optimization—theory and applications || closure and lower closure theorems under weak convergence

CHAPTER 10

Closure and Lower Closure Theorems under Weak Convergence

10.1 The Banach-Saks-Mazur Theorem

If X is a normed linear space over the reals with norm Ilxll, let X* be the dual of X, that is, the space of all linear bounded operators x* on X, the linear operation being denoted by (x*, x), or X* x X --+ R. A sequence [Xk] of elements of X then is said to be convergent in X to x provided Ilxk - xll--+ 0 as k --+ 00. A sequence [Xk] of elements of X is said to be weakly convergent in X to x provided (x*,x k) --+ (x*,x) as k --+ 00 for all x* E X*.

Both convergences are examples of (J-limits in the sense of Sections 2.15 and 8.1.

Let us mention here the following theorem:

lO.l.i (BANACH, SAKS, MAZUR). If X is any normed space over the rea Is with dual X* and norm Ilxll in X, if Xko k = 1,2, ... , is any sequence of elements Xk E X convergent weakly in X to an element x E X, then there is a system of real numbers Cki 2 0, i = 1, ... , k, k = 1,2, ... , with L~= 1 Cki = 1, such that, if Yk = L~= 1 CkiXj, then Ilh - xll--+ 0 as k --+ 00.

For this important theorem we refer to S. Mazur [1], or M. Day [1, p. 45], or K. Yosida [1, p. 120].

As a particular case, let G be a fixed measurable subset of points t =

(t1, ... ,r') E RV , v 21. Let X = L1(G) denote the space of all L-integrable functions h(t), t E G, with norm IIhl11 = SG Ih(t)1 dt. We know that X* = Loo(G), that is, the dual space of X is the space of all real valued functions y(t), t E G, essentially bounded in G, and with norm IIYlloo = esssup[ly(t)l, t E G]. The linear operation (y, h) is then (y, h) = SG y(t)h(t) dt. Then, a sequence of functions hk(t), t E G, k = 1, 2, ... , of L-integrable functions is said to be

325 L. Cesari, Optimization—Theory and Applications© Springer-Verlag New York Inc. 1983

326 Chapter 10 Closure and Lower Closure Theorems under Weak Convergence

convergent in Ll(G), or strongly convergent in L l, to an Ll-integrable function h(t), t E G, if

lim r Ihk(t) - h(t)1 dt = 0. k- 00 JG

A sequence offunctions hk(t), t E G, k = 1, 2, ... , of L-integrable functions is said to be weakly convergent in Ll(G) to an Ll:integrable function h(t), t E G, provided

lim r hk(t)y(t) dt = r h(t)y(t) dt k-oo JG JG

for all measurable and (essentially) bounded functions y(t), t E G. Such a function h is uniquely defined a.e. in G, and is said to be the weak limit of the sequence [hkJ in G.

A family {h(t), t E G} of L-integrable functions in G is said to be sequentially strongly [weakly J relatively compact in L l, provided every sequence [hkJ of elements of {h} contains a subsequence which is strongly [weakly J convergent in Ll(G) (to an element h E Ll(G) which need not be in {h }). Ifwe prescribe that any such strong limit [weak limit J h must be in {h}, then we say that the family is sequentially strongly compact [weakly compact].

Statement (10.1.i) for X = Ll(G) now yields: If hk(t), t E G, k = 1,2, ... ,is a sequence of L-integrable functions on G, and [hkJ converges weakly in Ll to an L-integrable function h(t), t E G, then there is a system of real numbers Cki ~ 0, i = 1, ... , k, k = 1,2, ... , with Lf=l Cki = 1 for all k, such that, if Hk(t) = L7=l Ckihi(t), tE G, k = 1,2, ... , then SGIHk(t) - h(t)ldt--+O as k --+ 00.

Definitions completely analogous hold for the spaces X = Ll (G, W) of all vector valued functions h(t) = (h l , ... ,hn), t E G, with Ll-integrable components, and the dual space X* = Loo(G, Rn) of all vector valued functions y(t) = (yl, ... ,yn), t E G, with essentially bounded components. Then Ilhlll =

SG Ih(t)1 dt, IIYlloo = ess sup[ly(t)l, t E GJ, (y, h) = SG y(t) . h(t) dt, where I I is the Euclidean norm and y . h = ylhl + ... + ynhn = y*h is the inner product in Rn.

If hk(t) = (h l , ... ,hn), t E G, k = 1, 2, ... , is a sequence of elements hk E

Ll(G, Rn) and [hkJ is weakly convergent to an element h(t) = W, ... ,hn), t E G, hE Ll(G, Rn), then there is a system of numbers Cki ~ 0, i = 1, ... , k, k = 1,2, ... , with Lf= 1 Cki = 1, such that, if Hk(t) = Lf= 1 Ckihi(t), t E G, k = 1,2, ... , then IIHk - hill --+ ° as k --+ 00.

10.2 Absolute Integrability and Related Concepts

A function x(t), a ::;; t ::;; b, is said to be AC or absolutely continuous, provided, given e > 0, there is b = b(x, e) > ° such that

N

L ix(pa - x(ocai ::;; e i= 1

10.2 Absolute Integrability and Related Concepts 327

for all finite systems of nonoverlapping intervals [lXd3;], i = 1, ... , N, in [a,b] with If= 1 (fJi - lXi) ::; i5.

A Lipschitz function x(t), a ::; t ::; b, is certainly AC. That is, if there is a constant L;::: 0 such that Ix(t) - x(t')I::; Lit - t'l for all t, t' E [a,b], then x is AC, namely, given e > 0, the requirement for absolute continuity above is satisfied for i5 = elL.

A family {x(t), a ::; t::; b} of AC functions is said to be equiabsolutely continuous if, given e > 0, there is i5 = i5(e) > 0 such that

N

I Ix(fJ;) - x(lXi)1 ::; e i= 1

for all functions of the family, and for all finite systems of nonoverlapping intervals [IX;, fJa, i = 1, ... , N, in [a, b] with If= 1 (fJi - IX;) ::; i5. For the sake of brevity, we shall often say the functions x of the family are equiabsolutely continuous.

Any equiabsolutely continuous family x is also equicontinuous, as we see by taking N=1.

If the functions x of the family are equi-Lipschitzian, that is, there is an L ;::: 0 such that Ix(t) - x(t')1 ::; Lit - t'l for all elements x of the family and all t, t' E [a, b], then the family is certainly equiabsolutely continuous (and hence equicontinuous).

We know that any L-integrable function h(t), a::; t::; b, is absolutely integrable, that is, given e > 0, there is a i5 = i5(h,e) > 0 such that hlh(t)ldt::; e for all measurable subsets E of [a, b] with meas E ::; i5.

A family {h(t), a ::; t::; b} of L-integrable functions is said to be equiabsolutely integrable provided that, given e > 0, there is i5 = i5(e) > 0 such that h Ih(t)1 dt ::; e holds for every element h(t), a ::; t ::; b, of the family, and all measurable subsets E of [a, b] with meas E ::; i5.

to.2.i. If {x(t), a ::; t ::; b} is a family of AC jimctions, then the family {x} is equiabsolutely continuous if and only if the family of derivatives {x'} is equiabsolutely integrable.

Proof. Let us assume {x'} to be equiabsolutely integrable, and let us prove that {x} is equiabsolutely continuous. Indeed, given e > 0, let i5 = i5(e) be the number given in the definition of equiabsolute integrability of the derivatives, and let [lXi' fJa, i = 1, ... , N, be any system of nonoverlapping intervals in [a, b] with Ii (fJi - IX;) ::; i5. If E =

U i [IX;, fJa, then meas E ::; i5 and

t Ix(fJ;) - X(IX;) I = t If:' x'(t) dtl ::; fE Ix'(t)1 dt ::; e,

and this holds for every element x E {x}. Conversely, assume that {x} is equiabsolutely continuous, and let us prove that {x'}

is equiabsolutely integrable. Given e > 0, let i5 = i5(eI6) be the number given in the definition of equiabsolute continuity of {x}. Let x(t), a::; t ::; b, be any element of {x}, and let E be any measurable subset of [a, b] of measure ::; i512. Let E + , E - be the subset of all tEE where x'(t) is defined and x'(t) ;::: 0, or x'{t) ::; 0 respectively. Then both E+ and E- have measures ::;i512. Since x'(t), a ::; t::; b, is L-integrable, and hence, by itself, absolutely integrable, there is a = a(e; x) > 0 such that h Ix'(t)1 dt ::; el6 for every measurable subset F with meas F::; a.1t is not restrictive to take a::; i512. Now E+, with meas E+ ::; i512, is certainly covered by some open set G with meas G < meas E+ + a. Let (1Xi> fJi), i = 1, 2, ... , denote the disjoint open intervals which are the components of


G, and note that

00 b b I (fJi - lXi) = meas G :5: meas E+ + (1 :5: - + - = b. i= I 2 2

Then, the same holds for the finite system (lXi' fJJ, i = 1, ... , N, whose union we denote by GN , N arbitrary. Thus, meas GN :5: b, meas(GN - E+) :5: (1, and for N large enough also meas(G - GN ):5: (1. Then

f£+lx'(t)ldt=(l+nGN + l+n<G_GN)x'dt

= (f£+nGN + IN-E+ )x' dt - fGN-£+ x'dt + f£+n(G-GN) x'dt

:5: r x'dt + r Ix'i dt + r . Ix'i dt JGN JGN-E+ JG-G N

N 8 8 8 8 8 8

:5: i~1 (x(fJJ - X(IXJ)+ 6" + 6":5: 6" + 6" + 6" = 2'

Analogously, we can prove that SE-lx'(t)1 dt:5: 8/2, and thus SE Ix'(t)1 dt :5: 8, and this holds for every measurable subset E of [a, b] with meas E < b/2. This proves (1 0.2.i). 0

to.2.ii. If {x(t)} is a class of equiabsolutely continuous functions x(t), a :5: t :5: b, to :5: a < b :5: T, to, T finite and fixed, then their total variations Vex; a, b] are equibounded.

Proof Take 8 = 1, and let b > 0 be such that for any element x(t), a :5: t :5: b, of the class and for any finite system of nonoverlapping intervals [lXi' fJJ c [a, b] with 'D(fJi - IXJ::::; 0 we also have Ii Ix(PJ - x(ai)1 ::::; 1. Now divide [to, T] into N equal parts each of length :5:0 by means of points to, t l , ... , tN = T. Finally, for any subdivision of [a, b] into arbitrary intervals [aj _ 1, a J by means of points of subdivision ao = a, ai' ... , am = b, we have V = sup S with S = Ijlx(aj- d - x(a)l. If we add the points ti to the points of subdivision aj, then we can associate the new intervals in at most N groups of consecutive intervals, each group covering an interval of length :5: b, and hence having a contribution :5: 1 in S. Thus S :5: N, and V[ x;a, b] :5: N. 0

Together with a family {x} of continuous functions x(t), a :5: t:5: b, we may consider the larger family {y} we obtain by adding to {x} all its limit elements in the p-metric. Thus, y(t), a:5: t:5: b, belongs to {y} if either y E {x} or (more inclusively) there is a sequence Xk(t), ak :5: t :5: bk, k = 1, 2, ... , of elements Xk E {x} with P(Xk, y) ~ 0 as k ~ 00.

In any case y is continuous in [a, b ].

to.2.iii. If {x(t), a :5: t :5: b} is an equiabsolutely continuous family, then the larger class {y} of all the limit elements of {x} in the p-metric is equiabsolutely continuous.

Proof Given 8 > 0, let b = b(8) be the number given in the definition of equiabsolute continuity of the family {x}. Let y(t), a :5: t:5: b, any element of {y} which is not already an element of {x}. Then there is a sequence xk(t), ak :5: t:5: bk, k = 1,2, ... , of elements of {x} with P(Xk, y) ~ 0 as k ~ 00. Hence, ak ~ a, bk ~ b as k ~ 00. Let [a" fJ.], s = 1, ... , N, be any finite system of nonoverlapping intervals contained in the open interval (a, b) with Is (fJs - as) :5: b(8). Then, for k sufficiently large, the intervals [as, fJ,] are contained

10.3 An Equivalence Theorem 329

in the intervals [ak, bk], and then Ls Ixk(Ps) - Xk(lXs)I ::; e for all such k, and by taking the limit as k -+ 00, also Is ly(Ps) - Y(lXs) I ::; e, and this relation holds for system of intervals [IX .. Ps] contained in (a, b). Since y is continuous in [a, b], the same relation holds for any such system contained in [a, b]. 0

10.3 An Equivalence Theorem

We state here the following composite statement, in which, only to simplify notation, we assume that G is a bounded measurable subset of RV, v ;?: 1.

lO.3.i (AN EQUIVALENCE THEOREM). Let {f(t), t E G} be a family of real valued L-integrable functions on a fixed bounded measurable subset G of RV. The following statements are equivalent:

(a) The family {f} is sequentially weakly relatively compact in L 1(G). (b) The family {f} is equiabsolutely integrable in G. (c) There is a constant M and a real valued function cjJ(,), 0:::; , < + 00,

bounded below, such that cjJ(')/' - + 00 as' - + 00, and SG cjJ(lf(t)j)dt:::; M for all h E {h} (growth condition (cjJ)).

(d) There is a real valued function 1/1(,), 0:::; , < + 00, bounded below, such that 1/1(01' - + 00 as , - + 00, and the family {I/I(lf(t)I), t E G} is equiabsolutely integrable.

In (c), (d) it is not restrictive to assume cjJ, 1/1 nonnegative, monotone nondecreasing, continuous, and convex in [0, + (0). Functions cjJ, or 1/1, as above are often called Nagumo functions. The equivalence of (a) and (b) was proved by Dunford and Pettis (see, e.g., Edwards [I, p. 274]). The implication (b) ~ (c) was proved by de la Vallee-Poussin (see, e.g., Natanson [I, p. 164]). The implication (c) ~ (b) was proved by Tonelli [I] for some particular cjJ, and then by Nagumo in the general case (see, e.g. McShane [I, p. 176]). Then, (d) ~ (c) as a consequence ofthe implication (b) ~ (c). Finally, (b) ~ (d) can be proved as (b) ~ (c) by a suitable construction of the function 1/1 so that I/I(I/I(e)) has the property of cjJ in (c) (see, e.g., Candeloro and Pucci [1 ], and also Cesari and Pucci [1 ] for an elementary proof of (10.3.i)) for v = 1. The property expressed under (c) is often called the growth property (cjJ).

The growth property (cjJ) and analogous ones, will be discussed in Section lOA below and will be shown in Section 10.5 to guarantee properties of upper semicontinuity of certain relevant sets Q*(x), that is, of certain set valued maps x - Q*(x). The equivalence of (a), (b), (c), (d), and the upper semicontinuity of the related sets Q*(x) will be then used in the proofs of the closure theorems in Section 10.6 and of the lower closure theorems in Section 10.7 all based on weak convergence. The same equivalence will also be used in the discussion of the sufficient conditions (A) in Section 10.7.


10.4 A Few Remarks on Growth Conditions

The implication (c) => (b) in Theorem (10.3.i) states that the growth condition, say (</J), in (c) implies equiabsolute integrability. When the functions for which the growth condition holds are the derivatives of AC functions x on an interval, then the functions x are equiabsolutely continuous. Moreover, there are analogous, though independent, growth conditions which have the same implication (say, below, (g1) = (</J), (g2), and (g3)). These conditions are rather common in the literature (see Bibliographical Notes) and are presented briefly below.

10.4.i. IJ {x(t), a :::;; t :::;; b} is any class oj AC Junctions x(t) = (Xl, ... ,x"), with - 00 < ao :::;; a < b :::;; bo < + 00, and (g1) (or (</J)) there is a scalar Junction </J(O, ° :::;; ( < + 00, bounded below, such that </J(W( --+ + 00 as ( --+

+ 00, and S~ </J(lx'(t)I) dt :::;; M Jor some constant M and any element x oj the class, then the class {x} is equiabsolutely continuous and the class {x'} is equiabsolutely integrable.

This is a corollary of (1O.3.i), implication (c) => (b), and (10.2.i).

10.4.ii. IJ {'1(t), x(t), a :::;; t :::;; b} is any class oj pairs oj Junctions, '1 scalar, L-integrable, x(t) = (Xl, ... ,x") AC, with - 00 < ao :::;; a < b :::;; bo < + 00,

ao, bo fixed, and with S~ '1(t) dt :::;; M Jor some constant M ~ ° and any element ('1, x) oj the class; and if (g2) given e > ° there is a locally L-integrable Junction t/1 e(t) ~ 0, which may depend on e, such that Ix'(t)1 :::;; t/1 it) + e'1(t), a :::;; t :::;; b, Jor every element ('1, x) oj the class, then the class {x} is equiabsolutely continuous and the class {x'} is equiabsolutely integrable.

ProoJ. First, for e = 1 we have IX'(t) I :::;; t/11(t) + '1(t), and hence '1(t) ~ -t/1I(t) for all t and all pairs ('1,x) of the family. Since t/11(t) is L-integrable in [ao,bo], we may take Mo = S~~ t/11(t)dt. Now, given e > 0, let a = min [1, e2- I(M + Mo + 1)-1]. The function t/1At) is L-integrable in [ao,bo]; hence, there is some D > ° such that S E t/1 a dt :::;; e/2 for every measurable set E with meas E < D. Now let '1(t), x(t), a :::;; t :::;; b, be any pair of the family, and take any measurable subset E of [a, b] with meas E :::;; D. Then

IE IX'(t) I dt :::;; IE [t/1 a(t) + a'1(t)] dt :::;; IE [t/1 a + a('1 + t/1 1)] dt

:::;; a I: '1 dt + a I: t/1 I dt + IE t/1 a dt

r e e :::;;a(Mo+M)+ JEt/1adt:::;;"2+"2=e. D

10.4.iii. IJ {'1(t), x(t), a :::;; t :::;; b} is any class oj pairs oj Junctions, '1(t) scalar, L-integrable, x(t) = (Xl, ... ,x") AC with - 00 < ao :::;; a < b :::;; bo < + 00, ao,

10.4 A Few Remarks on Growth Conditions 331

bo fixed, with J~ ,,(t) dt :::;; M for some constant M ~ 0 and any element of the class; and if (g3) for every vector P = (PI, . .. ,Pn) ERn there is a locally L-integrable function 4Jit) ~ 0 which may depend on P, such that ,,(t) ~ (p, x'(t)) - 4Jp(t), a :::;; t :::;; b, and for all elements (", x) of the class, then the class {x} is equiabsolutely continuous and the class of the derivatives {x'} is equiabsolutely integrable.

