THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 15 CLOSURE AND LOWER CLOSURE THEOREMS FOR MULTIDIMENSIONAL PROBLEMS Lamberto Cesari OR''Proie.t:. "024.0416 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1970

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8. Closure and Lower Closure Theorems for Multidimensional Problems 8.1. Closure Theorems for Orientor Fields Here G denotes as in (7.1) an open bounded subset of the t-space E, for V every t e cl G we denote by A(t) a given subset of the y-space E, and by A the set of all (t,y) with t e cl G, y E A(t). We assume A to be closed, and for every (t,y) E A we denote by Q(t,y) a given subset of the z-space E. We conr r 1 s sider here pairs z(t) = (z,...,z ), y(t) = (y,...,y ), t e G, of vector functions z e (L1(G))r, y E (L (G))s, such that y(t) c A(t), z(t) c Q(t,y(t)) a.e. in G. (8.1.1) We may say that the pair (z,y) is an abstract solution of the orientor field (8.1.1). In applications there are suitable functional relations between z and y, that here are not needed. (8.1.i) A first closure theorem. Let G, A(t), A, Q(t,y) be as above, G open and bounded, A closed, and let us assume that the sets Q(t,y) satisfy property (Q) in A. Let y(t), Yk(t), t E G, be s-vector functions, YYk E (Li(G))s, and z(t), zk(t), t c G, be r-vector functions, z,zk E (L1(G)), with yk(t) E A(t), zk(t) c Q(t,yk(t)) a.e. in G, k = 1,2,.... If Yk y+ strongly in (L1(G))s, and zk + z weakly in (L1(G))r, then y(t) E A(t), z(t) e Q(t,y(t)) a.e. in G. This theorem is a particular case of (8.2.i) below. 1

8.2 A Closure Theorem for Trajectories With Singular Components Let G be as above a bounded open subset of the t-space E, and let I be an interval containing cl G, G C cl G c I c E. It is not restrictive to assume O V I = [o,b], or [o,...,o,b,...,b] for some b > o. For every t e I, t = (t,...,tv), 1 -v let [o,t] denote the interval [o,...,o,t,...,t ]. Then, given any real-valued L-integrable function z(t), t e I, we may consider the function F defined by t F(t) = f z(T)dT = f z(T)dT t E I, (8.2.1) [o,t] o t where the integration in [o,t] will be often denoted simply by J, and where o d~ = dt... dT as usual. Note that F(t) is then a continuous function on I 0 with F(t) = o whenever t = (t,...,tV) E I, t..t = 0. We shall consider the countable system [t ) of points t = (t,...,tV) p P with t pjb, j = 1,...,v, and pj rational, and the countable system (I) of all intervals I = [a,] c I, ~ = (a,...,'), ) = (,''', ), h <, a, Eft ). o p For any given F(t), t E I, we may then consider the usual differences AIF of order v relative to the 2 vertices of I; in other words AIF = F(p) - F(a) if v = 1, A F = F(P1,2) F(o1,2) - F( 1, ) + F(a, a2) if v = 2, etc. If F is obtained by integration in [o,t] of a function z e L1(I ) as in (8.2.1), then AIF = I z(T)dT = J z(T)dT, I = [a,] c I, (8.2.2) [a, ] a 2

with the usual conventions. We recall here that for a function F as in (8.2.2) the following holds: For almost all t E I and hypercube of I, t o I, of side length h > o we have |I|-1 AIF + z(t ) as h + o+. An arbitrary function S(t), t E I, is said to be singular in I if for almost all t E I we have II -1A S + o as h + o+, where as above I denotes any o o I hypercube of side length h with t E I c I o Let G, A(t), A be as in (8.1.i). For every (t,y) E A we shall consider subsets Q(t,y) of the z-space E + z = (z,...,z ), r > o, a > o, with the following properties: 1. There is an L-integrable function 4(t) > o, t E G, such that if r r+l r+oQ z = (z z r z,.,z ) E Q(t,), (t,y) E A, then z > _(t), i + r +,...,r +. ~ -l -r -r+l -- r+a 2. If z = (z,...,z,z,..z ) Q(t,y) for some (t,y) E A, then -1 -r r+l r+a i -i also every point z = (z,...z, z,...,z ) with z >z, i = r + l,....,r + a, belongs to Q(t,y). (8.2.i) A second closure theorem. Let I, G, A(t), A as above, A closed, and for every (t,y) E A let Q(t,y) be a given subset of the z-space E satisfying r+a properties 1 and 2 above, and in addition satisfying property (Q) in A. Let I s I s s y(t) = (yl... y ), yk(t) = (yk,.-.,yk), t E G, be elements of (L1(G))S, let 11 I r+aT I 1 r+a r+a z(t) = (z%...,z ), zk(t) = (Zk...,zk ), t E G, be elements of (L1(G)) k = 1,2,..., and take Fi(t) = f z (T)dT, Fk(t) = f zk(T)dT for t E I, with lo k k oo

