JOURNAL OF OPTIMIZATION T H E O R Y AND APPLICATIONS: Vol. 19, No. l, MAY 1976
Weak Lower Semicontinuity of Integral Functionals
C.
OLECH
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 p a p e r is to give a proof of the following result
announced in Ref. 1, T h e o r e m 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.
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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 lul <~ 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
V z (see, for example, Morrey's book, Ref. 10).
The proof of T h e o r e m 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 representation 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))
uER
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
(g*)* = cl co g
may not hold.
Lemma 2.1.
Let gi : R " -~ R w { + oo}, i = 1, 2 . . . . . be arbitrary func-
tions. Then,
(lim inf gi(v))*(p) = lim lim sup g~*(p).
i-~oo
v-~u
N~oo
i-~
(3)
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JOTA: VOL. 19, NO. 1, M A Y 1976
Proof.
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))),
(4)
u
We note here that
h(u)= lira sup h(u)
sup
u
N~
(5)
JuI~N
and that
sup lim sup hi(v) = lira sup sup
t,N<N i-,~o
i-,~
tut~n
v-~u,lvl~<t¢
h~(u).
(6)
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).
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)).
i~eo
l)--~u
e>O
i~o~
(8)
lq--pt<e
However, for our purpose, (3) is satisfactory.
Relation (8) has a nice geometric counterpart. Namely, put
Pi = epi
gi = {(u, x)[x >Igdu)}
and
P~* - {(u, y)[ Y ~ g~(u)}.
Clearly,
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~oo
(9)
JOTA: VOL. 19, NO. 1, MAY 1976
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provided there is a bounded sequence {z~} such that z~ ~ P~. Above,
lim sup Pi = {z Ilira 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
~-measurable 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 ~-measurable 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 ~-measurable 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
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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)},
(11)
i=1
where
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.
F is measurable iff
Let F be a mapping from G into closed subsets of R".
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)).
tul~<N
(13)
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,
fN*(t, x(t), p) >!
iff there is u such that
lul N
and -f(t, x(t), u) +(u, p) ~>a.
Thus, the set
{tlfN*(t, x(t), p ) ~ }
JOTA: VOL. 19, NO. 1, MAY 1976
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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 measurable, 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.
R e m a r k 3.2.
Let
K = { w c Lllw(t) c Q(t)},
Q an arbitrary set-valued mapping. If (11) holds, then K is convex and
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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
x,(t)~Xo(t)
a.e. in G and
ul ~ Uo
IIx,-xollL,- 0,
(16)
(17)
weakly in LI,
I(x~, ui) = So f(t, xi(t), ui(t)) dt-~ a < +oo.
(18)
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.
N - - ~ o O i --~ oO
Put
A (N, k) = {t Isup fN*(t, x,(t), s(t)) >1¢~(t) + 1}.
Since
fN*(t, x,(t), p)<~ f°v+l)*(t, xi(t), p),
(20)
JOTA: VOL. 19, NO. 1, MAY 1976
11
therefore
A ( N + 1, k ) ~ A(N, k);
and since
sup fN*(t, xi(t), p) >1 sup fN*(t, xi(t), p),
i~k
i~k+l
thus
A(N, k + 1) c A(N, k).
From the above and (20), it follows that
m([') A(N, k)) = 0
(21)
for each N,
k
where m stands for Lebesgue measure.
Put
B(i, N) =
{t[lui(t)[ >
N}.
(22)
Clearly,
m((~ B(i, N) ) = O;
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 fu*j( t, Xk ( t), s( t) ) )
k~i
if t¢ B(i, Nj) w A(Nj, ij)= D~,
s,(t) = p,(t),
fli(t)
= ~(t) +
MllX,(t)[
(23)
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)
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JOTA: VOL. 19, NO. 1, MAY 1976
Because of (17), (ui(t), s(t)) is Lx-weakly convergent to
the second term either is zero or
[(u,(t), s,( t) -s(t))]
~
(Uo(t), s(t)),
while
]u~(t )]2M
if t ~ D~. Thus,
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, xi(t), ui(t))q~(t) dt-, Ic ~(t) ritz(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, respectively. Therefore, A~ is weak * convergent to a nonpositive measure ~, and
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JOTA: VOL. 19, NO. 1, MAY 1976
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)/dt + (Uo(t), s(t)) - fl(t) <~O,
where ~ stands for the absolutely continuous part of p~. Thus, we have the
inequality:
-dl~(t)/dt+(uo(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 fl(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
(30)
dtxa(t)/ dt >~f(t, xo(t), Uo(t)).
But
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
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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
I
v.(t) d t ~ M < + o o ,
there is VoC L1 such that
(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 L e m m a 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.
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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,
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6. SURYANARAYANA, M. B., Remarks on Lower Semicontinuity and Lower
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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 Conditions for Continuity and Semicontinuity of Functionals in Calculus of Variations,
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11. ROC'KAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, 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.