Proof. We have denoted by (a, b) the inner product in R". Let 4J(t), t/J(t), ao :::;; t :::;; bo, be the nonnegative L-integrable functions of assumption (g3) corresponding to the two unit vectors P = Ul = (1,0, ... , 0) and P = VI = (-1,0, ... ,0). Then

X,l(t):::;; ,,(t) + 4J(t), -X,l(t):::;; ,,(t) + t/J(t), a:::;; t :::;; b;

hence, IX,l(t)1 :::;; ,,(t) + 4J(t) + t/J(t), a :::;; t:::;; b. Thus,

(10.4.1) ,,(t) + 4J(t) + t/J(t) ~ 0, a:::;; t:::;; b.

Let M 1 = J~ (4J(t) + t/J(t)) dt. Now, given e > 0, let N be an integer such that N-lnM :::;; e/3, N-lnM 1 :::;; e/3. If Uj, Vj denote the unit vectors Uj = (c5 jj,

j= 1, ... ,n),vj=(-c5jj ,j= 1, ... ,n), then again by assumption (g3), for P = NUj and P = Nvj, there are functions cI>j(t) ~ 0, '1'j(t) ~ 0, L-integrable in [ao, bo] such that

N x,j(t):::;;,,(t) + cI>j(t), - N x,j(t):::;; '1(t) + '1'j(t),

and hence

a:::;;t:::;;b, i=l, ... ,n,

a:::;; t:::;; b,

for any pair ('1, x) of the class. Then cI>(t) = Li'= 1 cI>j(t), '1'(t) = Li'= 1 '1'j(t) are L-integrable in [ao, bo], and we also have

(10.4.2) Nlx'(t)1 :::;; n'1(t) + cI>(t) + '1'(t), a:::;; t:::;; b.

If E denotes any measurable subset of [a,b], we have, from (10.4.1) and (10.4.2),

IE Ix'(t)I dt :::;; N-ln IE ,,(t) dt + N- l IE (cI>(t) + '1'(t)) dt

:::;; N-ln IE [,,(t) + 4J(t) + t/J(t)]dt + N- l IE (cI>(t) + '¥(t»dt

:::;; N-ln lb ,,(t)dt + N-ln lb(4J(t) + t/J(t))dt

+ N- l IE (cI>(t) + '¥(t» dt.

Since cI> + '1' is L-integrable, there is c5 > 0 such that meas E :::;; c5 implies JE(cI> + '1') dt :::;; e/3, and then

IE Ix'(t)1 dt :::;; j + j + j = e.


We have proved the equiabsolute integrability ofthe class {x'}. The equiabsolute continuity of the class {x} follows from (1O.2.i). D

Remark 1. We shall prove below that the growth conditions (g2) and (g3) are equivalent.

Remark 2. Under the conditions of any of the statements (lO.4.i-iii), if in addition we know that the class {x} is also equibounded, then the same class {x} is sequentially relatively compact in the topology of the weak convergence of the derivatives (Section 2.14), that is, from any sequence of elements [x k ] of the class {x} there is a subsequence which converges uniformly (or in the p-metric), and whose derivatives are weakly convergent in L l . The compactness in C is the Ascoli theorem, the weak compactness of the derivatives is the Dunford-Pettis theorem.

The conditions (gl-3) above have been expressed directly in terms of functions ,,(t), x(t), x'(t), a ~ t ~ b. Actually, we may think of these functions as solutions of orientor fields, say (,,(t),x'(t» E Q(t,x(t», t E [a,b] (a.e.), for classes {(,,(t),x'(t)} with (t, x(t» E A, s: ,,(t) dt ~ M, A compact, and then the same properties (gl-3) can be derived from analogous geometric properties of the orientor field, that is, of the given sets Q(t,x). We express these properties as local properties, and, for the sake of simplicity, we call them local properties (gl-3) ofthe orientor field.

We say that the local growth property (gl) is satisfied at (I, x) E A, provided there are a neighborhood N ~(T, x) of (T, x) and a scalar function cp('), ° ~ , < + 00, bounded below, such that cp(W, --> + 00 as ,--> + 00, and (t, x) E NiT, x) n A, (y, z) E Q(t, x) implies y ;;::: cp(lzi).

We say that the local growth property (g2) is satisfied at (I, x) E A, provided there is a neighborhood N ~(T, x), and for every B > ° there is an L-integrable function 1/1 E(t) ;;::: 0, I - D ~ t ~ I + D, such that (t, x) E N iI, x) n A, (y, z) E Q(t, x) implies Izl ~ 1/1 E(t) + By.

We say that the local growth property (g3) is satisfied at (I, x) E A, provided there is a neighborhood N ~(I, x), and for every vector p E Rn there is an L-integrable function cpp(t) ;;::: 0, I - D ~ t ~ I + D, such that (t, x) E N~(I, x) n A, (y, z) E Q(t, x) implies y;;::: (p, z) - cpp(t).

We may say that any of these condition is satisfied in A if it is satisfied at every point (I, x) E A. For A compact, then some finite system of corresponding neighborhoods covers A, and for the classes {,,(t), x'(t), a ~ t ~ b} mentioned above, corresponding functions cp, or 1/1 .. or CPP can be found for the whole interval [a, b]. We leave this derivation as an exercise for the reader. In the following statements (lO.4.iv-vii) we refer to the local growth properties. The same holds for the properties in the large.

10.4.iv. (gl) implies (g2).

Let cp(') be the function as in (gl) such that y ;;::: cp(lzi) for all (y, z) E Q(t, x), (t, x) E

N ~(I, x) n A. Let L be a real constant such that cp(') ;;::: L for all " and thus y ;;::: L. Now, given B > 0, let ME > ° real be such cp(W, > B- 1 for all,;;::: ME' and take I/IE(t) = ME + BILl forI - D ~ t ~ I + D. Then, for (t, x) E N ~(T, x), (y, z) E (2(t, x) we have either Izl ~ ME and then Izl ~ ME + B(y - L) ~ I/IE + BY, or Izl ;;::: ME and then cp(lzi)/lzl ;;::: B- 1,

so Izl ~ Bcp(lzi) ~ I/IE(t) + By.

10.4.v. (g2) with I/IE constant implies (gl).

10.5 The Growth Property (rjJ) Implies Property (Q) 333

For e= 1 we have Izl:::;"'1 + y; hence y~ -"'I for all (y, z) E Q(t,x), (t,x) E N ~(Y,x)nA. If for all such t, x, y, z we have Izl :::; M for some constant M, then we can take 4>(0 =

- '" 1 for all 0 :::; ( :::; M, and take 4> arbitrary for ( > M, say 4>«() = - '" I + «( - M)2. If Izl can be as large as we want, then for every s = 1, 2, ... , there is a constant"', such that Izl:::; "', + S-Iy for all (y,z) E Q(t,x), (t,x) E NAY,x)nA, and it is not restrictive to take "'s ~ 0, ",,:::; "',+1> ",,-4 +00 as S-4 +00. For Izl ~ 1 we have then Y/lzl ~ s(s"',)/Izl, and hence Y/lzl ~ s/2 for Izl ~ 2"',. Now we take 4>«() = -"'I for (:::; 2"'1> and 4>(0 = (s/2K for 2"" :::; ( < 2", s+ I. Then 4> is bounded below, 4>«()f( -4 + 00 as ( -4 + 00, and Y/lzl ~ s/2 for 2"" :::; Izl :::; 2",,+ I implies y ~ 4>(lzlJ.

lO.4.vi. (g3) implies (g2).

Indeed, suppose uj, Vj denote the unit vectors Uj = (oij,j = 1, ... , n), Vj = (-Ojj, j = 1, ... , n), and for e > 0, let P = ne - I U j and P = ne -I Vj. Then by (g3), there are functions 4>j(t) ~ 0, "'j(t) ~ 0, L-integrable in [Y - oj> Y + OJ], such that for (t, x) E N ~i(Y' x) n A, (y,z) E Q(t,x), z = (Zl, ... ,2"), we have

y~ne-Izj-4>j(t), y~-ne-Izj-"'j(t), Y-oj:::;t:::;Y+Oj, i=I, ... ,n.

For 0 = min oj, we have then

ne-Ilzjl:::;y+4>m+"'j(t), Y-o:::;t:::;Y+o, i=I, .. ,n,

and for (jJ(t) = D= I 4>Jt), P(t) = D= I "'j(t), we also have

n

ne-1Izl:::; ne- I L Izjl:::; ny + (jJ(t) + P(t), Y - 0:::; t:::; Y + 0,

or Izl:::; ey + en-I«(jJ(t) + P(t)), Y - 0:::; t:::;Y + 0,

and this is (g2) for"', = en -I( (jJ + '1').

lO.4.vii. (g2) implies (g3).

Indeed, given P = (PI' ... , Pn) ERn, take N = Ipil + ... + IPnl, e = N- I . Then by (g2) we have Izl :::; "',(t) + ey, hence

y ~ e-1Izl- e-ItjJ,(t) = Nlzl- e-I"'.(t)

and finally for 4>p = e I", ,(t), also y ~ (p, z) - 4>p(t), and this is (g3).

10.5 The Growth Property (¢) Implies Property (Q)

The following theorem will be used in Sections 10.6 and 10.7 with the state variable x there having the same role as the variable x here. In some applications x may have the role of t or of (t, x).

lO.5.i (THEOREM; CESARI [6, 7J). Let A be any set of points x E Rh, and for every xEA, let Q(x) denote a set of points (y,z)=(y,zl, ... ,Zn)ER 1 +n


such that (a) (y, z) E Q(x), Y ::; y', implies (y', z) E Q(x). Let cp(O, ° ::; , < + 00,

be a real valued function, bounded below and such that cp(O!( --+ + 00 as ,--+ + 00. For some X E A let NaJx) be a neighborhood of x in A, and assume that (b) (y, z) E Q(x), x E N ao(x), implies y ~ cp(lzi}. If the sets Q(x) have property (K) at x, and the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

Alternatively, for every x E A let Qo(x) be a subset of the z-space W, let M ° denote the set M ° = [(x, z) I x E A, z E Qo(x)J, let To(x, z) be a real valued lower semicontinuous function on M 0, and let Q(x) denote the set Q(x) = [(y,z)IY~ To(X,Z),ZEQo(X)]. For some XEA and neighborhood Nao(x) of x in A, assume that x E N ao(x), Z E Qo(x) implies To(x, z) ~ cp(lzi). Then, if the sets Qo(x) have property (K) at x and the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

This statement is a particular case of the following one.

to.S.ii (THEOREM; CESARI [6, 7J). Let A be any set of points x E Rh, and for every x E A, let Q(x) denote a set of points (y, z) = (yO, l, ... ,i\ Zl, ... ,zn) E

R1+J1+n such that (a) (y, z) E Q(x), Y = (yO, ... ,yU), Y' = (yO', . .. ,iLf)j ::; yi', i = 0, 1, ... , jl, implies (ji', z) E Q(x). Let cp«(), 0::; , < + 00, be a real valued function, bounded below and such that cp(OJ( --+ + 00 as ,--+ + 00. For some X E A let N ao(x) be a neighborhood of x in A and L a real constant, and assume that (b) (ji,z) E Q(x), x E Nao(x) implies yO ~ cp(lzl), yi ~ L, i = 1, ... , jl. Then, if the sets Q(x) have property (K) at x and the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

Alternatively, for every x E A let Q(x) be a subset of the yz-space RJ1+n, let M denote the set M = [(x, y, z) I x E A, (y, z) E Q(x)], let T o(x, y, z) be a real valued lower semicontinuous function on M, and let Q(x) denote the set Q(x) = [(yO, y, z) Iyo ~ To(x, y, z), (y, z) E Q(x)] c R 1 +/dn, y = (yt, ... , it), Z = (zt, ... ,zn). For some X E A and neighborhood N ao(x) of x in A assume that x E N ao(x), (y, z) E Q(x) implies T o(x, y, z) ~ cpCizl), yi ~ L, i = 1, ... , jl, where cp is a function as above and L a constant. Then, if the sets Q(x) have property (K) at x and the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

Analogously, for every x E A let Qo(x) be a subset of the z-space W, let M ° denote the set M ° = [(x, z) I x E A, z E Qo(x)J, let Ti(x, z), i = 0, 1, ... , jl, be real valued lower semicontinuous functions on M 0, and let Q(x) denote the set Q(x) = [(yO, y, z) I yi ~ Ti(x, z), i = 0, 1, ... ,jl, Z E Qo(x)J, where y = (l, ... ,it). For some X E A and neighborhood N ao(x) let us assume that x E N ao(x), z E Qo(x) implies To(x, z) ~ cp(lzl), Ti(x, z) ~ L, i = 1, ... ,jl, for some function cp as above and some constant L. Then, if the sets Qo(x) have property (K) at x and the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

Remark 1. In the statements (1O.5.i, ii) we have implicitly assumed that the sets Q, Q are not empty, and the functions T ate finite everywhere on the sets M or M 0 as mentioned. Actually, the statements hold even if some of such sets are the empty set and

10.5 The Growth Property (cI» Implies Property (Q) 335

the functions T are extended functions whose values are finite or + 00, but now we assume that the same functions T are defined and lower semicontinuous in the whole space. Thus, whenever, say T(x) = + 00, it is required that T(x) --+ + 00 as x --+ x. Properties (K) and (Q) for maps x --+ Q(x) when we do not exclude empty sets are defined as usual, as mentioned in Section 8.5, Remark 1.

Proof of (1O.5.ii). Let Lo be a bound below for 4>(0. As usual, we denote by Q(x; b) the set UQ(x) where U is taken for all x in the b-neighborhood Nb(X) of x in A. We have to prove that if (y, z) E (ib>O cl co Q(x; b), then we also have (y, z) E Q(x). First we note that for any (y, z) E Q(x), y = (yO, y', ... ,it), we have yO ~ 4>(lzl), i ~ L, i = 1, ... ,/1,. By adding ILl + 1 to all i coordinates, i = 1, ... ,II, and adding ILol + 1 to all yO coordinates and 4>('), we make them all positive. We see that it is not restrictive to assume 4>(0) 0 and all i > 0, i = 0,1, ... , II, for (y,z) E Q(x), x E Nbo(x).

Now we have (y, z) E nb cl co Q(x; b) as stated, and hence ey, z) E cl co Q(x; b) for all b, 0 < b ::;; boo

Thus, there is a sequence of points (Yk' Zk) E co Q(x; bk) with 0 < bk ::;; bo, bk --+ 0, Yk --+ y, Zk --+ Z. By CaratModory's theorem (8.4.iii) there are v sequences of points 01/;, zk) E Q(xk), k = 1,2, ... , Y = 1,2, ... ,v, and numbers Ai with xi E A, xi --+ x as k --+ 00, Y = 1, ... , v, and

(10.5.1)

where Iy ranges over all y = 1, ... ,v, and we can take v = n + II + 2. Here Y =

(yO, yl, ... ,y~), Yk = (y~, yt, ... ,y~), yi = (y~y, yt Y, •.. ,yn y = 1, ... , v, k = 1, 2, ... , and moreover xi = (xt, ... ,xl?),zi = (zt y, ... ,z~Y).

Since 0::;; Ai ::;; 1 for all y and k, there is a subsequence, say still [k], such that Ai --+ AY

as k --+ 00, y = 1, ... , v, and then 0::;; AY ::;; 1. From the second relation (10.5.1) we derive that Ly ,1,> = 1, and thus at least one AY is positive. By a suitable reindexing, we may well assume that for some 0(,1 ::;; 0( ::;; v, we have 0 < AY ::;; 1 for y = 1, ... , 0(, while AY = 0 for y = 0( + 1, ... , v. We can even assume that 0 < 2~IAY::;; Ai::;; 1 for y = 1, ... , 0( and all k. Now in the third relation (10.5.1) all yL y~Y, Ak are nonnegative numbers, and y~ --+ yi as k --+ 00, i = 0, 1, ... , II. Thus, there is a constant M such that o ::;; yi, y~ ::;; M for all i and k, and from the third relation (10.5.1) we derive, for y = 1, ... , 0(, that r' Ayyiy ::;; Akyiy ::;; yi ::;; M, or 0::;; yiY ::;; 2W)~' M, a finite number, and this holds for y = 1, ... , 0(, i = 0, 1, ... ,II. Thus, there is a subsequence, say still [k], such that y~y --+ i y as k --+ 00, y = 1, ... , 0(, i = 0, 1, ... ,II. For y = 0( + 1, ... , v, again from the third relation (10.5.1) we derive 0 ::;; AkY~y ::;; M, and by a further extraction we may assume that ).kY~y --+ Aiy ~ 0 as k --+ 00, and this holds for y = 0( + 1, ... , v and i = 0, 1, ... , II.

If I',I" denote sums ranging over all y = 1, ... ,0( and y = 0( + 1, ... , v respectively, then the third relation (10.5.1) yields as k --+ 00

i = 0, 1, ... , II.

Now, for y = 1, ... ,0(, i = 0, and all k, we also have

(10.5.2) y = 1, ... ,0(, k = 1,2, ....