k ~ ~~~i i A(t), zk(t) E Q(t,yk(t)) a.e. in G, k = 1,2,..., that Yk + y strongly in L1(G), k kk i i = l,...,s, that Zk + Z weakly in L1(G), i = l,...,s, and that Fk(t) + F(t) + S (t) pointwise for all t E (t)}, i = r + l,...,r + a, ask- o, where S (t), t E I, are nonnegative singular functions on I. Then, y(t) E A(t), z(t) E Q(t,y(t)) a.e. in G. ~ 1 r+a Proof. We denote by N the sum N = r + a, and by z(t) = (z,...,z ~ r+a~ zk(t) = (zk...,zk ), t E Io, the same functions above with z(t) = zk(t) = o for t E I -G, k = 1,2,..., and we take also *(t) = o for t E I -G. Note that o o zk(t) > - 4(t), t E I, i = r + l,...,r + a, and that Zk(t) E Q(t,yk(t)) a.e. in G, k = 1,2,... 1 For every to E G, to = (t oIt, ), let 6 = 5 (to) > o denote the distance of t from aG, and let q = q = [t, t+h] denote any closed hypercube 0 0 g = [t < ti < t + h, j = l,...,v], with t, t + h E (t ), o < h < BO/v, t < tj < tj + h, j = 1,...,v. Then t e q C G, and we denote here, for the 0- - 0 sake of simplicity, by h also the v-vector (h,...,h). Since Yk + y strongly in Y. there is a subsequence which converges also pointwise a.e. in G. For the sake of simplicity, let us denote this sequence still [yk]. Then, for almost all to E G, we have Yk(to) + Y(to), (to,yk(t)) ~ A, k = 1,2,..., where A is a closed set. As k + o we obtain (t,yo(t )) E A, or go(to) E A(t ), and this relation holds for almost all t E G. o Also, for almost all to e G we have, as h + o+, Iq1l r z (t)dt + z (t), i = l,...,r + o, (8.2.3) (q a foz (t~dt +

I AS -1o, i r +,...r + a (8.2.4) q Because of the pointwise convergence Yk(t) + y(t) a.e. in G, we have Yk(t) - y(t) is a subset G of G with I G I = IGI, and we know that there are closed sets C., X = 1,2,..., withC C C G, C CC+1, ICI > I G I - -1, such that y(t) is continuous on C and Yk(t) + y(t) uniformly on Ck as k + ~ for each X = 1,2,.... Since G is bounded, and CA c G c G, each set CA is compact, and hence y(t) is uniformly continuous on each C. Let A be any fixed integer with A > IGI1, hence IC I > o. Let E > o be an arbitrary number. There is some 5' ='(~e,X) > o such that It-t'l < 5 0 O -0 tt' E C., implies Iy(t)-y(t')l < E/2. Also, there is some k(E,X) > o such that k > k(E,.), t E Ch implies lyk(t)-y(t)l < e/4. Then, for t,t' E CX, It-t'I < 5'(SE,), k > k(~s,),we have also lYk(t)-Yk(t')I < lYk(t)-y(t)l + ly(t)-y(t')I + ly(t')-Yk(t')l < < E/4 + E/2 + k/4 =. Let X (t), X*(t), t E G. be the characteristic functions of the sets CO and G-CO, so that XA + X* = 1 everywhere in G. All functions X (t) and X(t) zj(t), j = 1,..., are of class L (Cl), and for every t E Ck we have x (t ) =, X*(to) z(t)= o. Then, for almost all t E CX we have also, as h 0+ o, Ii-1 f X (t)dt + X (t ) = 1 (8.2.5) hIl f Xh*(t) z (t)dt X*(t) z (t) = o i = 1,...,r + o, (8.2.6) 11 f X*(t) 4(t)dt o. (8.2.7)

Let C? be the subset of C where this occurs. Then C' is measurable, cC C c G c G, and, if G' denotes the set UxCK, also IClI a IC-I > IGOI > O. IG'| IUC IG l = IGI (8.1 12) Let H and H* be the sets H = q C, H*= q - H = q - q n c - c. Let l > o be any positive number independent of ~. Let t be any point of o 1. r C let y = y(t 0 0 z (O ) = (ZoZ )...,ZO) = z(t ) and let 0 0 00 M1 = z(t)l 1. Let us fix h > o so small that h < E/v, h < 6/v, h < o/V, where 5 = 6 (t ), 6' = 8'(~,X), and also so small that 0 00 0 0 1' Z(t )-qj | Z (t)dtj < min lN1, 1], i = l,...,r + a, (8.2.8) o < 1 - q/IHI < min [N- M1-1,1], (8.2.9) |q-1l J X z(t) zi(t)dt < TN-1 i = 1,...,r, (8.2.10) I Iq- l < N-, i = +...,r + a (8.2.11) Iv-1 fq X*(t) *(t)dt < TN- (8.2.12) This is possible because of relations (8.2.3-7). For t E H, and k > k(~,x) we have now It-t I < vh < min [E,6,5'], lk(t)-Yo = Yk(t)-Yk(to)I + |yk(t )-y(to)i < ~ + ~ = 2, 06 -

and hence (t,yk(t)) c N 3(to,Yo) for t E H and k > k(~,X). Therefore, we have also zk(t) E Q(to,y(t ),32) for t E H and k > k(E,X). (8.2.13) and finally HI- fH zk(t)dt E cl co Q(to,yo; 3), k > k(e,X), (8.2.14) since the last set is convex and closed. 1 1 r Because of the weak convergence of zk(t) = (Zk...,zk) to z(t) = (z,...,z ) in (L1(G))r, we can determine an integer k' = k'(t,e,X,,,h) > k(E,.) such that, for k > k'(t,e,X,qh),we have ifH zk(t)dt - fH zH(t)dtl < iN-=r (8.2.15) Now, for k > k'(t,~,X,r,h) and i = 1,...,r, we have i -1 i I Z(to) - IHI fH zk(t)dtl _ I(t)-I flqZ (t)dtl + I (I H-1 ( )iz i(t)d + | II f ( z (t) - z(t))dtl + (I ql/IH-l) lI q-l z (t)dtl dl +d2 d + d4. (8.2.16) By (8.2.8) we have d <N Nr, by (8.2.15) we have d < N, and by (8-2.9), (8.2.10) we have d4 < 2N -. Also, by (8.2.8),(8.2.9) and the definition of M1, we have

