals, Transactions of the American Mathematical Society, Vol. 192, pp. 51-57, 1974.

16 JOTA: VOL. 19, NO. 1, MAY 1976

3. CESARI, L., Closure Theorems for Orientor Fields and Weak Convergence, Archive for Rational Mechanics and Analysis, Vol. 55, pp. 332-356, 1974.

4. CESARI, L., Lower Semicontinuity and Lower Closure Theorems without Seminormality Condition, Annati di Matematica Pura e Applicata, Vol. 98, pp. 381-397, 1974.

5. CESARI, L., A Necessary and Sufficient Condition j'br Lower Semicontinuity, Bulletin of the American Mathematical Society, Vol. 80, pp. 467-472, 1974.

6. SURYANARAYANA, M. B., Remarks on Lower Semicontinuity and Lower Closure, Journal of Optimization Theory and Applications, Voi. 19, No. 1, 1976.

7. CESARI, L., and SURYANARAYANA, M. g., Nemitsky's Operators and Lower Closure Theorems, Journal of Optimization Theory and Applications, Vol. 19, No. i, 1976.

8. POIAAK, B. T., Semicontinuity of Integral Functionals and Existence Theorems for ExtremaI Problems (in Russian), Mathematiceskii Sbornik, Vol. 78, pp. 65-84, 1969.

9. MOROZOV, S. F., and PLOTNIKOV, N. I., On Necessary and Sufficient Condi- tions for Continuity and Semicontinuity of Functionals in Calculus of Variations, Matematiceskii Sbornik, Vol. 57, pp. 265-280, 1962.

10. MORREY, C. B., Multiple Integrals in the Calculus of Variations, Springer- Verlag, New York, New York, 1966.

11. ROC'KAFELLAR, R. T., Convex Analysis, Princeton University Press, Prince- ton, New Jersey, 1970.

12. ROCKAFELLAR, R. T., Existence Theorems for General Control Problems of Bolza and Lagrange, Advances in Mathematics, Vol. 15, pp. 312-333, 1975.

13. ROCKAFELLAR, R. T., Measurable Dependence of Convex Sets and Functions on Parameters, Journal of Mathematical Analysis and Applications, Vol. 28, pp. 4-25, 1969.

14. OLECH, C., The Characterization of the Weak Closure of Certain Sets of Integrable Functions, SIAM Journal on Control, Vol. 12, pp. 311-318, 1974.