By the growth property of </> we conclude that the ct sequences [Zk, k = 1, 2, ... ], )! = 1, ... , ct, are bounded in Rn. By a further extraction we can well assume that Zk ..... zY as k ..... 00 where zY,)! = 1, ... ,ct, are ct points of Rn. Thus, for y); = (yZY, ... , yfY), yY = (yoY, ... ,yI'Y), )! = 1, ... , ct, k = 1,2, ... , we have y); ..... yY as k ..... 00, together with zi ..... zY, xi -+ x. Now (Yi, zi) E Q(xi) for all )! = 1, ... , ct, k = 1,2, .... Thus, given b > 0, we also have (Yi, zi) E Q(x; b) for all k sufficiently large. Hence, as k ..... 00, we have (yY, zY) E cl Q(x; b), )! = 1, ... , ct, and this relation holds for all b > 0. Thus, (yY, zY) E n~ cl Q(x; b), and by the property (K) of the sets Q(x) at x, we also have (yY, zY) E Q(x), )! = 1, ... ,ct.

For)! = ct + 1, ... , v, the sequence [y~Y, k = 1, 2, ... ] is either bounded or unbounded. Thus, by a suitable reindexing and extraction, we may well assume that, for some [3, 1 S ct S [3 S v, the [3 - ct sequences [y~Y, k = 1, 2, ... ], )! = ct + 1, ... , [3, converge to some yOY ERas k ..... 00, and that the v - [3 sequences [y~Y, k = 1, 2, ... ], )! = [3 + 1, ... ,v, diverge, or y~Y ..... + 00 as k ..... 00. Then, for)! = ct + 1, ... ,[3, the relations y~Y ~ </>{izkj) shows that the sequences [z);, k = 1, 2, ... ] also are bounded in Rn, and by a further extraction, they can be assumed to converge to some zY ERn, )! = ct + 1, ... ,[3. For )! = [3 + 1, ... , v, the same relations y~Y ~ </>(Izkj), where now y~Y ..... + 00, show that there are certain E); E Rn with z); = E);y~Y and E); ..... ° as k ..... 00. If we now denote by I"* and I'" summations ranging over all )! = ct + 1, ... , [3 imd )! = [3 + 1, ... , v respectively, we can write the fourth relation (10.5.1) in the form

where A); ..... AY > 0, z); ..... zY in the first sum, where A); ..... 0, z); ..... zY in the second sum, and where E); ..... 0, A);y~Y ..... AOy in the third sum. Thus, as k ..... 00, we derive z = I' AYZY. We have now

o < ,p S 1, tP, zY) E Q(x), )! = 1, ... , ct, 1 = I' ;'Y,

i = 0, 1, ... , II.

Since Q(x) is convex, we also have

(I'AYYY, I' AYZY) E Q(x),

and by the property (a) of the sets Q(x), also (y, z) E Q(x), with y = (yO, yl, ... ,yl'). We have proved that (y, z) E n~ cl co Q(x, b) implies (y, z) E Q(x). We have proved (lO.5.ii) under the hypotheses that the sets Q(x) have property (K) at x. Here we have assumed that the set n~>o cl co Q(x; b) is not empty. If this set is empty, then certainly Q(x) is empty, and property (Q) holds at x,

Let us now consider the alternate cases of (1O.5.ii). Assume that the sets Q(x) have property (K) at x. The argument is the same up to the statement (yl:, zk) E Q(Xk) for all )! = 1, ... ,ct, k = 1,2, ... ,with y); = (y~Y, yt Y, ... ,yfYj = (y~Y, yk). From here we derive that (y);,zk) E Q(Xk) c RI'+n, )! = 1, ... , ct,k = 1, 2, ... , with xk ..... x, y); ..... yY, z); ..... zY as k ..... 00. Hence, given b > 0, we also have (y);, zk) E Q(x; b) for all k sufficiently large; and (yY, zY) E cl Q(x; b). By property (K) we derive (yY, zY) E Q(x). On the other hand y~Y ~ To(x);, y);,zk), where To is lower semicontinuous. Hence yOY ~ To(x, yY,zY), and we conclude that (y, zY) = (yoY, yY, zY) E Q(x), )! = 1, ... , ct. The argument now proceeds as

before. Finally, we consider the case where the sets Qo(x) c Rn have property (K) at x.

Again, the argument proceeds as before, up to the statement (jil:, zk) E Q(xk),)! = 1, ... , ct,

k = 1, 2, .... From here we derive that z); E Qo(xk) c Rn for the same)! and k. Hence,

10.5 The Growth Property (cJ» Implies Property (Q) 337

by property (K) for these sets, we derive as before that zY E Qo(:x), y = 1, ... , IX. On the other hand, y~ ;::: Ti(X~, z~), i = 0, 1, ... ,Jl, and by the lower semicontinuity of these Th we derive yiY ;::: Tlx, zY), i = 0, 1, ... ,Jl. Thus, OP, zY) E Q(x), Y = 1, ... , IX, and the proof proceeds as before. Theorem (lO.S.ii) is thereby proved. 0

For Jl = 0 the main statement of (lO.S.ii) reduces to the main statement of (lO.S.i); for Jl = 0 the second and third alternatives of (lO.S.ii) coalesce into the alternative case of (lO.5.i).

It remains to prove the Remark 1. Now we allow the sets Q, Q to be the empty set, but we assume the functions T to be extended functions defined and lower semicontinuous in the whole space.

In the proof above of (lO.S.ii), first part, if for some x, Q(x) = 0 = no cl Q(x; c5), then the argument is the same up to the point where we state that (yY, Zl) E cl Q(x; c5), y = 1, ... , IX. This is a contradiction since the second member is the empty set. Thus, no point (y, z) can belong to no cl co Q(x; c5). In the second part we assume now that T o(x,y, z) is an extended function defined and lower semicontinuous in Rh+ll+n. If for some x, Q(x) = 0, then either Q(x) = 0, or Q(x) #- 0. If Q(x) = 0 = no cl Q(x; c5) the proof is the same up to the point where we state that (yY, zY) E Q(x), y = 1, ... , IX, a contradiction, since Q(x) is the empty set. If Q(x) #- 0, Q(x) = no cl Q(x; c5), then To(x, y, z) = + 00 for all (y, z) E Q(x), and again the proof is the same up to the statement y~Y ;::: To(x~, y~, zk), and now the second member approaches + 00 as k -+ 00, while y~Y -+ yaY, a finite number, a contradiction.

In the proof of the third part, we assume now that all Ti(x, z), i = 0, 1, ... , Jl, are extended functions defined and lower semicontinuous in Rh+n. Iffor some x, Qo(x) = 0, then either Qo(x) = 0, or Qo(x) #- 0· If Qo(x) = 0 = no cl Qo(x;c5), then the proof is the same up to the statement zY E Qo(x), y = 1, ... , IX, a contradiction. If Qo(x) #- 0, Qo(x) = no cl Qo(X; c5), then for every z E Qo(x) we must have Ti(x, z) = + 00 for at least one i, and then in the statement y~Y ;::: Ti(Xb Zk), i = 0, 1, ... , Jl, Y = 1, ... , IX, the second member approaches + 00 for at least one i, while y~ -+ yiY all finite numbers, again a contradiction. We have proved Remark 1.

Remark 2. Concerning the hypotheses of statements (lO.S.i) and (lO.5.ii), we note that the sets Q(x) may have properties (Q) and (K) without the sets Qo(x) having either property. Il!.deed, take A = [xIO:;; x:;; 1], Qo(02 = [zlz = 0], Qo(x) = [ziO < z:;; x], 0< x:;; 1, Q(O) = [(zO,z)lz = 0, 0:;; ZO < + 00], Q(x) = [(zO,z)lzo;::: (XZ)-l, 0 < z:;; x], o < x :;; 1. These sets are all convex; the sets Q(x) have property (Q) at every X, 0 :;; x:;; 1; and the sets Qo(x), 0 < x :;; 1, have neither property (K) nor (Q), since they are not closed.

Remark 3. The growth condition in (lO.S.i) and (lO.S.ii) can be simply expressed by saying: There is a neighborhood Vo of x such that, given Il > 0, there is also a constant N such that x E Va' Izi ;::: N, (yO, z) E Q(x) [or (yO, y, z) E Q(x)] implies Izi :;; Byo.

Theorems (10.S.i) and (10.S.ii) are properties of orientor fields, that is, concern problems of optimal control when the control parameters are eliminated. Let us see here some of their implications in terms of the control parameters.

Let x -+ U(x), X E A c W, U(x) c Rm be a given set valued map, and let M denote its graph, or M = [(x, u)lx E A, u E U(x)] C Rv +m• Let Jo(x, u), g(x, u) = (gl, ... , gil)' J(x, u) = (fl' ... ,f.) be functions defined on M. Let x be a point of A, and assume that


there is a fixed bo-neighborhood N~l'C) of x in A such that f and 1 are of slower growth than fo as lul--> 00 uniformly in N ~/x). By this we mean that:

(G) Given 8 > 0, there is N = N(8) 2 0 such that x E N ~o(x), U E U(x), lui 2 N implies If(x, u)1 ~ 8fo(x, u), 1 ~ 8fo(x, u).

Let yO, y, z denote the variables yO E R, y = (yl, ... , yU) E RI', Z = (ZI, ... , z") E

R", and for x E A let Q(x) denote the set of all (yO, y, z) with yO 2 fo(x, u), i 2 gi(X, u), i = 1, ... , /1, Zi = /;(x, u), i = 1, ... , n, u E U(x).

lO.5.ili. If M is closed, if the functions fo, gi, i = 1, ... , /1, are nonnegative and lower semicontinuous, if the functions Ii, i = 1, ... , n, are continuous on M, and if f and 1 are of slower growth than fo as lul--> 00 uniformly on N~o(x), then the sets Q(x) have property (K) at x, and if the set Q(x) is convex, then the same sets Q(x) also have property (Q) at x.

Proof. First let us prove that the sets Q(x) have property (K) at x. Let (yO, y, z) E

(]~> ° cl U[Q(x), x E N~(x)]. Then there are points Xk E N~(x) and points (y~, Yk, Zk) E

Q(Xk), k = 1, 2, ... , with Xk --> x, y~ --> yO, Yk --> y, Zk --> Z as k --> 00. Hence, there are also points Uk E U(Xk) with y~ 2 fo(xk, ud, y~ 2 gi(Xk, ud, i = 1, ... , /1, z~ = /;(Xk, Uk), i = 1, ... , n. Note that the sequence [Uk] must be bounded, since in the opposite case there would be a subsequence, say still [k], with IUkl--> 00, and hence fo(xk, ud --> + 00,

so y~ --> + 00, a contradiction. Since [Uk] is bounded, there is a subsequence, say still [k], with Uk --> Ii E Rm. Thus, Ii E n~>o cl U[U(x), x E N.(x)]. The set M is closed by hypothesis; hence the sets U(x) have property (K) at x by (S.5.iii) and subsequent Remark 2. The (x, Ii) E M. By the lower semicontinuity of fo and gi, and by the continuity off, we have, as k --> 00, yO 2 fo(x, Ii), i 2 g;(x, Ii), i = 1, ... , /1, Zi = J;(x, Ii), i = 1, ... , n. Hence (yO, y, z) E Q(x), and the sets Q(x) have property (K) at x. To prove property (Q) we have first to prove the growth condition of Remark 3 above. Indeed, for the neighborhood N/jo(x) of x, which we may suppose to be bounded, and 8> 0, there is N =

N(8) > 0 such that lui 2 N, U E U(x), X E N~o(x) implies 1 ~ 8fo, If I ~ 8fo. For lui ~ N, x E N~o(x), U E U(x), f is bounded, say If I ~ No. Thus, x E N~o(x), Izi = If I > No, U E U(x) implies lui 2 Nand Izl = If I ~ 8fo ~ 8yo. By Remark 3 above, and thus by (1O.5.i), the sets Q(x) have property (Q) at x. This proves (1O.5.iii). D

Condition (G) is often used. Theorem (l0.5.iii) holds also under analogous conditions, of equal practical interest, namely, either

(G') There are constants c, d 2 0 and a function cp(O, 0 ~ ( < 00, bounded below, with cp(W( --> + 00 as ( --> + 00, such that If(x, u)1 ~ clul + d and fo(x, u) 2 cp(lulJ for all x E Nix), u E U(x); or

(G") (a) There is a function cp«(), 0 ~ ( < 00, bounded below, with cp«()/( --> + 00 as ( --> + 00, such that fo(x, u) 2 cp(if(x, u)il for all x E N/j(x), u E U(x); and (b) either fo(x, u) --> + 00 as lul--> + 00 uniformly in Nix), or If(x, u)l--> + 00 as lul--> + 00 uniformly in N ~(x).

Remark 4. In Chapters 11-16, whenever we shall be concerned with the generalized solutions of Section 1.14, we shall need properties of the new functions

h

ft = j;(x, p, v) = L Pj/;(x, uU»), i = 0, 1, ... , n, j= 1

h

gt = gi(X, p, v) = L Pjgi(X, uU»), i = 1, ... , r, j= 1

f* = (fT, ... ,fn

g* = (gT, ... , g:),

10.5 The Growth Property (cfJ) Implies Property (Q) 339

where h is any fixed integer (h ~ n + 2), P = (PI, ... , Ph) E r, where r is the simplex [Pj ~ 0, j = 1, ... , h, PI + ... + Ph = 1], and where v = (U(I), ... , U(h)), u(j) E U(x), u(j) E R"',j = 1, ... , h, that is, v E (U(x) '/'. Thus, (p, v) is the new control variable and V =

r x Uh is the control space. The corresponding sets Q are here the sets

R(x)=[zO~f~, 3i~gr,i=1, ... ,r, z=j*, (p,V)ErXUh].

The growth condition on fo, f of (lO.S.iii) is not inherited by the functions f~, j*, as the following example shows: take fo = u2, f = u, n = 1, f~ = PI(U(1))2 + P2(U(2))2, f* = Pl ut l ) + P2U(2), h = 2, P = (PI,P2)' Then for PI = 0, P2 = 1, uti) = k, ut2) = 0, we have f~ = 0, f* = 0, while (0, 1, k, 0) -+ 00 as k -+ 00. However, we shall prove that the growth condition of (lO.S.iii) on the original functions fo, f still guarantees that the sets R(x) have properties (K) and (Q) at Xo.

First, let us prove that the sets R(x) have property (K) at Xo. To this purpose let (zo, 3, z) be a point of na cl R(xo, e). Then there is a sequence (z~, 3k, Zk), k = 1, 2, ... , with z~ -+ ZO, 3k -+ 3, Zk -+ Z as k -+ 00, z~ ~ Ij PjdO(Xk, u~)), 3~ ~ Ij Pjkgi(Xk, u~)), Zk = Ij Pjd(Xk, u~)), Xk E A, xk -+ Xo as k -+ 00, where 3 = (31, ... , 3'), 3k = (3t ... ,3;;),

- ( I . 0) _ ( I 0) Th ° 0 ° 0 i 0 j 0 d [ 0] . Z - Z , ... , Z , Zk - Zk,"" Zk . en Zk ~ , Z ~ , 3k ~ ,3 ~ , an Zk IS a bounded sequence. Here we can extract a subsequence, say still [k], and divide the indices j into two classes, according as the sequence [u~)] is or is not bounded, and then we can even assume that u~) -+ IP) E Rm as k -+ 00, or alternatively u~) -+ 00 as k -+ 00. We can extract the subsequence in such a way that we also have Pjk -+ Pj as k-+ 00, 0 :S Pj:S 1,j = 1, ... , h, and thus Ij Pj = 1. For any j of the second category we certainly have fO(Xk, u~)) -+ + 00; hence Pjk -+ Pj = 0, and pjdo(xk> u~)) bounded; hence, by another extraction, Pido -+ ci as k -+ 00. If I', In denote sums extended over the two categories of indices j, we have I' Pj = 1, and for the j of the second category f(Xk, u~)) = Ejdo(Xk, u~)), Eik E RO, with ejk -+ 0 as k -+ 00. Thus,

z = lim Zk = lim(I' + In)Pid(Xk, u~))

= lim I' Pjd(Xk, u~)) + lim In ejk(pjdo(Xb u~)))

= I' pj/(xo, uti)),

and analogously, since fo ~ 0, gj ~ 0 are lower semicontinuous, we also have

-0 > ", I" ( (i)) -i > ", I" ( (j)) . - 1 Z -L. PjJoxo,u, 3 -L. PjJoxo,u , 1- , ... ,r.

This shows that (zo, 3, z) E R(xo), that is, property (K) is proved. Let us prove now that the sets R(x) = co Q(x), which are necessarily convex, have

property (Q) at Xo. To this purpose we note that any set co R(x, 8) is the union of all points (ZO, 3, z) E Rl +r+o of the form (zo, 3, z) = Iy2y(z~,3y, Zy) with 2y ~ 0, y = 1, ... , V,

IyAy= 1, and (z~,3y,z)ER(xy), xyEN.(x), xEA. Thus, z~~n, 3~~gr, Zy=j*, where f~, g*, f* are computed at some (xy,P y, vy), Py E r, Vy E (U(Xy) t, and we can take any fixed integer v ~ 2 + n + r. For Py = (pyj, ... , PyJ, Vy = (uyl>' .. , Uyh), Ijpyj = 1, Uyj E U(xy),j = 1, ... , h, we have

Z~ ~ f~(xy, Py, vy) = I pyjfo(Xy, Uy), j

3~ ~ gr(xy, Py, Vy) = I Pyigj(xy, Uy), j

Zy = f*(xy,py, vy) = I Pyj/(xy, Uy), j

i = 1, ... , r,


and we can take any fixed integer h ~ 2 + n + r, say h = v. Then

ZO = L AyZ~ = L L Aypyjfo(xy, uy), y j

i = 1, ... , r, y i

y i

where Ly Li AyPYi = 1. In other words co R(xo, e) can be written in terms of the original functions fo, g, f instead ofthe functions f~, g*, f*, provided we take into consideration suitable convex combinations of v2 original points (z~i' 3Yi' Zy). By repeating the same argument as above, we prove that the sets R(x) have property (Q) at Xo. The details of the proof, which is similar to the one for (10.S.i), are left as an exercise for the reader.