= I- IHl q zi(t)dtl < N M1 M N. 2 s 1 1q Thus, (8.2.16) yields, for k > k'(t,E,X,r,h), Iz (t) - IHI IH Zk(t)dtl < 5N1, i = 1,...,r. (8.2.17) We shall now obtain analogous estimates for i = r + l,...,r + a. We know that zk(t) > - 4(t), t E G, i = r+l,...,r+a, and hence, by force of (8.2.12) we have for all k ql fH. zk(t)dt > -Iql J X*(t)4(t)dt > - N-1 i = q (8.2.18) Let F (t) = F (t) + S (t), t E I i r+l,..,r+a, so that Fr(t) + F (t) as k + m pointwise at all points t E {t }. By (8.2.2) and the definitions of Fk and F we have now 1l-l zikt t!ql-1 i -4 lJ 1 i I 1K qI|f Z (t)dt = qq Fk q (t)dt = r+l r+ (8.2.19) On the other hand, by F = F + S, we have it A F= i A F + iql A, i = r + 1,..., r + a. (8.2.20) q o q q V i i The 2 vertices of q are points of (t }, hence A Fk + A F as k * 0, and p q q o we can determine the number k'(t,e,X,qT,h) above so that, for k > k'(t,E,X,rl,h) we have also i -q1 i -1 Fi -1 qk ~qo --

Finally, (8.2.19), (8.2.20), (8.2.21), and (8.2.11) yield I l f -1 J (t)dt - I ql-1 J z (t)dtl q k.q I Il qi -A F Iqj1A F i < I 1 -l'A Fk I ql -1 F i + II qi1 Si — KK k qo q < N T + N1 Ti = 2N T, i = r + 1,...,r + a. (82.22) We have now, for i = r + 1,...,r + a and k > k'(t,E,X,T,h), zi(t ) - I H-1 H zk(t)dt > -i (1 ql - Iq I q z (t)dtj I( d/IHI)i - 1) f z )(t)dtd + d + d7 + d8. (8.2.23) By (8.2.8) we have d > -N -, by (8.2.9) and (8.2.22) we have d7 > -4N- b, 5 5 7 and by (8.2.10), (8.2.18) we have d8 > -2N-. Finally, by (8.2.8), (8.2.9) and the definition of M1, also 1 i -1 -1 -l d = - j(1-1 qL/Ij HI) q1 f zi(t)dt > -N M1 M1 = -N 1 Thus, (8.2.23) yields, for k > k'(t,,, T,h), 0-

- i -1 z (t) - I H| H zk(t)dt > - 8N r, i = r+l...,r+a. (8.2.24) Note that relations (8.2.17) and (8.2.24) can be written in vector form z(t) =k + k + Zk = I qj1 fq zk(t)dt k > k'(t,,,Tlh), (8.2.25) where < 8 = (+,, r r+a) r +l r+ > r+o hek[ _ 8 k -(o,..,,k k > o k > k. By force of (8.2.14) we have then Zk C cl co Q(to,yo, ~), hence, by property 2 of' the sets Q, also + k +k E cl co Q(to,yo, ~) and finally z(to Zk + 5k + k e (cl co Q(t,,'~ ) 8~ Since q is arbitrary and the set in parenthesis is closed, we have z(t ) c cl co Q(t yO,53E) and this relation holds for every E > o. Then, by property (Q) we have also z(t ) n> C1l co Q(t,yo3~) = Q(to,yo), for every point t E C. Thus, we have z(t) E Q(t,y,(t) ) for all t c G' = UCk, that is, almost everywhere in G. 10

8.3. A Third Closure Theorem for Orientor Fields We consider here the case where some or all of the r-components Zk, i=l,...,r of the vectors Zk, or Zk, of theorems (8.1.i), (8.2.i), converge strongly to z instead of weakly. As we shall see below if all components Zk, i = 1,...,r, converge strongly, then instead of property (Q) we need only require property (U). If, say, p components Zk. Zk converge weakly and the remaining r-p components Zk'.,Zk converge strongly, then the intermediate property (Q ) can be required. We discussed this property (Q ) in Appendix A, and we proved there that property (Q ) is equivalent to property (U), that property Qr is equivalent to property (Q), and that property (Q ) implies property (Q ), 1 < p < r. ~ r+a Let us mention here that, if z = (z,...,z ) is any point of the z-space E r > o, a > o, if p is any integer, o < p < r < r + a, and 5 any positive number, we denote by N (z,p,r) and N(z,p,r) the subsets of E + defined by N (z,p,r) = [z E Er+ Jllz -zi <, i = p + 1,...,r], 0 0 0 r-p N (z p,r) = N (Z ) E-p N(zp,r) = (z o) xE GvnG=EVAt 1AcEV+sr+cy the sets Q(t,y) satisfy property (Q ) at the point (to,y) E A provided, for 1 r+ - every point z = (zO..'z ) e E+ we have 11

Q(to,yo) n N(z,p,r) = nO>O Q(toYo;5) n N (zoP,r) o 0 0 y A No o (o, p (8.3.i) A third closure theorem for orientor fields. The same as (8.2.i) where the sets Q(t,y) c E r > o, a > o, satisfy property (Q ) for some p, o _ p < r < r +, and where the n-p components z (t), i = p + l,...,r, of zk(t), t E G, k = 1,2,..., converge strongly to the components z (t) of z(t), t E G, as k + o. Then the same conclusion holds as in (8.2.i). Proof. The initial steps are the same as for the proof of (8.2.i). Now Yk + Y strongly in (L(G)) z trongly in L1(G), i = p + l,...,r; hence, there is a subsequence, say still [k] for the sake of simplicity, such that' i Yk k+ Z i = p + 1,...,r, pointwise a.e. in G, say in a subset G of G with IG I = IG~. There are now closed subsets C, X = 1,2,.., with CA C G. CA Cc+ C, IC j > IG | -A, such that all y(t) = (y,...,y ), z (t), i = p + l,...,r, are continuous on C\, and Yk(t) + y(t), zik(t) z (t), i = p + 1,...,r, uniformly on CA as k + ~. As in the proof of (8.2. i) we take A > IGI, so that I C I > o, and two numbers ~ > o, B > o arbitrary. Then there is some 6' = 6'(E,,A) > o so that It-t'j < a,t,t'~ CA implies 0 0 0 (Iy(t) - y(t')l < ~/2,)Iz (t) - zi(t')l < P/2, i = p + l,...,r. Also, we choose k(~,A,B) > o such that k > k (F~,X,), t ~ C implies lyk(t)-y(t)j < E/4, Izik(t) - z(t) I < n/4. Then, for t, t'~ C%, It-t'I _< I (, ), k > k(~J,. ) we have, as in the proof of (5.2.i), Iy (t)-yk(t')l <, Izk(t) - zk(t')I <, i = p + 1,...,r. 12