10.6 Closure Theorems for Orient or Fields Based on Weak Convergence

In the closure theorems of the present section, it is convenient to treat t = (tl, ... ,tV) as v-dimensional, v ~ 1, varying in a fixed measurable subset G of RV. We shall only require the convergence in measure in G of certain measurable functions x(t) =

(Xl, ... ,x") that we shall call trajectories, and the weak convergence in LI (G) of certain functions W) = W, ... ,e') that will take the place of the derivatives. This added generality does not complicate either the statements or the proofs of the theorems. These theorems, which we shall use here for more existence theorems for one dimensional problems, will be used elsewhere for multidimensional problems. Thus, for every t E

G c W a subset A(t) of the x-space R' is assigned, and we take A = [(t,x)/t E G, x E

A(t)] C RV +". For every (t, x) E A a subset Q(t, x) of the z-space R" is assigned and we take Mo = [(t,x,z)/(t,x)EA, ZE Q(t,x)] c W+"+'. We shall need in this section the Banach-Sachs-Mazur theorem (1O.l.i).

1O.6.i. Let G be measurable with finite measure, and assume that, for almost all lEG, the set A(I) c R" is closed, and the sets Q(f,x) c R' are closed and convex, and have property (K) with respect to x in A(I). Let W), x(t), ek(t), Xk(t), t E G, k = 1, 2, ... , be measurable functions, e, ek E (LI(G»" and

(10.6.1) t E G (a.e.), k = 1, 2, ... ,

where ek -+ e weakly in (LI(G»' and xk(t) -+ x(t) in measure in G as k -+ 00. Then

(10.6.2) x(t) E A(t), e(t) E Q(t, x(t», t E G (a.e.).

Remark 1. Statement (1O.6.i) still holds even if we allow some ofthe sets A(t) and Q(t,x) to be empty. Precisely, we shall require that for almost all lEG the sets A(I) are not empty and closed, and we still require that for almost all lEG the map x -+ Q(f, x) has property (K) (with respect to x) in the closed set A(I), thus involving possible sets Q(f, x) which may be empty. Still we require Xk(t) E A(t), et(t) E Q(t, Xk(t» a.e. in G, and thus the sets Q(t,Xk(t» must be nonempty for a.a. t E G. The conclusion is still x(t) E A(t), W) E Q(t, x(t» a.e. in G, and thus the sets Q(t, x(t» must be nonempty for a.a.

10.6 Closure Theorems for Orientor Fields Based on Weak Convergence 341

t E G. As we mentioned in Section 8.5, Remark 1, the definitions of properties (K) and (Q) for set valued maps hold unchanged even if some of the sets are empty.

Proof of (1O.6.i). Let To be the possible set of measure zero of all t such that A(t) is not closed. By a suitable extraction we may well assume that xk(t) -> x(t) pointwise, a.e. in G. Let To be the subset of measure zero of all t E G where xk(t) does not converge, or it does not converge to x(t), or x(t) is not finite. Then, x(t) E A(t) for all t E G - (To u To). The sequence ~k(t), t E G, k = 1,2, ... , converges weakly in (LI(G»)' to ~(t). By the Dunford-Pettis theorem (1O.3.i) (equivalences of (a), (b), and (d», there is a function h«() ;;:: 0, ° ::::; ( < + 00, with h(W( -> + 00 as ( -> + 00, such that the sequence of scalar functions Pk(t) = h(l~k(t)l) ;;:: 0, t E G, k = 1,2, ... , is weakly convergent in L1(G), say to some function p(t) ;;:: 0, t E G, P E LI(G). As stated in Section 10.3, it is not restrictive to assume that h is monotone nondecreasing, continuous, and convex. Now, for any s = 1,2,3, ... , the sequence Ps+k' ~s+b k = 1,2, ... , converges weakly to p, ~ in (L1(G)r+ I. By the Banach-Saks-Mazur theorem (lO.1.i), there is a set of real numbers c~l;;:: 0, k = 1,2, ... , N, N = 1,2, ... , with If~ j c~l = 1, such that if

N N

pW(t) = I c~lPs+k(t), ~~)(t) = I c~l~s+k(t), t E G, N = 1, 2, ... , k~j k~j

then ~W(t) -> W) strongly in (Lj(G»)' and pW(t) -> p(t) strongly in Lj(G), and this is true for every s = 1, 2, .... Then, for every s, there is also a subset Ts of measure zero of points t E G, and a sequence of integers N~), A = 1,2, ... , with N~) -> 00, such that for t E G - T" ~(t) and ~(t) are finite, and ~~~(S) -> W) and p~~(S) -> p(t) as A -> 00. Let T denote the subset of measure zero in G which is the union of all sets To, To, T" s = 1,2, ....

Now letto be any pointto E G - T, and take Xo = x(to). Then (to, Xk(t O» -> (to, x o) E A, and given e > ° there is some So such that Ixs(to) - xol ::::; e for all s ;;:: So. For s ;;:: So we have now

~s+k(to) E Q(to,Xs+k(to», IXS+k(to) - xol::::; e,

We consider now the sets Q*(t, x) c: Rn + j defined by

Q*(t, x) = [(ZO, z) I ZO ;;:: h(izil, z E Q(t, x)],

k = 1,2, ....

(t,x) E A.

Since Q(to, xo) is closed and convex, and h is monotone nondecreasing, continuous, and convex, then the set Q*(to, x o) is also closed and convex. By (lO.S.i), second part, the sets Q*(to, x o) have property (Q) with respect to x at Xo. Now Pk(tO) = h(l~k(to)I); hence

(Ps+k(to), ~s+k(to» E Q*(to, XS+k(to»

(pW(to), ~~)(to» E co Q*(to; x o, e),

where Q*(to; x o, e) denotes the union of all Q*(to, x) with x E A(to), Ix - xol ::::; e, and the last relation holds for all Nand s ;;:: So. Finally, for N = N.! and A -> 00 we have

(p(to), Wo» E cl co Q*(to;xo, e)

where e > ° is arbitrary. Hence, by property (Q) of the sets Q*(to, x) with respect to x at X o, we also have

Hence,


We have proved that, for almost all t E G, we have

x(t) E A(t), ~(t) E Q(t, x(t», t E G (a.e.). o

The following statement, easier to prove than (1O.6.i), will also be used.

lO.6.ii. With the same notation as for (1O.6.i), let G be measurable and of finite measure, and let us assume that, for almost all Y, the sets Q(Y, x), x E A(T), are closed and convex. Let ~(t), x(t), ~k(t), ~k(t), t E G, k = 1,2, ... , be measurable functions, ~, ~b ~k E (Ll(G»', x(t) E A(t), ~k(t) E Q(t,x(t», t E G (a.e.), k = 1,2, ... , with ~k -+ ~ weakly in (L 1(G»" bk(t) = ~k(t) - ~k(t) -+ 0 weakly in (Ll(G»'. Then W) E Q(t, x(t», t E G (a.e.).

Proof. Since ~k -+~, bk -+ 0 weakly in (L 1(G»" then ~k -+ ~ weakly in (L 1(G»" and (~bbJ -+ (~, 0) in (L 1(G»2,. By (lO.l.i) there is a set of real numbers CNk ~ 0, k = 1, ... , N, N = 1,2, ... ,r.f= 1 CNk = 1, such that, if

N N

~~(t) = r. CNk~k(t), b~(t) = r. cNA(t), t E G, N = 1, 2, ... , k= 1 k= 1

then ~~(t) -+ ~(t), b~(t) -+ 0 strongly in (Ll(G»'. Thus, there is a subsequence [N).] such that ~~)t) -+ W), b~)t) -+ 0 pointwise in G. Let To denote the set of measure zero of all t for which this does not occur, or ~(t) is not finite, or Q(t, x(t» is not closed or is not convex. If ~~(t) = r.f= 1 CNk~k(t), t E G, then, for N = N)., t E G - To, we have ~~(t) = ~~(t) - b~(t), ~~(t) -+ ~(t) as N -+ 00. Since ~k(t) E Q(t, x(t» for all k, and Q(t, x(t» is a closed and convex set, we also have ~~(t) E Q(t, x(t» for N = N A and as A. -+ 00, also ~(t) E Q(t, x(t» for all t E G - To, that is, a.e. in G. This proves (1O.6.ii). 0

10.7 Lower Closure Theorems for Orientor Fields Based on Weak Convergence

We shall use essentially the same notation as in (10.6.i). Thus, points in RV ,

R", R1+', and R2 +, spaces will be denoted by t = (t1, ... ,tV), x = (Xl, ... ,x"), (ZO, z) = (ZO, Zl, ... ,z') or ('1,~) = ('1, ~1, ... ,~'), and (v, zO, z) or (p, '1, ~). Let G be any measurable subset of the t-space RV of finite measure, for every t E G let A(t) be a given nonempty subset of the x-space R", and let A = [(t, x) I t E G, x E A(t)J. For every (t, x) E A let Q(t, x) be a given subset of the zOz-space R'+ 1. We denote by x(t) = (Xl, ... ,x"), ('1(t), W» = ('1, ~1, ... ,~'), t E G, given functions from G to R", R'+ 1 respectively.

lO.7.i (A LOWER CLOSURE THEOREM FOR ORIENTOR FIELDS). Let G be measurable and of finite measure, and assume that for almost all Y E G, the set A(Y) is closed and that the sets Q(Y, x) are closed and convex, and have property (K) with respect to x in A(y)' Let ~(t), x(t), '1k(t), ~k(t), Xk(t), A(t), Ait), t E G, k = 1,2, ... , be measurable functions, ~, ~k E (L 1(G»" '1k E L1(G), with Xk ~ x in measure on G, ~k ~ ~ weakly in (L1(G»' as k ~ 00,

t E G (a.e.), k = 1, 2, ... ,

10.7 Lower Closure Theorems for Orientor Fields Based on Weak Convergence 343

(10.7.2)

(10.7.3)

- 00 < i = liminf l1]k(t)dt < + 00, k-+ 00 G

1]k(t) ~ Ait), A,Ak E L 1(G), Ak -d weakly in L1(G).

Then there is a function 1](t), t E G, 1] E L 1(G), such that

(10.7.4) x(t) E A(t), (1](t), ~(t» E Q(t, x(t», t E G, fG 1](t) dt ~ i.

Proof of (10.7.i). We give first the general proof in which certain auxiliary sets Q'*(t, x) are constructed which have property (Q) with respect to x. Immediately afterwards we sketch the drastically simpler proof for the case in which the original sets Q(t, x) already have property (Q) with respect to x.

Let To be the set of measure zero of all t E G for which A(t) is not closed. Let jk = J G 1]it) dt, k = 1, 2, .... By taking a suitable subsequence we may well assume that jk ~ i and Xk(t) ~ x(t) pointwise a.e. in G as k ~ 00. Here - 00 < i < + 00, so that if bs denotes the maximum of Ijk - il for k ~ s + 1, we have b. ~ 0 as s ~ 00.

Let To be the subset of measure zero of all t E G for which A(t) is not closed, or where Xk(t) does not converge, or it does not converge to x(t), or x(t) is not finite. Then x(t) E A(t) for all t E G - (To u To). The sequence Ak(t), ~k(t), tE G, k = 1,2, ... , converges weakly to A(t), ~(t) in (L 1(G)y+1. By the equivalence theorem (1O.3.i) there is a function h(O ~ 0, 0 ~ ( < + 00,

with h(W( ~ + 00 as ( ~ + 00, and such that the sequence of scalar functions Pk(t) = h(l~k(t)1 ~ 0, t E G, k = 1, 2, ... , also is weakly convergent in L 1(G), say to some function p(t) ~ 0, t E G, P E L1(G).

It is not restrictive to assume that h is monotone nondecreasing, continuous, and convex. Now, for any s = 1, 2, ... , the sequence Ps+k' AS+k' ~s+k' k = 1,2, ... , converges weakly to p, A, ~ in (L 1(G»),+2. By the BanachSaks-Mazur theorem (10.l.i), there is a set of real numbers c~t ~ 0, k = 1, ... , N, N = 1,2, ... , with ~J=1 c~t = 1, such that if

N N

pW(t) = L C~~PS+k(t), A~)(t) = L C~~AS+k(t), k=1 k=1

N

~W(t) = L C~~~s+k(t), t E G, N = 1, 2, ... k= 1

then (pW,AW,~W)~(p,A,~) strongly in (L 1(G»r+2, and this is true for every s = 1, 2, .... Then, for every s, there is also a subset Ts of measure zero of points t E G, and a sequence of integers NIS), I = 1, 2, ... , with NlS) ~ 00,

such that for t E G - T., p(t), A(t), ~(t) are finite and (with simplified notation) (p~~(t), A~~(t), ~~~(t» ~ (p(t), A(t), W» as I ~ 00. Let T denote the subset of measure zero in G which is the union of all To, To, T., s = 1, 2, .... Let us take

N

1]~)(t) = L C~~1]s+k(t), k=1


and note that

Pk(t) = h[lek(t)I], l1k(t) ~ 2k(t), t E G, fG l1k(t) dt= jk,

so that, for all S = 1,2, ... , N = 1,2, ... , we also have

k=1, 2, ... ,

(10.7.5) l1W(t) ~ AW(t), t E G, i - bs ::; fG l1~)(t) dt ::; i + bs·

For N = N1S) and I- 00, the relations (10.7.5) and by Fatou's lemma (S.7.i) imply

l1(S)(t) = lim inf l1~:(t) ~ 2(t), tE G - T, 1-+ 00

S = 1,2, ....

Thus, l1(S)(t) is finite a.e. in G and of class Ll(G). Let T~ denote the set of measure zero of all points t E G where l1(s)(t) is

not finite. Finally, if

l1(t) = lim inf l1(s)(t), t E G, s-+ 00

then again we have l1(t) ~ 2(t), t E G, SG l1(t)dt::; i. Note that, for t E G - T., 2(t) is finite. Thus, for 1 sufficiently large, say

1 ~ lo(t, s), we certainly have l1~:(t) ~ 2(t) - 1. In other words, we may drop from the sequence NjS), 1 = 1, 2, ... , enough initial terms (finitely many, depending on t and s) in such a way that the relation l1W(t) ~ 2(t) - 1 holds for I. Also, l1(t) is finite a.e. in G and of class Ll(G). Let To denote the set of measure zero of all points t E G where 11 is not finite.

Let T* denote the set of measure zero in G which is the union of all sets To, T~, To, T., T~, s = 1,2, .... Let to be any point to E G - T*, and take Xo = x(to). Then (to, Xk(tO» - (to, xo) E A, and, given e > 0, there is some So such that IXs(to) - xol ::; e for all s ~ So. For s ~ So we have

(l1s+k(to), es+k(to» E Q(to, XS+k(to», IXS+k(to) - xol ::; e, k = 1, 2, ....

We consider now the sets Q*(t, x) c W+ 2 defined by

Q*(t, x) = [(v, y, z) v ~ h(lzl>, (y, z) E Q(t, x)],

and we also need the sets

(t,x) E A,

Q'(t,x) = [(y,z)ly ~ 2(t) - 1,(y,z) E Q(t,x)] c Rn+l, Q'*(t, x) = [(v, y, z) v ~ h(izl), y ~ 2(t) - 1, (y, z) E Q(t, x)] c Rn+ 2.

For each t fixed, the sets Q'(t, x), Q'*(t, x), X E A(t), are the intersections of Q(t, x), Q*(t, x) with the fixed closed sets [(y,z)ly~}(t)-1,ZERn], [(v, y, z)ly ~ 2(t) - 1, (v, z) E Rn+ 1]. Then certainly the se~s Q'(to, x), x E A(to), have property (K) with respect to x at Xo, since the sets Q(to, x) already have this property. Since h is monotone nondecreasing, continuous, and convex, the sets Q*(to, xo) and Q'*(to, xo) are convex. Finally, for t = to, we can apply


the second statement of (10.5.ii) to the sets Q'*(to, x) with Jl = 1, To(x, y, z) = h(Jzi). The present variables v, y, z ,Eeplace the variables (yO, y, z) ofthe second part of (10.5.ii); the present sets Q'(to, x) (for to fixed) replace the sets Q(x); and the present sets Q'*(to, x) for to fixed replace the sets Q(x). Also, the present function h(izi) and constant A(to) - 1 replace the function q,(izi) and constant L of the second part of (1O.5.ii). Here T o(x, y, z) = h(jzi) is constant with respect to x, y and continuous in z, and thus certainly lower semi continuous in (x, y, z) as required in (10.5.ii). We conclude that the sets Q'*(to, x), x E A(to), have property (Q) with respect to x at x = Xo. By (10.7.1) and the definitions of Pk and Q*(t, x) we have now, for S ~ So,

(P.+k(to), I1.+k(to), e.+k(to)) E Q*(to, x.+k(to)),

and hence

(tl c~lp.+k(to), Jl C~l11.+k(to), ktl c~le'+k(to)) E co Q*(to;xo,e).

Finally, for N = N\'), 1 ~ lo(to, s), we have I1~)(to) = If= 1 C~l11.+k(to) ~ A(to) - 1, and hence

(10.7.6) (Jl c~lp'+k(to), Jl C~l11.+k(to), ktl c~le'+k(to)) E co Q'*(to;xo,e).

As 1--+ 00, the points in the first member of this relation form a sequence possessing (p(to), I1(')(to), Wo)) as an element of accumulation in Rn+ 2

(all p(to), I1(')(to), Wo) finite). Thus

(10.7.7) (p(to), I1(')(to), Wo)) E cl co Q'*(to; Xo, e), s ~ So.