We proceed now as for (8.2.i): we define the sets C', we take any t E C', and we fix h > o so that h < E/v, h < 5 /v, h < 8'/v, where 6 = 5 (t ), 6' = 5',(E,X,p), and so that the corresponding hypercubes q of side length h and 0 0 the sets H = q n Ck satisfy relations (8.2.8-12). Thus, relations (8.2.13) holds now for all t C H and k > k(~E,X,). On the other hand, for t E H = q n AC, k > k(~,X,5), we certainly have zk( ) -z (t)l < 5 |z1(t)-z (t )I < 5, i p + 1,...,r, and hence Izk(t) - zi(t)I < 2, i = p + Thus, relation (8.2.13) can now be rewritten in the stronger form zk(t) E Q(toyo;3~)) n N2 (z(t0),r,p), and consequently (8.2.14) becomes HI1J f zk(t)dt ccl co [Q(t,yo;3~) A N02(z(t0),rp)] (8.2.27) for all k > k(E,X,). Note that strong convergence in L (G) implies weak convergence. Thus the 1 arguments of (8.2.i) concerning the components Zk, i = l,...,r + a, hold without change. In other words, we can determine some number k'(t,E,X,rl,h) > k(e~,X,) such that, for k > k'(t,E,Xk,rIh), we have i 1 (t)dt < 5N-1 i -= 1,.r, 13

i -1 (t)dt > 8N-1 z (t ) - I HI f z(t)dt > - 8N, i = r + 1,...,r + a. O H - Thus, the same relation (8.2.25) holds Z(t) = Zk = HI- iH zk(t)dt, as in the proof of (8.2.i), where Ikl > 8r, Sk > 0 i = r + l,...,r + C, k k -, k > k'(t,~E,X,f,r,h), and where the point z is now in the right hand set (8.2.27). Thus, instead of (8.2.26) we have z(to) E (cl co [Q(t o,Yo;3) n N2 (z(tO),rp)])8, where E,~,Tl are arbitrary positive numbers. Finally z(to) E cl co [Q(to,Yo 3E) n N 2(z(to),r,p)] where E and p are arbitrary. By property (Q ) we have z(to) cn E>0 n cl co [Q(t,Yo;3E) n N 2(z(tO),r,p)] z(t) E Q(tO,yO) N(z(tO),r,p) Thus, z) E (t ) (t,y(t )) for every t c C C where X is arbitrary, and finally z(t) E Q(t,y(t)) a. e. in G. 8.4 Closure Theorems for Control Problems We shall use here again the notation of (7.1). Thus, G is a bounded open subset of the t-space E, t = (t I...,t ), and for every t E cl co G a subset A(t) of the y-space E is assigned. Let A denote the set of all (t,y) with t c cl G, y c A(t). For every (t,y) c A a subset U(t,y) of the u-space 14

E is assigned, and M denotes the set of all (t,y,u) with (t,y) E A, u E U(t,y). m Let f(t,y,u) = (fl,...,fr) be a given vector function on M. For every (t,y) E A let Q(t,y) denote the set Q(t,y) = [z E Er | z = f(t,y,u), u E U(t,y)] c E We first state and prove the implicit function theorem for orientor fields which we need below and which is similar to statement (1.6.i). (8.4.i) An implicit function theorem for orientor fields. Let G, A(t), A, Q(t,y) be as above, G open and bounded, A and M closed, f continuous on M. If z(t) = (z l...,z ), y(t) = (y...,y ), t e G, are any two measurable vector functions such that y(t) E A(t), z(t) E Q(t,y(t)) a.e. in Cr, then there is some measurable vector function u(t) = (u,...,u ), t E G, such that y(t) E A(t), u(t) E U(t,y(t)), z(t) = f(t,y(t), u(t)) a.e. in G. Proof. The proof is only a modification of the one for (1.6.1) in Section 1. First, M is a (closed) subset of the Euclidean space E; hence, a V4t sTIm metric space which is the union of countably many compact subsets. Let F: M+N be the continuous mapping defined by (t,y,u) + (t,y,f(t,y,u) so that N = F(M) is the subset of all (t,y,z) E E with (t,y) E A, z = f(t,y,u), u c U(t,y), v+s+r or (t,y) E A, z c Q(t,y). Finally, let a: G -+ N be the measurable map defined by t -* (t,y(t),z(t)). If cp: N - M denotes the partial inverse of F defined in (1.7.iii) such that Fp is the identity map on N, then the map V: G + M defined by 4 = cp is a mapping t - (t,y(t),u(t)) such that z(t) = f(t,y(t),u(t)) a. e 15