Note that I1(to) = lim inf.--+ 00 I1(')(to) is finite, so that (p(to), 11 (to), Wo)) is a point of accumulation of the sequence in the first member of (10.7.7), while the second member is a closed set. Thus,

(P(to),l1(to),Wo)) E clco Q'*(to;xo,e).

Since e > 0 is arbitrary, by property (Q) we have

(P(to),I1(to), Wo)) E n clco Q'*(to;xo,e) = Q'*(to,xo). e>O

By the definition of Q'*(to, xo) we then have

(I1(to), Wo)) E Q(to, xo).

We have proved that for almost any t E G we have x(t) E A(t), (I1(t), W)) E

Q(t,x(t)), and that 11 E Ll(G) with JG I1dt::;; i. Theorem (10.7.i) is thereby proved. 0

Second proof of (10.7.i). We assume here that for almost all I the sets Q(I, x) have property (Q) with respect to x. We proceed as in the proof above omitting the recourse to the equivalence theorem. Then we apply the BanachSaks-Mazur theorem to the sequence A.+k, e.+ko k = 1,2, ... , obtaining AW(t),


~W(t) as before, and then the functions 11W(t) and 11(t). Now relations (10.7.6) hold as before, and then

(tl c~l11s+k(to), Jl c~l~'+k(to») E co Q(to;xo,e).

From here we derive in succession as before

By property (Q) then

(11(s)(t), Wo» E cl co (2(to; xo, e),

(11(to), Wo» E cl co (2(to; xo, e).

(11(to),Wo»E n clco Q(to;xo,6) = Q(to,xo) 8>0

which proves the theorem. o Remark 1. Statement (1O.7.i) still holds even if we allow some ofthe sets A(t) and Q(t,x) to be empty. Precisely, we shall require that for almost all lEG the sets A(I) are not empty and closed, and we still require that for almost all 7 E G the map x ..... Q(T,x) has property (K) (with respect to x) in the closed set A(I), thus involving possible sets Q(T, x) which may be empty. Still we require xk(t) E A(t), (P/k(t), ~it» E Q(t, xit» a.e. in G, and thus the sets Q(t,Xk(t» must be nonempty for a.a. t E G. The conclusion is still x(t) E A(t), (p/(t), W» E Q(t, x(t» a.e. in G, and thus the sets Q(t, x(t» must be nonempty for a.a. t E G. (Cf. the analogous Remark 1 of Section 10.6).

In Section 10.8 we shall prove a partial inverse of (1O.7.i) showing, in particular, that (10.7.3) is not only a sufficient condition for lower closure, but also essentially a necessary one.

In this formulation of (10.7.i), Ak ...... A weakly in L 1(G); hence IIAkill is a bounded sequence, and the part i> - 00 of the requirement (10.7.2) is actually a consequence of(1O.7.3). The lower closure theorem (10.7.i) will be used in situations where it is known that - 00 < i < + 00 and where functions A, Ak satisfying (10.7.3) are easily found. Here we list simple alternative conditions, (Al)-(A7), under each of which functions A, Ak satisfying (10.7.3) can be immediately derived. Here we denote by (a, b) the inner product in Rft.

Ap There is a real valued function t/J(t) ~ 0, t E G, t/J E L 1(G), such that for every (t, x) E A, (y, z) E (2(t, x), we have y ~ - t/J(t).

Indeed, then we have 11k(t) ~ Ak(t) = - t/J(t), t E G, k = 1,2, ... , A = - t/J.

A1• There is a real valued function t/J(t), t E G, t/J E L 1(G), and a constant y ~ ° such that for all (t, x) E A and (y, z) E Q(t, x) we have y ~ - t/J(t) - ylzl.

Indeed, then we have 11it) ~ Ak(t) = - t/J(t) - YI~k(t)l, t E G, k = 1,2, .... Since ~k ...... ~ weakly in (L 1 (G»' by hypothesis, by the Dunford-Pettis theorem (cf. Section 10.3) we know that the same functions ~k are equiabsolutely


integrable in G. Hence, by the same Dunford-Pettis theorem, the sequence /~k(t)/, t E G, k = 1,2, ... , is weakly compact in L l(G), and there is, therefore, a subsequence [ksJ such that Ak,(t) = - t{!(t) - Y/~ks(t)/, t E G, s = 1,2, ... , is weakly convergent in Ll(G) to some function -t{!(t) - yO"(t), O"(t) ~ 0, t E G, and we can apply (1O.7.i).

A 3• ~k E (Lq(G))', 1 ~ q ~ + 00, ~k ~ ~ weakly in (LiG))" and there are a real valued function t{!(t) ~ 0, t E G, t{! E L l(G), and an r-vector function ¢(t), t E G, ¢ E (Ls(G))', lis + 11q = 1, such that for all (t, x) E A and (y, z) E

Q(t, x) we have y ~ - t{!(t) - (¢(t), z).

Note that if ~k ~ ~ weakly in (Lq(G))', then Ak(t) = - t{!(t) - (¢(t), ~k(t)) converges weakly in Ll (G) to - t{! - (¢, ~).

A4 • There are constants rx ~ [3 real and y > ° such that (a) for every (t, x) E A and for every /z/ ~ y there are points (y, z) E Q(t, x), and for all such points y ~ [3; (b) for every (t, x) E A there is some point (Yo, 0) E Q(t, x) with Yo ~ rx.

In other words, for every (t, x) E A the projection Q(t, x) on the z-space Rn of Q(t, x) contains the whole ball /z/ ~ y, and for all (y, z) E Q(t, x) with /z/ ~ y we have y ~ [3. Moreover, for every (t, x) E A there is some point (Yo,O) E Q(t, x) with Yo ~ rx. Now, if (y, z) is any point of Q(t, x) with /z/ > y, then Z1 = yzl/z/ has distance y from the origin and is interior to the segment Oz between 0 and z, with Z1 = (1 - 0")0 + O"Z, where 0" = yl/z/. Then, by the convexity of Q(t, x) there is some Y1 such that (Y1,zd E Q(t, x) and [3 ~ Yl ~ (1 - O")Yo + O"Y, or

y~ 0"-1Yl - 0"-1(1- O")Yo ~ [3y-l/Z/- rxy-l/ z/(l- y/Z/-1)= rx + y-1([3 - rx)/z/,

and we are in the situation discussed under (A z).

As. x, Xk E (Lp(G)f, I/Xk - xl/ p ~ ° as k ~ 00 for some p, 1 ~ P < + 00,

~'~k E (L l (G»', ~k ~ ~ weakly in (L 1(G»', and there are a real valued function t{!(t) ~ 0, t E G, t{! E L 1(G), and constants y, y' ~ ° such that for all (t,x) E A, (y,z) E Q(t,x) we have y ~ -t{!(t) - y'/x/P - y/z!.

The argument is similar to the one under (A 3 ), since I/Xk - xl/ p ~ ° implies 1/ /Xk/ P - /xl p l/ l ~ 0, and then the sequence Ak(t) = - t{!(t) - ilxk(t)/P -Y/~k(t)l, t E G, k = 1,2, ... , certainly possesses a weakly convergent subsequence in Ll(G). Instead ofthe requirement x, Xk E (Lp(G)t, I/xk - xl/ p ~ 0, we may require xL Xi E Lpi(G), I/x~ - XiI/Pi ~ ° as k ~ 00 for different Pi' 1 ~ Pi < 00, i = 1, ... ,n. This remark holds throughout the present and next chapters.

A6 • Xk E (Loo(G)t, I/Xkl/oo ~ Lo, ~k E (Loo(G))', I/~kl/oo ~ L1 for given constants Lo, Lb and there are a real valued function t{!(t) ~ 0, t E G, t{! E L 1(G),


and a real valued monotone nondecreasing function a(~), 0 :::;; ~ < + 00,

such that for all (t, x) E A, (y, z) E Q(t, x) we have y;;::: - !/J(t) - a(ixi + izi).

The argument is similar to the one above.

11 7, Let 1:::;; p < + 00, 1 < q :::;; + 00, Xk E Lp(G), ~k E LiG), iixkiip:::;; Ll ,

ii~kiiq :::;; L2 for some constants L t , L 2 , and assume that there are a constant 1> ;;::: 0, a real valued function !/J(t) ;;::: 0, t E G, !/J E Lt(G), and a Borel measurable function p(t,x):G x R"-+Rr such that for all (t,x)EA, (y,Z)EQ(t,X), and l/s + 1/q = 1, we have

y ;;::: - !/J(t) - 1>ixiP - (p(t, x), z) and ip(t, x)i S :::;; 1>ixip + !/J(t).

Here we assume that Xk -+ x strongly in (Lp(G)t and that ~k -+ ~ weakly in (Lq(G))'.

Since Xk -+ x strongly in (Lp(G)t, then the functions iXk(t)jP are equiabsolutely integrable, and so are the functions ip(t, Xk(t) )is. Since ~k -+ ~ weakly in (Lq(G))r, then ii~kiiq is a bounded sequence, and by the Holder inequality, the sequence (p(t, Xk(t)), ~k(t)), t E G, k = 1, 2, ... , is also equiabsolutely integrable, and so is the sequence Ak(t) = - !/J(t) - 1>iXk(t)jP - (p(t, xk(t)), ~k(t)), t E G, k = 1, 2, . . . . Thus, [AkJ contains a subsequence which is weakly convergent in Ll(G).

Remark 2. Note that in view of (10.3.i), whenever ~k --> ~ weakly in L l , there is some real valued Nagumo function 'P(O, ° :::;; , < + 00, with 'P(Og --> + 00 as ,-> + 00, such that 'P( I~k[) is weakly convergent in L l , and then we could require that ZO ;<: -Iji{t) -c5lxlP - 'P(lzil·

Remark 3. The reader should note that there is a statement similar to (As) which is not true. Indeed, let us assume in (As) that ~k E (LiG»)' for some q> 1, and that (*) y;<: -1jJ(t) - c5'lxIP - c5lzlq. Under this assumption we cannot conclude lower closure. This can be seen by the following example. Take G = (0,1), A(t) = R, n = r = 1, q > 1, 1/s+1/q=1,f0(t,x,Z)=S-llt-IXls+t-Ixz, and Q(t,x)=[(y,z)ly~fo,zER]. By the elementary relation rxf3:::;; s-lrxs + q-I{3'l, 11., f3 ~ 0, we derive that for 11. = It-lxi, f3=lzI, that s-IICIXls+q-Ilzlq~lt-IXllzl, and hence fO=S-llt-IXls+t-IxZ~ s-llt-lxls - t-1lxllzl ~ _q-Ilzlq. This shows that the relation (*) holds for IjJ == 0, c5' = 0, c5 = q - I.

On the other hand, let us take Xk(t) = tk l /s, ~k(t) = - k1/q for 0< t:::;; k- l, and Xk(t) = 0, ~k(t) = ° for k- l < t < 1. Then Xk --> ° uniformly in (0,1), and ~k --> ° weakly in Lio, 1). Then I'lk(t) = fo(t, Xk(t), ~k(t» = - q-Ik for ° < t:::;; k- l, I'/k(t) = ° for k- l < t < 1, and Sb I'/k dt = _q-l < 0, for all k, while I'/o(t) = fo(t, 0, 0) = 0, and Sb 1'/0 dt = 0. Note that here ~k -> ~o weakly in Lq does not imply that I~klq --> I~olq weakly in Ll. On the other hand, if ~k --> ~o strongly in Lq, then of course I~klq -> I~olq strongly in LI. The inequality (*) would be sufficient with q = 1.

Remark 4. In Theorem (1O.7.i) the requirement (10.7.3) cannot be disregarded, even if we replace (10.7.2) by the stronger requirement I'/k(t) :::;; Mo. This can be shown by the


following simple example. Take v = n = r = 1, 0;5; t;5; 1, 0;5; x;5; 1; Q(t,x) = [(zO,z)lzO ~ 0, z = 0] if 0;5; t < 1, 0;5; x;5; 1, t + x < 1; Q(t, x) = [(zO,z)lzO ~ -x-I, z = 0] if 0;5; t < 1, 0 < x;5; 1, t + x ~ 1; Q(I,x) = R x {O}. Then, all sets Q(t, x) are closed half straight lines, or lines, and have property (K), and even property (Q) with respect to x everywhere. Let us take ~k(t) = ~(t) = 0, Xk(t) = k- 1, x(t) = 0, 0;5; t ;5; 1, 'Ik(t) = 0 for 0;5; t ;5; 1 - k- 1 , and for t = 1, 'Ik(t) = - k for 1 - k- 1 < t < 1. Then fA 'Ik(t)dt = -1, k = 1,2, ... , i = -1. For x(t) = 0,0;5; t;5; 1, we must have 'I(t) ~ 0 for all 0;5; t < 1. Hence, fA 'I(t) dt ~ 0, and the last relation (10.7.4) cannot be satisfied.

Remark 5. Note that in the lower closure theorem (1O.7.i) no property (Q) was required for the given sets Q(t, x) in R" + 1. This is possible because we have assumed the weak convergence in Ll(G) ofthe functions ';k(t), t E G, k = 1,2, ... , and this implies, by the equivalence theorem (10.3.i), implication (a) ~ (c), that there exists some function rf>(C), o ;5; C < + 00, bounded below, with rf>(WC -+ + 00 as C -+ + 00, and fG rf>(I~k(t)j) dt ;5; M for all k. In tum, having assumed that the functions ek(t), t E G, k = 1, 2, ... , have their values in a finite dimensional space R', Caratheodory's theorem (S.4.iii) holds, and by our theorem (1O.5.i), the auxiliary sets Q'*(t,x) c Rn+2 have property (Q) with respect to x in A(t) (for almost all t).

We have shown that the proof of (10.7.i) is very much simplified if we know that the original sets Q(t,x) already have property (Q) with respect to x.

The following statement, easier to prove than (1O.7.i), will also be used:

lO.7.ii. With the same notation as for (10.7.i), let G be measurable and of finite measure, and let us assume that, for almost all I E G, the sets Q(I, x), x E A(I), are closed and convex. Let W), x(t), 'Ik(t), iMt), ~k(t), ~k(t), A(t), Ait), t E G, k = 1,2, ... , be measurable functions, e, ';t, ~k E (L 1(G»" 'Ik' "Fik E L 1(G), with ';k(t) -+ W) weakly in (L 1(G»" bk(t) = ';k(t)~k(t) -+ o weakly in (L 1(G»" b~(t) = 'It(t) - ift(t) -+ 0 weakly inLl(G) as k -+ 00, x(t) E A(t), (,fk(t), 'it» E Q(t, x(t», t E G (a.e.), k = 1, 2, ... ,

- 00 < i = lim inC r 'It(t) dt < + 00, k-oo JG

Then there is a function "I(t), t E G, 'I E L 1(G), such that

('I(t), ';(t» E Q(t, x(t», t E G (a.e.),

Proof· Here ~k -+~, bk -+ 0 weakly in (L 1(G»" Ak -> A weakly in Ll(G); thus (~k>bk' b~,AJ->(e,O,O,A) weakly in (L 1(G»2r+2, and by (10.l.i) there is a set of real numbers CNk ~ 0, k = 1, ... , N, N = 1, 2, ... , ~J= 1 CNk = 1, such that, if

N

(';~(t),b~(t),b~*(t),A~(t» = I CNk(';kA,b~,At), t E G, N = 1,2, ... , k=1

then (';;,b~,b~*,A~)->(';,O,O,A) strongly in (L 1(G»2,+2. Then, there is also a subsequence N;, such that (';:A,b~A,b~~,At)-+(.;,O,O,A) pointwise a.e. in G as A-+ 00. Let To be the set of measure zero where this does not occur, or ';(t), or A(t) are not finite, or Q(t,x(t» is not convex or not closed.If~~(t), ~(t) denote the functions I:= 1 CNkr.k and I:= 1 cNkiik respectively, then for t E G - To and N = N;, we have ~; = .;~ - b~ -+ ';(t), f1~(t) = 'I~(t) - b~*(t) ~ A~(t) - b~*(t) with A~ - b~* -> A strongly in Ll(G). By the


remark after Fatou's Lemma (S.7.i), we know that ,,(t) = lim infA_ex> ftNA(t) is L-integrable in G with ,,(t) ~ A(t), SG ,,(t) dt :5: i. On the other hand, (iMt), ~k(t» E Q(t, x(t» for all t and k, where Q (t, x(t» is a closed convex set. Thus (ft~(t), ~~(t» E (2(t, x(t» for all t E G - To and all N. For N = N A and as A --> 00 we have now (,,(t), W» E (2(t, x(t», and this relation holds a.e. in G. 0

10.8 Lower Semicontinuity in the Topology of Weak Convergence

A. Lower Semicontinuity of Integrals under Weak Convergence

From the lower closure theorem (lO.7.i) in terms of orientor fields we shall now immediately derive a lower semicontinuity theorem (10.8.i) for the integral

(to.8.1) I = fG F o(t, x(t), ~(t» dt, ~(t) E Q(t, x(t», t E G (a. e.),

directly in terms of the function F 0'

The statement (to.8.i) actually concerns lower semicontinuity properties of multiple integrals. This added generality does not complicate the statement of the theorem.

We shall use essentially the same notation as before; in particular, the independent variable t, which is a v-vector t = (t1, ... ,n, v ~ 1, ranges over a bounded domain G ofthe t-space RV. For every t E G let A(t) be a nonempty subset of the x-space Rn, x = (Xl, ... ,xn), and let A be the set A = [(t, x) I t E G, x E A(t)] C Rv + n, whose projection on the t-space is G. For every (t, x) E A let Q(t, x) be a given subset of the z-space R r , z = (Zl, ... ,z'), and let M be the set M = [(t, x, z)l(t, x) E A, z E Q(t, x)] c Rv+n+r. Let F o(t, x, z) be a given real valued function defined on M, and for every (t, x) E A let Q(t, x) denote the set Q(t,x) = [(ZO, z)lzO ~ F o(t, x, z), z E Q(t, x)]. We may extend F ° in Rv+n+r by taking F o(t, x, z) = + 00 for (t, x, z) E Rv+n+r - M. Then F ° is said to be an extended function. For most applications it is sufficient to assume.