in G, u(t), t C G, is measurable, and _ _ M' N u(t) E U(t,y(t)) a.e. in G. F G Let Y = (Ll(G))s, Z = (L1(G))r, and let T denote the set of all measurable 1 1 vector functions u(t) = (u1,...,u ), t e G. (8.4.ii) A closure theorem in G. Let G be open and bounded, A and M closed, f continuous on M, and let us assume that the sets Q(t,y) satisfy property (Q) in A. If Yk' Zk, uk, k = 1,2,..., is a sequence of elements Yk E Y, zk E Z, Uk e T, with Yk(t) E A(t), uk(t) E U(t,yk(t)), zk(t) = f(t,yk(t), uk(t)) a.e. in G, k = 1,2,..., and Yk + y strongly in Y and zk - z weakly in Z, then there is at least one element u E T such that y(t) e A(t), u(t) E U(t,y(t)), z(t) = f(t,y(t), u(t)) a. e. in G. Proof. First the elements Yk' Zk' k = 1,2,..., satisfy the orientor field relations Yk(t) E A(t), zk(t) e Q(t,yk(t)) a. e. in G, k = 1,2,.... Hence, by force of closure theorem (8.1.i) we have also y(t) C A(t), z(t) c Q(t,y(t)) a. e. in G. Finally, by force of the implicit function theorem (8.4.i) there is a measurable vector function u: G - E, or u c T, such that u(t) E U(t,y(t)) and z(t) = f(t,y(t), u(t)) a. e. in G. Closure theorem (8.4.ii) has an analogous version for boundary controls. Let r c aG be a set which can be decomposed into finitely many nonoverlapping parts l'...,rN, each Fj being the image under a transformation I F. of class K of the unit interval I of dimension v-l, so that a natural area measure 16

function a is defined on r. As usual we assume that for any t E r a subset B(t) of the y-space E' is assigned, and we denote by B the subset of E of all (t,y) with t E r, V+S' y B(t). For every (t,9) E B let V(t,9) be a given subset of the v-space E,, 0 and let M be the subset of E of all (t,~,v) with (t,y) E B, v e V(t,y). V+S I +m1' o Let g(t,y,v) = (gl,...,gr,) be a given function on M, and for every (t,y) E B let R(t,j) denote the set of all M = (a,..., with z = g(t,j,v), v ( V(t,r). ~o S o Let Y = (L1(F)), Z = (Ll())r, and let T be the set of all a-measurable vector functions v(t) = (v,...,v ), t e r. (8.4.iii) A closure theorem on r. Let'be as above, r,B,M closed, g continuous 0 on M, and let us assume that the sets R(t,y) satisfy property (Q) in B. If O 0 Yk' Zk, vk' k = 1,2,..., is a sequence of elements Yk E Y, zk Z, vk E T, with yk(t) E B(t), vk(t) E V(t, Yk(t)),.k(t) = g(t,yk(t), vk(t)) o-a.e. in o 0 r, k = 1,2,..., and Yk + Y strongly in Y and k + Z weakly in Z., then there is Yk y k 0 O at least one element v E T such that y(t) E B(t), v(t) E V(t,y(t)), ~(t) = g(t,y(t),v(t)) o-ate. in r. Proof. Here r is the finite union of sets r.. We can limit ourselves to 1 one set r.. Note that r. is the image of an interval I c E under a map 1 1 v-l i.: I - ri' say t = t(T), T E I, of class K. Thus, sets of a-measure zero on F. and sets of measure zero on I correspond. Note that yk(t) - y(t) strongly 17

in Ll(F) (with respect to the measure a) if and only if yk(t(T)) + O(t(T)) strongly in L (I); analogously, zk(t) + i(t) weakly in Ll(r) if and only if zk(t(T)) +.(t(T)) weakly in L1(I). The problem under consideration, which concerns the orientor field y(t) E B(t), z(t) c R(t,g(t)) a-a.e. in r, where R(t,-) = g(t,y,V(t,y)), is now transformed into a problem concerning the orientor field y(t(T)) c B(t(T)), Z(t(T)) E R(t(T)), ~(t(T)) a.e. in I. We have proved in (App. A) that, the sets R(t,g) have property (Q) in Fi if and only if the sets R(t(T),y) have property (Q) in I. We can now, therefore, apply theorem (8.4.i), or, equivalently, first closure theorem (8.1.i) for orientor fields, and then implicit function theorem (8.4.i). 8.5. Lower Closure Theorems Let G, A(t), A, U(t,y), M be as in (8.4), and let f(t,y,u) = (fl,...,f ), f (t,y,u) be continuous functions on M. For every (t,y) E A let Q(t,y) denote the set Q(t,y) = [(z,z) E E Iz > f (t'y), z f(t,y,u), u E U(t,y)]CuE r +1 r+1 As before let Y = (L (G))s, Z = (L (G))r, and let T be the set of all measurable vector-functions u(t) m (u, M, t E G. We say that the property of lower closure holds in G provided the following occurs: For every sequence yk, Zk) uk, k = 1,2,..., of elements Yk e Y, Zk c Z,

Uk E T, with Yk(t) ~ A(t)(t ) E t) E U(t,yk(t)), zk(t) = f(t,yk(t),uk(t)) a.e. in G, with zk(t) = f (t,yk(t),uk(t)) E L (G), and Yk + Y strongly in Y, zk + Z weakly in Z, and lim /G zk(t)dt < + c, then there is at least one element u E T such that y(t) E A(t), u(t) E U(t,y(t)), z(t) = f(t,y(t),u(t)) a.e. in G, z0(t) = f (t,y(t), u(t)) E L (G), and fG Z (t)dt < lim fG zk(t)dt. o 1 G (8.5.i) A lower closure theorem in G. Let G be open and bounded, let A(t), A, U(t,y), M as in (8.4), A and M closed, let f, f be continuous on M, and let us assume that the sets Q(t,y) have property (Q) in A, and that f (t,y,u) >*(t) for all (t,y,u) c M and some t > o, 4 E L, (G). Then, the lower closure property holds. Proof. First, let us consider an interval I containing G. It is not restrictive to assume I- =[o < tj < b, j = 1,...,v], or I = [o,b], where o and b denote the v-vectors (o,...,o), (b,...,b). We shall now introduce the auxiliary variables z and u, the vector 0 M ( u = (u,u,..,u ) = (u,u), the vector function f(t,y,u) = (f,f,...,f ) with f = u, the control space U(t,y) = [u = (u,u)Iu > fo(t,y,u), u c U(t,y) c EM+l2 and the functional relations z(t) = f(t,y(t),(t)t), z (t) = f = u (t) a. e. in G, 0 1 s 1 r where y(t) = (y...y ) z(t) = (z...*z ), t E G, and we shall take t F~(t) = J z (T)dT, o 19

where the integration ranges over the interval [o,t] c I, and z0(T) = o for T E I - G. 0 In the new notations the usual sets Q(t,y) c E, Q(t,y) c E have now r'~ G. r+l the interpretation Q(ty) = [z = (z,z) I z > f (t,y,u), z = f(t,y,u), u E U(t,y)] = f(ty (t,,y, y)) Er+' Q(t,y) = [zIz = f(t,y,u), u E U(t,y)] = f(t,y, U(t,y)) c E It is convenient to define f and f outside G by taking f = f = o for t E I - G 0 Note that f (t,y,u) > - *(t) for all (t,y,u) E M; hence IG z (t)dt = IG f (t,y(t),u(t))dt > - IG t(t)dt =- L GG 0 oG where L > o is a fixed number. Hence, if xk, uk, k = 1,2,..., is any sequence of admissible pairs with xk + x weakly in S, lim IG z (t)dt < + c as k -*, and we denote by i this lim, then -L <i< + oc, that is, i is finite. We take a subsequence, say still [k] for the sake of simplicity, so that /G zk(t)dt - i as G k k oo, and we can even assume that IG zk(t)dt < i + k- < i + 1 for all k. We take zk(t) = f(t) = f (t,yk(t),uk(t)), t E G, and k o k 0 0k k t t Fk(t) = J Uk(T)dT = J fo(T,yk(T), Uk(T))dT, t c I, k = 1,2,... 0 0 o o Then zk + z weakly in Z, Yk + y strongly in Y, and 20

k(t) A(t), (z (t), Zk (t)) E (t,yk (t)) E a.e. in G, k = 1,2,... Ykt ) k Qt" k' k Yk r+l 0+ o+ Let Zk (t) = f (t,yk(t),uk(t)) + r(t) for t E G, zk (t) = o for t c I -G, O k O o+ o+ + and note that zk (t) > o in I, and zk (t)dt < L + i + l. Let Fk(t), t E I, k - o I k - kIo + t o+ 0 be defined by Fk(t) = f zk (T) dT, t E I. Let z-(t) = *(t) for t E G, z-(t) = o for t E I -G, and let F-(t), t E I, be defined by F (t) = J Z-(T)dT. Then o + - Fk(t) = J f (T,yk(T), Uk(T))dT = Fk(t) - F-(t), t E I k 0 ukk o o Here F-(t) is a fixed nonempty continuous function of t in I, while the functions Fk(t) are nonnegative continuous functions of t in I with a common bound + + 0+ o < Fk(t) < Fk(b) = J Zk (t)dt < L + i +. + For every interval I C R we denote as usual by A Fk the differences of + v order v of Fk relative to the 2 vertices of I; hence + 0+ AIFk J zk (t)dt >o, AIF = | (t)dt > o Of course, each of the interval functions AIZk is also absolutely continuous, but this does not play an essential part below. It interests here to know that the nonnegative interval functions AIZk are nonnegative, additive, and of uniform bounded variations, the total variations being all < L + i + 1. Let us consider the countable lattice point ft }, or t = bP E I, P = (P' p ), where pl,.,Pv are arbitrary rational numbers, o < pj < 1, j = l,...,v. Let (I) be the countable system of intervals I c R whose vertices 21

are points t e (t ). We may order the points t into a sequence. Since the functions zk(t), t e R, are uniformly bounded in R, (and, hence, at each t = t ), k P we can successively select subsequences which are convergent at t - t, and then, by the diagonal process, we can select a unique subsequence, say [k ], of integers k, such that the limits z (t)' F (t) as s - X exist for every t E It ]. tg s ks sc h t lms k P + + Then, the interval functions AIzk -+ AIF also have limits as s s c for every interval I of the countable collection {I). The limits A z is a nonnegative additive interval function for I e {I), and AIz has bounded variation, namely a total variation < L + i + I (for I describing the collection {I)). For every t E I and interval I E (I) with t E I we may consider the quotients AF F/eas L We know that for I a hypercube, I E (I), t E I, and almost all t E I, the limit AIF /meas I - z(t ) exists as diam I + o [App. B.6.i, in conjunction with B.1. Remark 2, case 3]1. Moreover, z(t) is finite almost everywhere in I, nonnegative, L-integrable in I, and zero in I-G. Furthermore, if t F(t) = f z(T)dT, t E I, 0 o hence AIF = fI a(t)dt, I c I + + then o < A F < AiF for every I e {I}, and the difference AIS = AIF - AiF is a nonnegative interval function for I E (I), of bounded variation, and singular, that is, AIS/II] - o as diam I - o for almost all t E I (with t E I c I, I E [I}, I a hypercube). Equivalently, F possesses a Lebesgue decomposition 22

F (t) = F(t) + S(t), F an integral function, S singular, S > o, A S > o. By subtracting the fixed function F we have F0(t) = F+(t) - F'(t) = (F +(t) - F (t)) + S(t), or F0(t) = F(t) + S(t), + - where F = F - F is an integral function, S is singular, (S > o, AIS > o for I E (I), and t F(t) = f (z(T) - V(zT))dT, t E I f 0 0 We can now apply closure theorem (8.2.i) with a = 1, N = r + 1. We conclude that y(t) c A(t), (z (t), z(t)) E Q(t,y(t)) a.:e. in G Then, by implicit function theorem (8.4.i), there is a measurable function u(t) = (u,u) = (u,u,...,u ), t c G, such that y(t) c A(t), u(t) E U(t,y(t)), u (t) > z 0((t)> f (t,y(t), u(t)) >- (t), z(t) = f(t,y(t),u(t)). I Z0(t)dt < f (t,y(t),u(t)dt < IG u(t)dt = I f dt 23 2.5