C. A closed, M closed, F o(t, x, z) continuous on M.

However, for the proof that follows, the following rather general assumption suffices:

C*. For every e > 0 there is a compact subset K of G such that (a) meas(G -K) < e, (b) the extended function Fo(t,x,z) restricted to K x Rn+ r is Bmeasurable, and (c) for almost all Y E G the extended function F oCf, x, z) of (x,z) has values finite or + 00, and is lower semicontinuous in Rn+r.

Under hypothesis (C), and for any pair of measurable functions x(t) = (Xl, ... ,xn), ~(t) = (~l, ... ,~r), t E G, with x(t) E A(t), ~(t) E Q(t, x(t», t E G

10.8 Lower Semicontinuity in the Topology of Weak Convergence 351

(a.e.), then the function F oCt, x(t), W)) is finite a.e. in G and measurable in G (cf. Hahn and Rosenthal [I, p. 122]). Under hypothesis (C*) and measurable functions x(t), ~(t), t E G, as above, again F oCt, x(t), W)) is measurable in K. Since meas(G - K) < e and e is arbitrary, we conclude that Fo(t,x(t),~(t)) is measurable in G.

Remark 1. Under hypothesis (C*) we may simply assume that F 0 is a given extended function in W+ n+', whose values for almost all t E G are finite or + 00, and that for almost all IE G the set A(I) = [x E Rn\ F o(I, x, z) 1= + 00] =1= 0. For any (I, x) let Q(I, x) =

[zER'\Fo(T,x,z) =1= +00]. Then A is any set of points (t,x) whose projection on the t-space is G and whose sections for almost all t E G are the sets A(t).

Remark 2. If Fo(t, x, z) is a Caratheodory function on G x Rn+" that is, Fo is measurable in t for every (x, z), and continuous in (x, z) for almost all t, then, by (8.3.v), for every '1 > 0 there is a compact set K c: G with meas(G - K) < '1 such that Fo is continuous in K x R"+'. This shows that Caratheodory functions Fo certainly have property (C*). A condition slightly more restrictive than (C*) is often used for the same purpose, namely (Cn the same as (C*) where instead of (b), (c), F o(t, x, z) is required to be lower semicontinuous in K x R"+' as a function of (t, x, z). This more restrictive condition (Cn as proved by Eke1and and Temam [I, p. 216] and by Rockafellar [4, p. 176], is an equivalent form for the "normality" conditions required by these two authors.

In Section 10.7, as a comment on the lower closure theorem (lO.7.i), we noted that the abstract condition (10.7.3) is certainly satisfied under the practical and easily verifiable alternative conditions (AJ Here, in terms of the functional (lO.8.1), some of the assumptions (Ai) are replaced by the following straightforward alternative assumptions (Li):

(LI) There is a real valued function l/!(t) ~ 0, t E G, l/! E LI(G), such that F oCt, x, u) ~ - l/!(t) for (t, x, u) E M and almost all t.

(L2) There is a real valued function l/!(t) ~ 0, t E G, '" E LI(G) and a constant C ~ 0 such that F oCt, x, u) ~ - l/!(t) - qui for (t, x, u) E M and almost all t.

(L3) There is a real valued function l/!(t) ~ 0, t E G, l/! E LI(G) and an r-vector function cP(t) = (cPI""'cP,), tEG, cPiELoo(G), such that Fo(t,x,u)~ - l/!(t) - (cP(t), u) for (t, x, u) E M and almost all t.

(L4) There are constants oc ~ fl real and y > 0 such that (a) for every (t, x) E A the set Q(t, x) contains the ball Izi ~ y in R'; and (b) F oCt, x, u) ~ fl for all (t, x) E A, lui ~ y, and F o(t, x, O} ~ oc. Here we assume explicitly that the sets Q(t, x) are convex and that F o(t, x, u) is convex in u.

Under anyone of these hypotheses (Li)' and for all measurable functions x(t), t E G, and L-integrable ~(t), t E G, as before with x(t) E A(t), ~(t) E Q(t, x(t)), t E G (a. e.}, then F oCt, x(t), W)) is not only measurable in G but also not below some L-integrable function in G. Indeed, under (LI) we have Fo ~ -l/!; under (L2) we have Fo ~ -l/!(t) - q~(t)l; under (L3) we have Fo~-l/!(t)-(cP(t), W»)~-l/!(t)-lcP(t)II~(t)l, so Fo~-l/!(t)-LI~(t)l, where L = ess suplcP(t)l. Under (L4), for every Z E R', Z E Q(t, x), Izl ~ y (if any), we take Zl = YZllzl, so that IZII = y and Fo(t,X,ZI) ~ fl. Moreover, for (J = Y/lzl,


° < (I < 1, and by the convexity of Q and F 0 we have Fo(t,x,ztl::; (1 - O'}Fo(t,x,O) + O'Fo(t,x,z),

or Fo(t,x,z) ~ O'- 1Fo(t,x,Z1) - 0'-1(1- O'}Fo(t,x,O)

~ y-1lzlfJ - y-1Izl(1 - Yl zl-1)a

= a - y-1(a - mlzl, and we are in the situation (L2)' Note that under any of the hypotheses (LJ with x measurable and e E (L 1(G»" the Lebesgue integral JG F o(t, x(t), e(t»dt exists, finite or + 00.

For G bounded and closed, A closed, x(t), t E G, continuous, and xk(t), t E G, k = 1, 2, ... , converging uniformly to x in G, then it is enough to verify the conditions above only for (t,x) E An rlj, (t,x, u) E M, where rlj is a closed bounded neighborhood of the graph r of x in Rv +n• In particular, if F 0 is continuous, then condition (b) of (L4) is always satisfied, since F 0 is bounded in the compact set (A n r Ij) x [lui::; y], say IF 01 ::; c, and we can take a = c and fJ = -c. More particularly, if Q(t, x) = R" for every (t,x), M = A x Rn, then Rn certainly contains the ball lui ::; y, condition (a) of (L4) is also satisfied, and (L4) itself is satisfied.

lO.S.i (A LOWER SEMICONTINUITY THEOREM). Let condition (C) or (C*) be satisfied, and assume that for almost all t E G and all x E A(t) the extended function F o(t, x, z) be convex in z (in R' and hence the set Q(t, x) is convex). Assume that anyone of the conditions (Li) holds. Let W), x(t), ek(t), Xk(t), t E G, k = 1,2, ... , be measurable junctions, e, ek E (L1(G»', such that Xk -+ x in measure in G, ek -+ e weakly in (L 1(G», as k -+ 00, and Xk(t) E A(t), ek(t) E Q(t, Xk(t», t E G (a. e.), k = 1, 2, .... Then, x(t) E A(t), W) E Q(t, x(t», t E G (a.e.), and

(10.8.2) r F o(t, x(t), W» dt ::; lim inf r F o(t, xk(t), ek(t» dt. JG k-+oo JG

Proof. The integrals above exist, finite or + 00. Let i denote the second member of (10.8.2). If i = + 00 there is nothing to prove. Assume i < + 00.

Let us prove that i > - 00. Under condition (L1) this is evident. Since ek -+ e weakly in (L 1(G»" then IlekilLI is bounded, say IlekilLI ::; N. Under condition (L2) then F 0 ~ - '" - Cek' and JG F 0 dt ~ - JG '" dt - CN. Under conditions (L3) we have Fo ~ -'" - Llekl, where L = ess supl"'l, and then JGFodt ~ - Jt/I dt - LN. Under condition (L4) we have again F 0 ~ y-1(fJ - a)lekl + a, and again JG Fodt ~ y-1(fJ - a)N + a meas G.

By taking '1k(t) = FO(t,xk(t),ek(t»,tE G, we have now ('1k(t),ek(t»E Q(t, Xk(t» with Q(t, x) = [(ZO, z) I ZO ~ F o(t, x, z), z E Q(t, x)]. In order to apply (10.7.i) with cl A(t) replacing A(t), we need only prove that, for almost all IE G, these sets Q(f, x) have property (K) with respect to x in the closed set cl A(I). Indeed, under condition (C) and for all Y E G, the set M(Y) = [(x, z)1 x E A(Y), z E Q(T, x)], the section of M with the hyperspace t = T, is closed, and then

M(Y) = [(x, y, z) I x E A(I), y ~ F oCt, x, z), z E Q(Y, x)]


is closed, since F off, x, z) is continuous on the closed set M(f) (hence lower semi continuous on Rn+ r ), and Met) is closed because of (8.5.v). The closed set Met) is the graph of the sets Q(I, x) as x describes the closed set A(I), and then the map x -> Q(I, x) has property (K) (with respect to x) on A(f) by virtue of (8.5.iii) and subsequent Remark 2. From the orient or field relation xk(t) E A(t), (t/k(t), ~k(t» E Q(t, xk(t)), t E G (a.e.), k = 1,2, ... , by applying (10.7.i) we derive that there is an L-integrable function '1(t), t E G, such that x(t) E A(t), ('1(t), ~(t» E Q(t, x(t», t E G (a.e.), and SG '1(t) dt ::; i.

Under condition (C*) and for almost all IE G, the set

A(I) = [x E Rn I F 0(I, x, z) 1= + CXJ]

is not empty by hypothesis, and the sets Q(I, x) = [z E Rr I F o(I, x, z) #- + CXJ] are not empty for x E A(I). Now cl A(I) is not empty and closed, but the sets Q(I,x) for x E (cl A(I) - A(I) are empty. Again, for almost all IE G, the extended function F o(I, x, z) is lower semicontinuous in Rn+r; hence by (8.5.v) the sets Q(I, x) = epi F o(I, x, z) are closed. Again by (8.5.iii) and subsequent Remark 2, the map x -> Q(Y, x) has property (K) (with respect to x) on the closed set cl A(f), (and this involves also the empty sets Q(I, x) for x E (cl A(t) - A(t»). From the orientor field relations

t E G (a.e.), k = 1, 2, ... ,

by applying (1O.7.i) with cl A(t) replacing A(t) we derive that there is an L-integrable function '1(t), t E G, such that x(t) E cl A(t), ('1(t), ~(t» E Q(t, x(t», t E G (a.e.), and SG y/(t)dt::; i. Now for x E (clA(t) - A(t» the set Q(t,x) is empty. Thus, for almost all t E G we must have Q(t, x(t» nonempty, hence x(t) E A(t), t E G, (a.e.).

In any case F o(t, x(t), ~(t»::; Y/(t), t E G (a. e.), and we know that F o(t, x(t), ~(t» is measurable and not less than some L-integrable function in G. That is, Fa is between two L-integrable functions, and then L-integrable. Moreover

fG Fo(t,x(t),~(t»dt::; fG '1(t)dt::; i.

Theorem (1O.8.i) is thereby proved. o

Remark 3. Note that, under the conditions of (lO.8.i), the function F o(t, x(t), W)) is certainly measurable, and because of the conditions (L i), the Lebesgue integral I[ x,~] =

JG Fa dt on the left hand side of (10.8.2) is either finite or + 00. Theorem (1O.8.i) can be completed with the statement that, if I[ x,~] = + 00, then the relation (10.8.2) is still valid in the sense that on the right hand side necessarily we have lim I[xk, ~kJ =

+ 00 as k -> + 00. Indeed, otherwise, there would be a subsequence, say still [k], with - 00 < i = lim I[ x k , ~kJ < + 00, and by (1O.7.i) there would be an L-integrable function ry(t), t E G, with JG ry(t) dt :-s; i, and ry(t) 2: F oft, x(t), W), a.e. in G, a contradiction.

As a particular case, the theorem (1O.8.i) contains, for v = 1, the case of integrals of the form


We need consider the situation in which Xk(t), ~k(t), tlk ::::;; t::::;; t2k , may be defined on different intervals, x(t), ~(t), tl ::::;; t::::;; t2, and that tlk --. t1, t2k --. t2 as k --. 00. We shall assume that ~k --. ~ weakly in L 2 , and by this we understand that all~, ~k are extended to some large [to, T] (containing all [tlk' t2k]) by taking them equal to zero outside their original intervals of definition, and that ~k --. ~ weakly in (L 1[to, T])'. We shall also assume that X k --. x in measure, and by this we understand that we have performed an analogous extension, and that Xk --. x in measure in [to, T]. Alternatively, we may take x(t) = X(tl) for t::::;; t1> x(t) = x(t2) for t ~ t2, and analogously for Xk' This extension is more natural when all x, Xk are continuous in their intervals of definition and the convergence is in the p-metric. With these conventions, which will be used from now on, the following theorem holds.

IO.S.ii (A LOWER SEMI CONTINUITY THEOREM FOR AC TRAJECTORIES IN THE

TOPOLOGY OF THE WEAK CONVERGENCE OF THE DERIVATIVES). Let condition (C), or (C*) be satisfied, and assume that for almost all t and all X,E A(t) the extended function F o(t, x, z) be convex in z (in R" and hence the set Q(t, x) is convex). Assume that any of the conditions (Li) holds. Let x(t), W), tl ::::;; t::::;; t2, Xk(t), ~k(t), tlk ::::;; t::::;; t2k , k = 1,2, ... , be measurable functions with tlk --. t1> t2k --. t2 as k --. 00, and Xk(t) E A(t), ~k(t) E Q(t, Xk(t», t E [tlk' t2k] (a. e.). Assume that Xk --. x in measure, and ~k --. ~ weakly in L l . Then x(t) E A(t), ~(t) E Q(t, x(t», t E [tl' t2] (a. e.), and

it2 Fo(t,x(t),W»dt::::;; liminf i t2k FO(t,Xk(t)'~k(t»dt. Jt] k-+ 00 Jtlk

In particular, if the functions Xk are AC in [tlk' t2k] and converge in the pmetric to a continuous function x(t), tl ::::;; t ::::;; t2, if ~k(t) = x;'(t), t E [tlk' t2k] (a.e.) and ~k --. ~ weakly in L l , then x is AC and W) = x'(t), t E [t1' t2] (a. e.).

Proof. First we note that, in case ofa fixed interval (t lk = t l , t2k = t2 for all k), then (l0.8.ii) is an immediate corollary of (10.8.i) with G = [tl> t2l

In general, with tlk --. t1> t2k --. t2, we note that for any 15, 0 < 15 < 2 -1(t2 - t l ), the interval [tl + 15, t2 - DJis contained in all intervals [tlk' t2k] with k sufficiently large. Under any of the conditions (Li) we have seen that Fk = FO(t,Xk(t), ~k(t» ~ -I/I(t) - Cj~k(t)1 for some constant C. Since ~k converges weakly in L l , we know from (1O.3.i) (implication (a) => (b» that the sequence [~k] is equiabsolutely integrable. Thus, given 8> 0, we can take 15 > OsufficientlysmallsothatIkl = J:::<l Fkdt ~ -8,lk2 = J:~~DFkdt ~ -8

for all k sufficiently large. For Ii. = J:~~~ Fkdt, we have now, as k --. 00,

-8 + lim inf Ii. - 8 ::::;; lim inf Ikl + liminf Ii. + lim inf Ik2

::::;; liminf(Ikl + Ii. + Id = i,

or lim inf Ii. ::::;; i + 28. By the above we have now

itrD Fo(t,x(t),~(t»dt::::;; liminf ftrli Fo(t,xk(t)'~k(t»dt::::;; i + 28, JtJ+D JtJ+D


where b > 0 can be taken as small as we want. Again, F o(t, x(t), ~(t)) ~ -tfJ(t) - q~(t)I, an L-integrable function, so the limit of the first integral as b ~ 0 + exists, is finite, and equals the L-integral I[ x J. Since 8 > 0 is arbitrary, we have I [x] ~ i. The last part of (1O.8.i) follows from the remark that tlk ~ t1, tZk ~ tz, and if t* is any point of (t1,tZ)' then tlk < t* < tZk for all k sufficiently large. Hence, Xk(t) = Xk(t*) + S:. ~k(r) dr, and as k ~ 00, we obtain x(t) = x(t*) + n. ~(r) dr. Thus, x is AC, and ~(t) = x'(t) a.e. in [t1> tzJ.

Remark 4. Theorems (1O.8.i) and (1O.8.ii) are clearly corollaries of (10.7.i). If we know that for almost all t the sets Q(t, x) have property (Q) with respect to x, then the simpler version of (10.7.i) is needed as mentioned at the beginning of the proof of (1O.7.i).

Remark 5. As an example we see that I[ x] = S:; ix'ip dt, p ~ 1, is by (10.8.ii) a lower semicontinuous functional. Note that if x, X k are AC, X k ---> x in the p-metric and xl. ---> x' weakly in L 1, then I[ x] :::; lim inf I[ Xk], and this is true for any p ~ 1. Thus, if x' is not Lp-integrable, then I[ x] = + 00, and lim I[ Xk] = + 00 (cf. Remark 3 above).

Remark 6. If A is closed in W+ 1, if Q(t, x) = Rn for every (t, x), and thus M = A x Rn,

if F o(t, x, z) is continuous on M and convex in z for every (t, x) E A, then condition (L4) is certainly satisfied as we mentioned above as a comment on condition (L4)' From (lO.8.i) we conclude that the integral (10.8.3) is lower semicontinuous in the topology of the weak convergence of the derivatives (mode (b) of Section 2.14). Thus we have also proved here the sufficiency part in the statement (2.18.i).

B. Continuity of Linear Integrals under Weak Convergence

Before considering the question of the continuity oflinear integrals, we shall prove two simple closure theorems for linear differential systems, which we shall also use in Section 11.4.

Let us consider the linear relation

y(t) = Ao(t, x(t)) + B(t, x(t) )~(t),

y(t) = (yt, ... ,l), x(t) = (Xl, ... ,xn), ~(t) = (~t, ... , ~r),

where as usual all vectors x, y, ~ are thought of as column vectors, where Ao(t, x) is an h x 1 matrix, B(t, x) is an h x r matrix, and all entries are defined in a subset A of [to, T] x R". We may assume as usual that

(CL) A is a closed set and all entries are continuous in A. However, for what follows, the following much weaker assumption suffices:

(CL *) For every 8> 0 there is a compact subset K of [to, TJ such that (a) meas([to, TJ - K) < 8, (b) the set AK = [(t, x) E A I t E KJ is closed, and (c) all entries are continuous in AK •

For instance the entries of the matrices A and B could be Caratheodory functions in [to, TJ x Rn, measurable in t for all x, and continuous in x for almost all t.

to.8.m (A LEMMA). Let xk(t), ~k(t), tlk ~ t ~ tZk , k = 1,2, ... , and x(t), ~(t), t1 ~ t ~ tz, be given functions, x, Xk measurable, ~, ~k Lp-integrable for some


p ~ 1, such that tlk -+ t1, t2k -+ t2, Xk -+ x in measure, ~k -+ ~ weakly in Lp as k -+ 00, and

k = 1,2, ....

Assume that for some functions c/>(t) ~ 0, c/> ELl, and t/J(t) ~ 0, t/J E Lq, lip + l/q = 1, we have

iB(t, xli ~ t/J(t),

with t/J E Loo if p = 1, q = 00. Then Yk -+ Y weakly in L1 with

(10.8.3) y(t) = A(t, x(t» + B(t, x(t) )~(t),

As agreed upon in Section (1O.8A) we take W) and all entries of the matrices Ao(t,x(t», B(t,x(t» to be equal zero outside [t1,t2]. Analogously we take ~k(t) and all entries ofthe matrices A(t,xk(t», B(t,xk(t» to be equal zero outside [tlk' t2k]. .

Proof. Since iAo(t,xk(t»i ~ c/>(t), c/> ELl, it is only an exercise to prove that AO(t,Xk(t» -+ Ao(t,x(t» pointwise as well as strongly in L1. Analogously, since iB(t, xk(t»i ~ t/J(t), t/J E Lp, it is an exercise to show that B(t, Xk(t»-+ B(t,x(t» pointwise as well as strongly in Lp. Finally, B(t,Xk(t»~k(t)-+ B(t, x(t) )~(t) weakly in L1. Thus, Yk -+ Y weakly in L10 and (10.8.3) holds. Also note that J:~~ Yk(t) dt -+ J:~ y(t) dt. D

Let us consider now a system of linear differential equations of the form

x'(t) = Ao(t, x(t» + B(t, x(t) )~(t),

x(t) = (xl, ... ,xn ), ~(t) = (~1, ... , ~r), (10.8.4)

where Ao(t, x) is an n x 1 matrix and B(t, x) is an n x r matrix whose entries are defined on a subset A of [to, TJ x Rn under the same general assumptions (CL) or (CL*). We assume that for given functions ~k(t), tlk ~ t ~ t2k , ~k E Lp, system (10.8.4) has certain AC solutions Xk(t), tlk ~ t ~ t2k> and we prove under mild assumptions that there is subsequence, say still k, such that the sequence xk , ~k has limit elements x, ~ satisfying (10.8.4).

lO.S.iv (A LEMMA). Let Xk(t), ~k(t), to ~ tlk < t2k ~ T, k = 1,2, ... , be given functions, Xk AC, ~k E Lp for some p > 1, (t,Xk(t» E A for t E [tlk' t2kJ (a. e.), satisfying (10.8.4), such that ii~kiip ~ J1. and iXk(tt)i ~ N for some tt E [t lk, t2kJ and constants N, J1.. Assume that for some constants c, C ~ 0, p, q > 1, lip + 1/q = 1, and scalar functions c/>(t), t/J(t) ~ 0, c/> E L 1[to, TJ, t/J E Lq[to, TJ, we have

iAo(t,x)i ~ c/>(t) + cixi, iB(t,x)i ~ t/J(t) + CJxi· Then there are functions x(t), ~(t), tl ~ t ~ t2, X E AC, ~ E Lp[t1, t2J, and a subsequence, say still [kJ, such that tlk -+ t1, t2k -+ t2, Xk -+ x in the p-metric, ~k -+ ~ weakly in Lp, x~ -+ x' weakly in L 1, and x, ~ satisfy (10.8.4).


The same statement holds for p = 1, q = 00, provided t/I E L oo ' and the condition II~kllp ~ J1 is replaced by the assumption that the sequence [~k(t)] is equiabsolutel y integrable.

Proof. First, let us take a subsequence, say still [k], such that t lk ----> t 1, t2k ----> t2, tt ----> t*, xk(tt) ----> X* as k ----> 00. Now, since the sequence II~kllp is bounded, there is some function ~(t), t1 ~ t ~ t 2, and a further subsequence, say still k, such that ~k ----> ~ weakly in Lp- Now we have

(10.8.5)

where

x~(t) = Ao(t, xk(t)) + B(t, xk(t) )~k(t),

Ix~(t)1 ~ 4>(t) + cIXk(t) I + (t/I(t) + qxk(t)I*k(t)1

= (4)(t) + t/I(t)l~k(t)i) + (c + q~k(t)I)lxk(t)l,

114> + t/l1~kII11 ~ 114>111 + Iit/lllqll~klip ~ 114>111 + 11t/lllqJ1 = M 1, Ilc + q~k(t)lllp ~ c(T - to)1/P + ql~kllp ~ c(T - to)1/p + CJ1 = M 2'

Thus

Xk(t) = xk(tt) + r: x~(t)dt, Ixk(tt) I ~ N, Jtk (10.8.6)

Ixk(t)1 ~ N + M 1 + i; (c + q~k(r)i)lxk(t)1 dr,

and by Gronwall's lemma (18.l.i),

Ixk(t)1 ~ (N + Md exp(i~ (c + q~k(t)l)dt ~ (N + M 1) exp(c(T - to) + C(T - t)1/q J1) = M 3'

Consequently, IAo(t,Xk(t))1 ~ 4>(t) + cM3, IB(t,xk(t))1 ~ t/I(t) + CM3, and

Ix~(t)1 ~ (4)(t) + cM 3) + (t/I(t) + CM 3*k(t)I,

where 4> E L 1, t/I E L q , ~k E L p , hence x~ E L1 with norm and

Ilx~111 ~ 114>111 + cM 3(T - to) + (ilt/lllq + CM 3(T - to)1/Q)J1 = M 4 ·

Moreover, x~ is equiabsolutely integrable. Indeed if E is any measurable subset of [tlk' t 2k] then

IE Ix~(t)1 dt ~ IE 4> dt + cM 3 meas E + (IE (t/I(t) + CM 3)q dty/q J1,

and the second member approaches zero as meas E ----> 0 uniformly with respect to k. By the Dunford-Pettis theorem (1O.3.i), there is an L1-integrable function a(t) and a subsequence, say still [k], such that x~ ----> a weakly in L1•

Then the sequence [Xk] is equibounded and equiabsolutely continuous (1O.2.i), and by Ascoli's theorem (9.l.i) there are an AC function x(t), t1 ~ t ~ t2, and a further subsequence, say still [k], such that Xk---->X in the


p-metric. From (10.8.6) we derive now that x(t) = x* + J~ u(7:)d7:, and hence x'(t) = u(t) a.e. Now we see that in the first relation (10.8.5) we have IAo(t, Xk(t) I ::; 4>(t) + cM 3, IB(t, Xk(t) I ::; I/I(t) + CM 3, with 4>(t) + cM 3 E L 1, I/I(t) + CM 3 E Lq , and by (1O.8.iii) we derive that x~ -+ x' weakly in L1 and hence x'(t) = A(t, x(t» + B(t, x(t»~(t), t E [tl> t2] (a.e.). For p = 1, and [~k] equiabsolutely integrable, by Dunford-Pettis (10.3.i) there is a subsequence, say still [k], such that ~ converges weakly in L1• The proof is now analogous.

o Remark. By a theorem of Krasnoselskii ([I, p. 27] and [1]), the requirement IAo(t,x)1 ~ ¢(t) + clxl, ¢ E Lh is the necessary and sufficient condition in order that Ao(', x(·» be Ll whenever x(·) is L l . Analogously, the requirement IB(t,x)1 ~ t/!(t) + qxl, t/! E L q ,

lip + 11q = 1, is the necessary and sufficient condition in order that B(',x('» is Lq whenever x( . ) is L q•

We consider now linear integrals of the form

l[x,~] = l:2 (Ao(t,X) + it1 Ai(t,X)~)dt, where the scalar functions Ai(t, x), i = 0, 1, ... , n, are defined in a subset A of [to, T] x R", under the general assumptions (CL) or (CL *). For W) = x'(t), these integrals reduce to the usual linear integrals

lEx] = l:2 ( Ao(t,x) + it1 Ai(t,X)X')dt.

lO.8.v (A CoNTINUITY THEOREM FOR LINEAR INTEGRALS). Let xk(t), ~k(t),

tlk ::; t::; t2k> k = 1,2, ... , and x(t), ~(t), t1 ::; t::; t2, be given functions, x, Xk continuous, ~, ~k Lp integrable for some p ;;:: 1, such that Xk -+ x in the p-metric, ~k-+~ weakly in Lp. Under condition (CL) we have l[xk'~k]-+l[x,~] as k -+ 00. (b) Under condition (CL *) the same is true even if we know only that x, Xk are measurable, and Xk -+ x in measure, provided the graphs of x and Xk are in a subset Ao of A where IAo(t, x)1 ::; 4>(t), IAi(t, x)1 ::; I/I(t), i = 1, ... , n, for some functions 4> E L 1, 1/1 E Lq in [to, T] (I/I(t) a constant if p = 1, q = (0).

Thus, in particular, if the functions x, Xk are AC, if Xk -+ x in the p-metric, and x;' -+ x' weakly in Lp, p;;:: 1, then l[xk] -+ lEx] under the assumptions (CL). Under the assumptions (CL *) the same is true provided the graphs of x and all Xk lie in a subset Ao of A where we know that IAo(t, x)1 ::; 4>(t), IAi(t, x)1 ::; I/I(t), i = 1, ... , n, 4> E Lh 1/1 E Lq (1/1 a constant if p = 1, q = (0).

Proof of (1O.8.v). Under conditions (CL) and all x, Xk continuous with Xk -+ x in the p-metric, then for any compact neighborhood N /J of the graph r of x, then all Xk have their graphs in N /J for k sufficiently large, and Ao and all Ai are bounded in AnN /J. Then lemma (10.8.iii) applies. Under conditions

10.9 Necessary and Sufficient Conditions for Lower Closure 359

(CL *), the specific assumptions of the statement above make it possible to apply (10.8.iii) straightforwardly.

Another proof of (1O.8.v) is as follows. By (1O.8.ii) we derive that I[x,~] is lower semicontinuous. But the same holds for - I[ x, ~], that is, I[ x,~] is both lower and upper semicontinuous, that is, I[ x,~] is continuous.

We shall use lemma (1O.8.iv) in Section 11.4 to prove existence theorems for linear problems of optimal control.

Exercise

Formulate a limit theorem for the integrals I[xJ when the trajectories are solutions of a differential system (lO.8.4) containing arbitrary functions ~ E Lp, p > 1 (controls).

10.9 Necessary and Sufficient Conditions for Lower Closure

A. Partial Converse of Lower Closure Theorem (lO.7.i)

In Theorem (lO.7.i) the requirement (lO.7.3) not only cannot be disregarded, as the example in Remark 4 of Section lO.7 shows, but it is essential, as we shall see by proving a partial converse of (lO.7.i). To this end a few comments and definitions are needed.

First, it is not restrictive to assume in (10.7.i) that Xk -+ x pointwise almost everywhere in G, since the present requirement that X k -+ x in measure implies the existence of a subsequence convergent pointwise almost everywhere.

Analogously, in (10.7.3) we could merely require that the sequence [Ak] be only relatively weakly compact in L 1(G), since again we can extract a subsequence which is convergent weakly to some A E Ll(G).

Concerning the sets Q(t,x), clearly it is enough that we assume them to be convex for x = x(t), t E G, only. Given any real valued function '1(t), t E G, we take as usual '1+(t) = 1- 1<1'11 + '1), '1-(t) = 2- 1(1'11- '1), t E G, '1+, '1- ~ 0, '1 = '1+ - '1-.

lO.9.i. In (lO.7.i) the requirement '1k ~ Ak, Ak -+ A weakly in Ll(G) is equivalent to the requirement that the sequence '1;;(t), t E G, k = 1,2, ... , be relatively weakly compact in Ll(G).

Proof. If ['1;;] is relatively weakly compact in L 1(G), then we can find a subsequence such that '1;; converges weakly in Ll(G) toward some function '1-(t), t E G, '1-(t) ~ 0, '1- EL1(G), and then we simply take Ak= -'1;;. Conversely, suppose that '1k~Ak' Ak -+ A weakly in Ll(G). Then by (lO.3.i) the sequence [Ak(t), t E G, k = 1,2, ... J is equiabsolutely integrable. Then the same occurs for the sequence A;;(t), and since ° :::;; '1;;(t) :::;; l;;(t), we see that the sequence ['1;; (t), t E G, k = 1,2, ... ] also is equiabsolutely integrable, and thus weakly relatively compact in Ll(G). This proves (lO.9.i). 0


Given a pair of measurable functions xo(t), ~o(t), t E G, ~o E L1(G), a sequence (xk(t), ~k(t), 'Ik(t)), t E G, k = 1,2, ... ,of measurable functions on G is said to be admissible relatively to xo, ~o provided (a) Xk(t) E A(t), ('7k(t), ~k(t)) E Q(t, xk(t)), t E G (a.e.), k = 1,2, ... ; (b) - 00 < i = liminfJG '7k(t)dt < + 00; (c) Xk -+ Xo in measure in G, and ~k -+ ~o weakly in L1(G).

Again, given a pair of measurable functions xo(t), ~o(t), t E G, ~o E L1(G), we say that the lower closure property holds at xo, ~o provided, for any sequence Xk' ~k' '7k admissible relatively to xo, ~o, there exists some '70 E Ll(G) such that (17o(t), ~o(t)) E

Q(t,xo(t)) a.e. in G, and fG 17o(t)dt ~ i = liminfJG 17k(t)dt. Also, we shall need the real valued function T(t,x,z) defined in Section 1.13 for

given sets Q(t,x) in Rn+l, (t,x) E A; also, we assume that Q(t,x) = epizT(t,x,z). As before, let xo(t), ~o(t), t E G, ~o E L1(G), be a given pair of measurable functions,

and let us suppose this time that T(t,xo(t), ~o(t)) E Ll(G). We say that the lower compactness property holds at xo, ~o provided for every sequence Xk' ~k> 17k> which is admissible relatively to xo, ~o, the sequence 17k(t), t E G, k = 1,2, ... , is weakly relatively compact in Ll(G). We can state now the following proposition:

lOS.ii (THEOREM). If xo(t), ~o(t), t E G, ~o E L1(G), is a given pair of measurable functions with T(t, xo(t), ~o(t)) E L1(G), then the lower compactness property holds at xo, ~o if and only if the lower closure property holds at xo, ~o.

Proof. If the sequence [17k] is weakly sequentially compact in L1 , then there is a subsequence, say still [k], such that 17k -+17- weakly in L 1, and then, for At = -17k, the relation (10.7.3) holds, and statement (1O.7.i) proves the lower closure property. Conversely, assume that lower closure property holds, let [Xl' ~ko 17k] be any admissible sequence relative to xo, ~o, and let us prove that ['7k] is weakly relatively compact. The argument is by contradiction. Suppose that [17k] is not weakly relatively compact. Then [17k] is not equi-absolutely integrable. Thus, there is a l) > 0, and for each integer s = 1, 2, ... , there is another integer ks ~ s and a measurable subset Es of G, such that meas Es ~ k;l, 17k.<t) ~ 0 for tEE .. and k 17k.<t)dt ~ -l) < 0 for all s = 1,2, .... Let us define the sequence x.(t), ~.(t), ifs(t), t E G, s = 1, 2, ... , by taking

(xs(t), e.(t), ifs{t)) = {(xo(t), ~o(t), T(t, xo(t), ~o(t))) for t E G - E .. (xdt), ~k,<t), 17k.<t)) for tEEs'

Then Xs -+ Xo in measure, es -+ ~o weakly, and

I if.{t)dt = I T(t, xo(tHo(t)) dt + I '1dt)dt ~ I Tdt -l), JG JG-Es JEs Ii JG-E. and the last member approaches fG T dt - l) as s -+ 00. Also, if 170 denotes the function corresponding to (Xk' ~k' 17J guaranteed by the lower closure property, then we have

I ifsdt ~ I Tdt + IE 170 dt JG JG-Es JJ II

and the second member approaches fG T dt as s -+ 00. Thus, lim inf fG ifsdt as s -+ 00 is finite, say j. Now X .. t, ifs is admissible relative to xo, ~o, and by the lower closure property there is a corresponding function if E Ll(G) such that fG if(t)dt ~ j, and (if(t), ~o(t)) E Q(t,xo(t)) a.e. in G. By the definition ofTwe also have fG if dt ~ fG T dt, so that

fG Tdt ~ fG ifdt ~j ~ liminf fG ifsdt ~ fG Tdt -l),

a contradiction, since l) > O. Theorem (1O.9.ii) is thereby proved. o


In view of (10.9.i) and (1O.9.ii) we see that requirement (10.7.3) in Theorem (1O.7.i) becomes essentially a necessary and sufficient condition. Theorem (1O.7.i) with the remarks (10.9.i) and (1O.9.ii) contains a result proved by loffe [1].