= F(b) = F0(b) - S(b) = lim Fk (b) = S(b) < i - S(b) < i. S-0o S Theorem (8. 5.i) is thereby proved. Statement (8. 5.i) has an analogous version for boundary controls. Let r c aG, B(t), B, V(t,), M, MY, Z, T as in (8.4.iii). Let g(t,y,v) = (gl,...,gr'), go (t,y,v) be continuous functions on M. For every (t,y) E B let R(t,y) denote the sets R(t,y) = [(z,z) E E +1 Iz > g (t,y,v), z = g(t,y,v), v E V(t,y)] c E'+ - r'+l We say that the property of lower closure holds in r provided the following occurs: For every sequence Yk' Zk vk' k =1,2,..., of elements Yk E Y' vk(t)) o-a.e. in r, with zk(t) = g (t,Yk(t), vk(t)) E Ll(r), and Yk + o 0 0 strongly in Y, zk + z weakly in Z, and lim fr zk(t) da < +, then there is at least one element v T such that (t) B(t), v(t) (t(t) ) V(t) = J z (t)do < lirm zk(t)do. O O (8.5ii) A lower closure theorem on aG. Let r, B(t), B, V(t,y), M as in (8.4.iii), o B, M closed, let g,g be continuous on M, and let us assume that the sets R(t,y) have property (Q) in F, and that the following condition holds (4): go(t,y,v) > - Y(t) for all (t,y,v) E M and some t > o, < E Ll(F). Then, the lower closure property on F holds. Proof. As in the proof of (8.4. iii) we consider only one set Fi as the image of an interval I c Er+1 under a transformationT-i of class K, or t = t(T), 24

T E I. Then fr z (t)do = fI z (t(T)) J(T)dT, -1 where J(T) (Jacobian) is a measurable function on I with 0 < K < J(T) < K < + 0 for some constant K > 1. Thus, the given problem on r can be transformed into an analogous problem on I concerning the integral (7, 1. 1) and the functional relations'(t(T)) E B(t,T), Z (t(T))J(T) = f(t(T), 9(t(T)), v(t(T))J(T), V(t(T)) E V(t(T), 0(t(T)) a.e. in I Now, sets of a-measure zero in Fi and sets of measure zero on I correspond. Also, yk(t) + y(t) strongly in Ll(r) (with respect to the measure a) if and only if gk(t(T)) + y(t(T)) strongly in L (I); and zk(t) (t) z(t) weakly in L (r) if and only if zk(t(T))J(T) - z (t(T))J(T) weakly on L1(I). The problem under consideration, which concerns the orientor field y(t) c B(t), (z (t), z (t)) E R(t,y(t)) o-a.e. in r, where R(t,y) = [z > g (t,y,v), z = g(t,y,v), v E V(t,y), is now transformed into a problem concerning the orientor field y(t(T)) E B(t(T)), (Z (t(T))J(T), z(t(T)J(T)) E R*(T,y(t(T))), a.e. in I, where R*(T,y(t(T)) = [z > g(t(T),y,V) J(T), Z = g(t(T),Y,V)J(T), V E V(t(T),Y)]. We have proved in (App.A ) that the sets R(t,y) have property (Q) in r. if and only if the sets R(t(T),4) have property (Q) in I, and this in turn if and only if the sets R*(T,Y) have property (Q) in I. We can now, therefore, apply theorem 25

Remark. Under the same hypotheses of (8.3.i) the following is true: If Xk, Uk, k = 1,2,..., is any sequence of admissible pairs with xk + x weakly in S and lim I [Xk,uk] < + o, then there is some u E T such that the pair x,u is admissible, and for every open set G C G we have o G f (t,y(t),u(t))dt - k-+ Im f (t (t) uk(t))dt O O In particular, for G = G this relation reduces to I[x,u] < lim I[xk,uk]. o k To prove this stronger form of (9.3.i) we must first consider the family (J) of all cubes J G, J = [bp,ba], p = (P1,...,pv), a = (al...-av)' pj < aj, pj,cj rational. This collection is countable. Then, by the diagonal process, we can extract the subsequence, say now [k ], in the proof of (8.3.i) in such a way that lim 1 ft lik if f (t),u (t))dt = f f ( (t) (t))dt s+o- T0tk5 k5 k-oo Jotykk(t)u(d= as well as lim 1im Slm o G f (ti'Yk((t),uk (t))dt k= - f (t,yk( k(,u(t))dt We can proceed now as in the proof of (9.3.i), and obtain an element u E T lim lim such that x,u is admissible, and I[x,u] < I[xksuk] = k- I[xkuk] If G is any open subset of G, then G is the union of countably many disjoint (closed) intervals J E (J). Given ~ > o there is a finite set of these intervals say J1,'..' JN such that 26

f If (t,y(t), u(t))Idt < E, f 4(t)dt < E G -U J G -U J o S S o S s Hence JG f (t,y(t),u(t))dt < Z f f dt + =O O S lim lirm = lrn Y f dt + (f +) dt + E S-0o 0 S-o30 O UJ G -U J S O S S <ki fj f (tYk(t),u (t))dt + 2E, k-k oo G o0 kt k 0 where ~ > o is arbitrary. This proves the statement. Remark. Condition (4) in statement (9.3.i) is certainly satisfied if, for instance, f (t,y,u) > o for all (t,y,u) E M, or f (t,y,u) > v for all (t,y,u) E M, where v is some real constant. Nevertheless, condition (4) in (9.3.i) can be reduced. A generalization of (9.3.i) will be given below (9.3.ii), where we shall use the following weaker assumption. Condition (4*). For every t c cl G there are a neighborhood N(t) of t in cl G, an L-integrable function *(t) > o, and real numbers bl,...,b (all bl,...b and 4 may depend on t and N(t)) such that f(t,y,u) = f (t,y,u) - r b f (t,y,) -4 (t) (8-5.2) o j=l j for all (t,y,u) E M with t E N(t), with exception perhaps of a set of points (t,y,u) whose t-coordinate lies on a set of measure zero in G. 27

We shall note that, under condition ( t*), it is natural to consider the sets Q(t,y) = [(z,z)lz > f (t,y,u), z = f(t,y,u), u E U(t,y)], or the analogous sets Q*(t,y) = [(Z,Z)JZ0 > f (t,y,u), z = f(t,y,u), u E U(t,y) It is easy to see that the sets Q are closed, or convex, or satisfy property (Q) if and only if the same occurs for the sets Q*. Indeed, the sets above are transformed into one another by the fixed affine transformation = z -b-z, Z = z (8.5.ii) The same as (8.3.i) with hypothesis (*t*) replacing (4t). Proof. The set cl G C E is closed and bounded, hence compact. For every t E cl G there is a neighborhood N (t) and constants b = (bl,...,b ) as stated in assumption (t*). These neighborhoods form a covering of cl G. Thus, finitely many of the same neighborhoods cover G, say N5 (t ), Y = 1,...,N. Let b 1.,b be the corresponding constants, so that r oy (t,y,u) = f0(t,y,u) = -> b.f.(tyu)> (t) j=lJ for all (t,y,u) E M, t e NE (t ), where * (t) is L-integrable in N5 (t ), and U=1 Nb (t) Dcl G. 28

The set H = (tI It-t I = 5, = 1,...,N) is closed and has measure zero in E. Hence, G-H is open, meas (G-H) = meas G, and each component of G-H is also open. If we now take G1 = (G-H) n (int N5 (tl)) 1 1 G = (G-H) n (int No (t)) - G1 -... - G1 = 2,...,N, we see that G-H has been divided into r disjoint open sets G, 7 = 1,...,N, G c N5 (t ), and each component of G-H belongs entirely to one and only one set G. Possibly empty sets G above can be eliminated by suitable reindexing. 7 7 Since meas H = o we have now N fG zk(t)dt = fG f (t,y(t))dt = f f dt = Z 1 fG f dt G-H 7 N r (8.5.3) 7Y=1 JG[f (t, k(t),uk(t))dt bf(t,(t)) dt'Yj=l Let L = lim fG zk(t)dt as k + c, L < + oo On each open set G c N5 (t ) we have f (t,y,u) >-. (t) for all (t,y,u) E M, t E G, r e L(G ), and we denote by L the constant L = GI dt, y = 1,...,N. Y 7 7 7 Y G 7 Y Also, vk + v weakly in V as k + oo. Thus, v E L(G), and if X (t), t E G, denotes the characteristic function of G in G, then 7 fG Zk(t)dt + fG z(t)dt, fG X (t)Zk(t)dt f GX (t)z(t)dt, 7 Y =!,...,r, as k + o, where zk(t) = f(t,yk(t),uk(t)) a.e. in G. Hence JG f(tYk(t),uk(t))dt + fG z(t)dt, y = 1,...,Ni (8.i54) 29

Again, for every y = 1,...,N, we have fG T (tYk(t),uk(t))dt = IG (f -b f)dt Y Y =f f dt - f dt b f f dt G o S~7 G o j Tj G j s 7 < Gf (tY k (t)u (t))dt + s fG dt - Ej b G f (t,yk(t),u(t) )dt. s 7 As k +, we have now L < lim JG fo(tYk(t),uk(t))dt < L + Z L -L b j fG z (t)dt, Y Y for every 7 = 1,...,N. Since L < + oo, each of these lim is finite. In addition certainly L > - oo, and thus L is finite. The sets Q*(t,y) are closed, convex, and satisfy property (Q) in A, as pointed out in Remark before (8.5.ii), and where A = [(t,Yy) t ~ G, (t,y)E A]. 7 7 We can now apply (8.5.i) to the integrals fG T dt, y = 1,...,N. By N successive applications of (8.5.i) and successive extractions, we obtain a subsequence which we shall still call [k] for the sake of simplicity, and measurable functions U (t), t E G, such that u (t) c U(t,y(t)), z(t) = f(t,y(t),u (t)), a.e. in G, JG fo(ty(t),u (t))dt < lim G f (t y(t)uk(t))dt, ) = 1,...N. 7 ( 575) If now u(t), t E G, is defined by taking u = u in G, and u arbitrary in H n G, then 3o

u(t) E U(t,y(t)), z(t) = f(t,y(t),u(t)) a.e. in G, fG fo(t,y(t),u(t))dt < lim fG f (t,yk(t),uk(t))dt, y = 1,...,N. 7 7 (8.5.6) Now relation (8.5.4) becomes fG f(t,y(t),u(t))dt = lim fG f(t yk(t),uk(t))dt, y = 1,...,N. Y7~~~~~~ Y7~~~~~ ~(8.5.7) By relations (8.5.3), (8.5.5), and (8.5.7), we have IG z (t)dt = IG fo(t,y(t),u(t))dt = Z f f dt G G 7G o [fG [ o(t,y(t),u(t))dt + Ej b f f (t y(t),u(t))dt] 7 7 < Z [lim f T (t,y (t),uk(t))dt + Zj b lim f f (t,y(t),uk(t))dt] - G o k k'7j-G k k 7 7 < lim [Z G f dt + Z bj f f dt] lim f f (t y (t)u(t))dt = im fG zk(t)dt. This proves statement (8.5.ii). 31

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