B. Lower Semicontinuity at Every Trajectory

Let G be a measurable subset of R, and let f(t, x) be a CaratModory function on G x R", that is, f(t, x) is measurable in t for every x and is continuous in x for almost all t. For any measurable function x(t), t E G, let us consider the function f(t, x(t», t E G; in other words, let F denote the Nemitskii operator defined by x(t) ..... f(t, x(t», or F[ x Jet) =

f(t, x(t», t E G.

lO.9.iii (NEMITSKII). If f is a Caratheodory function, then the N emitskii operator F maps measurable functions into measurable functions. Moreover, if Xk ..... Xo in measure in G, then F[xk ] ..... F[xo] in measure in G.

We are interested in the case where F maps functions x E (LiG»" into functions F[ x] E Lq(G) for given p, q, 1 :$ p, q < + 00. For the sake of brevity we shall write Lp for (Lp( G»" for any p and n.

lO.9.iv (NEMITSKII, KRASNOSELSKII, VAINBERG). If f is a Caratheodory function, and F:Lp ..... Lq, then F is continuous and bounded in the topologies of Lp and Lq • Moreover, F:Lp ..... Lq if and only if there are a constant M and a function I/J(t) 2 0, t E G, I/J E Lq ,

such that If(t, x)1 :$ I/J(t) + MlxI P/q.

For proofs of statements (1O.9.iii) and (1O.9.iv) we refer to Krasnoselskii [I], where also ample bibliography is given on these important questions.

Whenever f(t, x(t», t E G, has a Lebesgue integral on G, finite or + 00, then also the following functional H: Lp ..... R is defined:

H[x] = fGf(t,x(t»dt,

This functional H is said to be lower semicontinuous in Lp provided (a) H[ x] # - 00,

and (b) X k ..... x in Lp implies H[ x] :$ lim infk~ '" H[ xk]. An analogous definition holds for weak lower semicontinuity.

lO.9.v (POLJAK [1]). If P is any fixed number, 1 :$ P < 00, and f(t, x) is a Carathiodory function, then the following statements are equivalent:

(A) H[x] is lower semicontinuous in Lp; (B) H[ x] is defined and # - 00 for all x E Lp; (C) H[x] is defined in Lp and H[x] 2 rx - Pllxll~for suitable constants rx, P; (D) f(t, x) 2 - aft) - blxlP for some constant b 2 0 and function aft) 2 0, a E Lp.

Proof. Obviously, (D) => (C) => (B). Now assume that (B) holds, and let f- denote f-(t,x) = min [OJ(t, x)]. Let F _ denote the Nemitskii operator relative to f-. Then, for x E Lp, we have - 00 :$ f-(t, x(t» :$ 0, f-(t, x(t» E LJ, that is, F _ maps Lp into LJ, and by (l0.9.iv), If-(t, x)1 :$ aft) + blxlP for some constant b 2 0 and function aCt) 2 0, a E Lp. Thus,

f(t, x) 2 f-(t, x) 2 - aCt) - blxlP, (t,x) E G x R".


We have proved that (B) ~ (D). Now assume that (A) holds. By definition of lower semicontinuity in Lp , we see that (A) ~ (B). Finally, assume that (D) holds and that Xk -+ Xo in Lp- For f*(t, x) = f(t, x) + a(t) + blxlP, we have f*(t, x) ~ 0 for all (t, x) E

G x R". From Xk -+ Xo in Lp, it follows that Xk -+ Xo in measure in G, and by Nemitskii's theorem (1O.9.iii), also I*(t, Xk(t)) -+ f*(t, xo(t)) in measure in G. By Fatou's lemma (S.7.i)

Thus,

IG I*(t, xo(t)) dt :s; lim inf I f*(t, Xk(t)) dt. J( k-.oo JG

H[xo] = fG f(t, xo(t))dt = fGu*(t,xo(t)) - a(t) - blxo(t)jP]dt

:s; liminf I f*(t,Xk(t))dt - I a(t)dt - bllxo(t)II~ k-oo JG JG

= lim inf I f(t, Xk(t)) dt + b [lim Ilxkll~ - Ilxoll~J k-oo JG 1-00

= li~inf I f(t,Xk(t))dt = liminf H[Xk]. 1-00 JG 1-00

We have proved that (D) ~ (A). Statement (lO.9.v) is thereby proved. D

Let us now return to functions T(t,x,z), (t,x,z) E G x R" x Rr, G measurable with finite measure in R', T a Caratheodory function (that is, T measurable in t for all (x, z), and continuous in (x, z) for almost all t E G). We are interested in the functional

I[x,z] = fG T(t,x(t),z(t))dt

defined for all measurable functions x(t), z(t), t E G, for which the measurable function T(t, x(t), z(t)) has a Lebesgue integral, finite or + 00. For the sake of simplicity we shall say that I[x, z] is (s, s) lower semicontinuous in L t x L t if I[ x, z] is defined and =F - 00

for all x, z E L t, and I[ x, z] :s; lim infk~ <Xl I[ Xk, Zk] whenever Xk -+ x, Zk -+ Z in Lt. Statement (1O.9.v) for p = 1 then becomes:

lO.9.vi. If T(t, x, z) is a Caratheodory function, then the following statements are equivalent:

(a) I[ x, z] is (s, s) lower semicontinuous in L t x Lt ; (b) I[ x, z] is defined and =F - 00 for all x, z E L t ; (c) I[x,z] ~ IX - Pllxlh - YllZllt for suitable constants IX, p, Y; (d) T(t,x,z) ~ -a(t) - blxl- clzl in G x R" x R' for suitable constants b, c ~ 0 and a

function a(t) ~ 0, a E Lt.

We may consider now sequences (Xk' Zk) of functions Xk' Zk E Lp , with Xk -+ x strongly in Lp and Zk -+ Z weakly in Lq• We shall say that (Xk' Zk) is (s, w)-Lp,q convergent to (x, z) in Lp x Lq•

lO.9.vii. If T( t, x, z) is a Caratheodory function, then of the four statements

(a*) I[x, z] is (s, w) lower semicontinuous at every (x, z) E L t X Lt, (b) I[x,z] is defined and =I--ooforall(x,z)ELt x Lt. (c) I[x, z] ~ IX - Pllxlit - Yllzllt for suitable constants IX,P, y,


(d*) T(t, x, z) ~ - a(t) - blxl- clzl in G x R n x R', and T(t, x, z) is convex in z for almost all t E G and all x ERn,

(a*) and (d*) are equivalent, and if T is convex as stated in (d*), then (a*), (b), (c), (d*) are equivalent.

Proof. If (a*) holds, that is, I[ x, z] is (s, w) lower semicontinuous in Ll x L 1, then certainly I[x, z] is (s, s) lower semicontinuous in Ll x Llo and hence (b), (c) holds as well as the first part of (d*) by (1O.9.vi). 1t remains to prove that (a*) implies the convexity of T(t, x, z) in z, and this is essentially Tonelli's theorem (2.l9.i) concerning convexity as a necessary condition for lower semicontinuity on every trajectory. The proof in the present situation can be obtained by the same argument as in (2.l9.i) via Scorza-Dragoni's statement (8.3.v). We leave the details of the proof as an exercise for the reader. Conversely, if (d*) holds, then certainly I[x,z] is (s,s) lower semicontinuous in Ll x Ll by (1O.9.vi), and we have only to prove that I[ x, z] is (s, w) lower semicontinuous in Ll x L 1• First, since T(t, x, z) is convex in z for almost all t E G and all x ERn, then I[ x, z] is convex in z E Ll for every x ELl' Now let us assume that I[ x, z] is not (s, w)-L 1,1 lower semicontinuous. Then there is an e > 0 and a sequence (Xk' Zk),

k = 1, 2, ... , which is (s, w)-L 1.1 convergent to some (x, z) E Ll X Ll and such that I[ Xk, Zk] ~ I[ x, z] - e for infinitely many k. By selection and relabeling we may assume that I[ Xko Zk] ~ A. - e for k = 1, 2, ... , where A. = I[ x, z J. By (1O.l.i) there is a system of real numbers CN. ~ 0, S = 1, ... , N, N = 1, 2, ... , such that I:= 1 CN. = 1 and I:= 1 cN.z. -+ z strongly in Ll as N -+ 00. Let Nk, k = 1,2, ... , be any sequence of integers with N k -+ 00 as k -+ 00. Then (Xk' I:! 1 CN •• Z.), k = 1, 2, ... , is (s, s)-L 1,1 convergent to (x, z) as k -+ 00. Hence, by the just stated (s, s)-lower semicontinuity of I[ x, z] and the convexity of I[ x, z] with respect to z, we have

N.

A. = I[ x, z] ~ lim inf I [Xk' I CN •• Z.] k-+oo s= 1

N.

~ lim inf L CN •• I[ Xk' z.J ::<=;; A. - e, """'00 5=1

a contradiction. We have proved that (d*) implies (a*). Statement (1O.9.vii) is thereby proved. []

Remark. Theorem (1O.9.vii) shows that the conditions (A) of Section 10.7, or (L) of Section 10.8, are not only sufficient but also, in a sense, necessary for lower closure. For more work along the lines of the present section we refer to Poljak [1], already cited, and to Rothe [1], Olech [1], and Ioffe [1] (see bibliographical notes for further references). The statements in Section 10.9 concern only strong and weak convergence in L 1• Analogous statements hold for strong and weak convergence in Lp. We shall present the material above more extensively, and including the analogous statements in Lp and in Sobolev spaces, in [IV], in connection with optimization with partial differential equations, where this material is essential. Let it be mentioned here that for (s, w) convergence in Lp x Lq, then the statement of (1O.9.vii) becomes

T(t, x, z) ~ (p(t, x), z) - clxlp - a(t),

where a(t) ~ 0, a E Llo and the map t -+ Ip(t, x(tW', l/q + l/q' = 1, is relatively weakly compact in L 1•


C. Exercises

1. Prove statement (10.9.v) with weak convergence in Lp in (A) and f(t, x) convex in x for almost all t E G.

2. Prove in detail that, in (10.9.vii), (a*) implies the convexity of T(t, x, z) with respect to z as stated in (d*).

3. Prove Theorems (1O.6.ii) and (l0.7.ii) by assuming that the relevant sets have the weak form of property (Q) stated in Remark 1 of Section 8.5. (The conclusions of (1O.6.ii) and (1O.7.ii) will be used in Chapter 14 under hypotheses which indeed imply this weak form of property (Q) as well as the hypotheses in the abovementioned theorems.)

Bibliographical Notes

References on the Banach-Saks-Mazur theorem (10.1.i) and on the Dunford-PettisNagumo theorem (actually, the composite equivalence theorem (1O.3.i» have been already given in the text. Growth condition (gl), or (cp), in Section lOA is the usual Tonelli-Nagumo condition. Growth condition (g2) is slightly less demanding and was proposed by L. Cesari [35] (see also previous remark in L. Cesari, J. R. LaPalm, and T. Nishiura [1]). Condition (g3) has been proposed by R. T. Rockafellar [2], and as we have proved (1004. vi-vii), (g3) and (g2) are equivalent. Growth condition (gl) or (cp), which is so closely related to weak convergence in LI as stated by the equivalence theorem (1O.3.i), implies property (Q) of certain related set valued functions. This is the essence of theorems (1O.5.i), (1O.5.ii), which were proved by Cesari in various forms in [6,7]. The same holds for statement (1O.5.iii), and the analogous implications hold for the convex sets which occur with generalized solutions as proved in Remark 4 of Section 10.5. The closure and lower closure theorems (1O.6.i) and (1O.7.i), both based on weak convergence in LI of the "derivatives", are now a consequence of the BanachSaks-Mazur theorem, of the Dunford-Pettis theorem, and of the implication stated by theorems (1O.5.i-ii). Restricted forms of these closure and lower closure theorems have been proved independently and about at the same time by L. Cesari [13], M. F. Bidaut [1], and L. D. Berkovitz [1].

For further results on lower closure theorems, see L. Cesari [12, 13, 16, 17], L. Cesari and M. B. Suryanarayana [8], and C. Olech and A. Lasota [1,2].

The very general lower closure theorem (1O.7.i) concerns abstract integrals J,,(t)dt under mere orientor field constraints (,,(t), ¢(t» E Q(t, x(t» for given convex subsets Q(t, x) in R"+ I. The rather abstract requirement (10.7.3) in the lower closure theorem (1O.7.i) is certainly verified under any of the alternative conditions (Ai) of Section 10.7, which are easy to verify and have therefore practical significance. Of these conditions (AI) and (A 2 ) were noted by all of the aforementioned authors, condition (A3) by L. D. Berkovitz (this condition is contained in (A 2», and condition (A4) by Cesari in [16]. The conclusion which has been reached recently-and which is embodied in statements (1O.6.i), (1O.7.i)-is that, in connection with "weak convergence of the derivatives" (weak convergence in HI,I) there is no need for explicitly requiring any property (Q) (of the sets Q in R", or Q in R" + I) since the weak convergence implies a growth property by (1O.3.i), and this in turn, by (1O.5.i-ii), implies the property (Q) for certain auxiliary

Bibliographical Notes 365

sets (the sets Q* in R"+l or Q'* in R"+2). The lower closure theorem (1O.7.i) and its immediate corollary, the lower semicontinuity theorem (lO.S.i), are of course the main tools in the proof of the existence theorems of Chapter 11 based on "weak convergence of the derivatives". The required conditions in the lower closure theorems (10.7.i) and lower semicontinuity theorems (lO.S.i,ii) are proved to be necessary in Section 10.9.

On the other hand there are existence theorems (see Chapter 12) for which mere "uniform convergence of the trajectories" is used, and for these theorems the corresponding closure and lower closure theorems based on mere uniform convergence (S.6.i), (S.S.i,ii) are needed and it appears that in these theorems some form of property (Q) is needed. Thus, the presentation, based on geometrical considerations, of lower closure and lower semicontinuity theorems in these Chapters Sand 10 has a certain degree of uniformity.

Closure and lower closure theorems (1O.6.ii) and (1O.7.ii) have been proved by L. Cesari and M. B. Suryanarayana [1], and they will be used in connection with property (0) for the existence theorems of Chapter 13 covering a number of situations of practical significance some of which had been indicated by E. H. Rothe and L. O. &rkovitz. Each one of the specific conditions in the existence theorems of Chapter 13 implies property (0) and this in tum implies the weak form of property (Q) mentioned in Remark 1 of Section 10.5, which could be used in an alternate proof of the same existence theorems.

The lower semicontinuity theorems (lO.S.i,ii) are corollaries of(1O.7.i). The integrand function F o(t, x, z) is assumed either continuous on M, (condition (C) of Section 1O.S), or briefly "measurable in t and lower semicontinuous in (x, z)", specifically, satisfying condition (C*) of Section 1O.S. This condition is rather general among those proposed so far for such lower semicontinuity theorems (namely it is equivalent to the various forms of "normality" conditions used by R. T. Rockafellar [2] and by I. Ekeland and R. Temam [I], as proved by these authors).

The conditions (Ai) of practical interest in Section 10.7 are replaced in Section 1O.S by corresponding conditions (Li)' also easy to verify. Of these, condition (L4)' which corresponds to (A4), is especially relevant, since (L4) is always satisfied in classical free problems of the calculus of variations in one independent variable (Remark 3 of Section 10.SA). Thus, from (lO.S.ii) and (2.19.i) it follows that if F o(t, x, z) is continuous in A x R", then SF 0 dt is lower semicontinuous with respect to weak convergence of the derivatives (weakly in H I •1) if and only if F o(t, x, z) is convex in z, x = (xl, ... , x"), z = (zl, ... , z"), n ~ 1 (L. Cesari [17], and also, in this book, (2.lS.i) and Remark 6 of Section 1O.S). In particular, linear integrals, i.e. SFodt, with n ~ 1, Fo = P(t,x) + LI=1 Qi(t, X)Zi, P, QI, ... , Q" continuous on A, are continuous in the topology of the weak convergence ofthe derivatives (namely, weak convergence in H 1•1).

For n = 1 and F o(t, x, z) continuous in A x RI together with F 0 .. F Oz .. F Ot ..

Tonelli proved that J F 0 dt is lower semicontinuous with respect to uniform convergence of the trajectories if and only if F Ozz ~ 0 (Tonelli [I, p. 400]). Tonelli also gave an example [I, p. 392] of a linear integral SFodt, with n = 1 and Fo = P(t,x) + Q(t,x)z, P, Q continuous on A, which is not continuous in the uniform topology, and McShane [5, p. 211] gave an example of a linear integral JFodt with n = 2 and Fo = P(t,x) + Q(t,X)ZI + R(t,X)Z2, P, Q, R of class Coo, which is not continuous in the uniform topology. Tonelli [13] showed later that OQ/OX2 = oR/oxl is a necessary condition for continuity in the uniform topology for such integrals.

M. Vidyasagar [1] proved a lemma analogous to (lO.S.ii) requiring also a uniform Lipschitz condition on Ao(t, x) and B(t, x) with respect to x and uniform in t.


Section 10.9 is a brief presentation of further results concerning necessary and sufficient conditions for lower closure on all trajectories. These results are essentially based on the precise characterizations of properties of boundedness and continuity of Nemitskii and other operators, for which we refer to M. A. Krasnoselskii [I], to M. A. Krasnoselskii, P. Zabreiko, E. I. Pustylnik, and P. W. Sobolevskii [I]; to M. M. Vainberg [I], and to further work ofL. V. Sragin [1] and ofB. T. Poljak [1]. The part concerning the necessity of the convexity condition is of course Tonelli's theorem on lower semicontinuity on all trajectories (2.19.i). For further work on necessary and sufficient conditions for lower closure and lower semicontinuity theorems we refer to C. Oleach [1], A. D. loffe [1], V. I. Kazimirov [1], and V. S. Morozov and V. I. Plotnikov [1 ].

For independent work on lower semicontinuity in Lp spaces and Sobolev spaces we mention here also G. Fichera [1], and E. H. Rothe [1].

optimization—theory and applications || closure and lower closure theorems under weak convergence

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